Mounisha Reddy

Big Ideas Math Book 4th Grade Answer Key Chapter 3 Multiply by one – Digit Numbers answer key is helpful for students who are willing to do their best in exams and can Acknowledge these topics by downloading this pdf for free of cost. In this chapter, each and every question was explained in clean and clear cut manner to help the students in a better way to understand these Math problems easily. Big Ideas Math Answers Grade 4 Chapter 3 explains different types of questions with examples and references by which one can grasp different methods in easy and fun way.

Big Ideas Math Book 4th Grade Answer Key Chapter 3 Multiply by One-Digit Numbers

The chapter 3 Multiply by one – digit numbers covers the various topics such as  Multiplying the one or more digit numbers with on digit numbers , Rounding the numbers, Estimating the product of Two  numbers , Finding the products using the different methods such as Expanded form , Distributive property , commutative property, Associative property, Using partial products to find the final product of two numbers with examples to make it easy for the students who finds math as complex task, This chapter is useful for students for easy Growth in their Academics.

Lesson 1 Understand Multiplicative Comparisons

• Lesson 3.1 Understand Multiplicative Comparisons
• Understand Multiplicative Comparisons Homework & Practice 3.1

Lesson 2 Multiply Tens, Hundreds, and Thousands

• Lesson 3.2 Multiply Tens, Hundreds, and Thousands
• Multiply Tens, Hundreds, and Thousands Homework & Practice 3.2

Lesson 3 Estimate Products by Rounding

• Lesson 3.3 Estimate Products by Rounding 
• Estimate Products by Rounding Homework & Practice 3.3

Lesson 4 Use the Distributive Property to Multiply

• Lesson 3.4 Use the Distributive Property to Multiply   
• Use the Distributive Property to Multiply Homework & Practice 3.4

Lesson 5 Use Expanded Form to Multiply

• Lesson 3.5 Use Expanded Form to Multiply
• Use Expanded Form to Multiply Homework & Practice 3.5

Lesson 6 Use Partial Products to Multiply

• Lesson 3.6 Use Partial Products to Multiply
• Use Partial Products to Multiply Homework & Practice 3.6

Lesson 7 Multiply Two-Digit Numbers by One-Digit Numbers

• Lesson 3.7 Multiply Two-Digit Numbers by One-Digit Numbers
• Multiply Two-Digit Numbers by One-Digit Numbers Homework & Practice 3.7

Lesson 8 Multiply Three- and Four-Digit Numbers by One-Digit Numbers

• Lesson 3.8 Multiply Three- and Four-Digit Numbers by One-Digit Numbers
• Multiply Three- and Four-Digit Numbers by One-Digit Numbers Homework & Practice 3.8

Lesson 9 Use Properties to Multiply

• Lesson 3.9 Use Properties to Multiply
• Use Properties to Multiply Homework & Practice 3.9

Lesson 10 Problem Solving: Multiplication

• Lesson 3.10 Problem Solving: Multiplication
• Problem Solving: Multiplication Homework & Practice 3.10

Multiply by One-Digit Numbers Performance Task

• Multiply by One-Digit Numbers Performance Task
• Multiply by One-Digit Numbers Activity
• Multiply by One-Digit Numbers Chapter Test
• Multiply by One-Digit Numbers Cumulative Practice
• Multiply by One-Digit Numbers STEAM Performance Task 

Lesson 3.1 Understand Multiplicative Comparisons

Explore and Grow

Model the counters. Draw to show your model.

There are 20 counters. Five of the counters are yellow. The rest are red.

How many more red counters are there than yellow counters?
How many times as many red counters are there as yellow counters?
Answer: model

15 red counters, 3 times the yellow counters.

Explanation:
Totally there are 20 counters in which 5 of them are yellow counters and the remaining 15 of them are red counters.
So, red counters are three times the yellow counters.

Structure
Explain how you can use an addition equation or a multiplication equation to compare the numbers of yellow counters and red counters.
Answer:  Number of yellow counters = 5 and number of red counters =15,
There are red counters as 3 times of yellow counters that is,
3 × 5 = 15  or
5 + 5 + 5 = 15.

Explanation: Given the total number of yellow counters and the number of red counters so, simply we should use our basic addition rule and basic multiplication rule to explain the mathematical equation for the number of yellow counters and number of red counters.

Think and Grow: Understand Multiplicative Comparisons

You can use multiplication to compare two numbers.
Example
Write two comparison sentences for 24 = 4 × 6.

You can compare two numbers using addition or multiplication.
• Use addition to find or how many more how many fewer
• Use multiplication to find as much. how many times

Answer : 24 is 6 times as many as 4 or 24 is 4 times as many as 6
Example
Write an equation for each comparison sentence.

Answer: 12 = 8 + 4 ,  12 = 3 × 4

Show and Grow

Write two comparison sentences for the equation
Question 1.
15 = 3 × 5
Answer:  15 is 3 times as many as 5.
or
15 is 5 times as many as 3.

Explanation: we should multiply the numbers 3 and 5 ,
to get the multiplication result as 15.

Question 2.
32 = 4 × 8
Answer: 32 is 4 times as many as 8.
or
32 is 8 times as many as 4.

Explanation: we should multiply the numbers 4 and 8 ,
to get the multiplication result as 32.

Draw a model for the comparison sentence. Then write an equation.
Question 3.
21 is 14 more than 7.
Answer:

21 = 14 + 7.

Explanation: The term more than is referred  to addition ,
we should add the numbers 14 and 7,
to get the addition result as 21.

Question 4.
40 is 8 times as many as 5.
Answer:

40 = 8 × 5.

Explanation: The term as many as is referred  to multiplication ,
we should multiply the numbers 5and 8 ,
to get the multiplication result as 40.

Apply and Grow: Practice

Write two comparison sentences for the equation.
Question 5.
48 = 6 × 8
Answer: 48 is 6 times as many as 8. or
48 is 8 times as many as 6.

Explanation: The term as many as is referred  to multiplication ,
we should multiply the numbers 6and 8 ,
to get the multiplication result as 48.

Question 6.
63 = 7 × 9
Answer:  63 is 7 as many as 9. or
63 is 9 as many as 7.

Explanation: The term as many as is referred  to multiplication ,
we should multiply the numbers 7and 9 ,
to get the multiplication result as 63.

Write an equation for the comparison sentence.
Question 7.
20 is 2 times as many as 10.
Answer: 20 = 2 × 10.

Explanation: we should multiply the numbers 2 and 10 ,
to get the multiplication result as 20.

Question 8.
18 is 10 more than 8.
Answer: 18 = 10 + 8.

Explanation: we should add the numbers 10 and 8,
to get the addition result as 18.

Question 9.
35 is 7 times as many as 5.
Answer: 35 = 7 × 5.

Explanation: we should multiply the numbers 7and 5 ,
to get the multiplication result as 35.

Question 10.
16 is 4 times as many as 4.
Answer: 16 = 4 × 4.

Explanation: we should multiply the numbers 4 and 4 ,
to get the multiplication result as 16.

Question 11.
Earthworms have four more hearts than humans. How many hearts do earth worms have?
Answer: 5.

Explanation: Humans have one heart and question implies that,
Earthworms have four more hearts than humans.
So 1 + 4 = 5.

Question 12.
Ants can lift 50 times their body weight. An ant weighs 5 milligrams. How much weight can the ant lift?

Answer:  250 milligrams.

Explanation: Ant body weight = 5 milligrams and it can lift 50 times their body weight.
So, ant can lift 50 times the 5 milligrams,
that is 5 × 50 = 250 milligrams.

Question 13.
Writing
Explain how you know the statement “32 is 8 times as many as 4”is a comparison involving multiplication.
Answer:  Multiplication is used to find as  much or how many times,
and the two factors 8 &  4 gives the product of 32 .
So, 32  = 8 × 4.

Question 14.
Number Sense
Write an addition comparison statement and a multiplication comparison statement for the numbers 8 and 24.
Answer: Addition comparison statement  for 24 = 16 + 8  is ,
24 is 16 more than 8.
multiplication comparison statement for 24 = 3 × 8 is ,
24 is 3 times as many as 8.

Think and Grow: Modeling Real Life

Example
You perform a science experiment and use4 times as much hydrogen peroxide as water. You use a total of 10 tablespoons of liquid. How many table spoons of hydrogen peroxide do you use?

Draw a model.

Answer: tablespoons of liquid,
8 tablespoons of hydrogen peroxide is used in the solution.

Explanation: Given, hydrogen peroxide used in this solution is 4 times as many as water ,
so let us say that , Hydrogen peroxide is represented as H and water as W,
then we have H = 4W, from the model given we can say that
Hydrogen peroxide + water = 10 tablespoons of liquid,
we get H + W = 10
from above details we can write it as,
4W + W = 10
5W = 10
W = 10/5
w = 2. Replace this value in assumed equation H = 4W that is
H = 4(2)
H = 4 ×2
H = 8.
Hydrogen peroxide (H) = 8
Water (W) = 2, in total
8 + 2 = 10.
Given equation is balanced.
so here we are using 8 tablespoons of Hydrogen peroxide and 2 tablespoons of water to get the required 10 table spoons of liquid.

Find the number of tablespoons of water.
The model shows ______ equal parts. There are ______ tablespoons of liquid in all.
Answer: The model shows 2 tablespoons of water . There are 10 tablespoons of liquid in all.

5 × ? = 10 Think: 5 times what number equals 10?
Answer: 5 × 2 = 10.
Explanation: 10 is 5 times as many as 2.

You use _______ tablespoons of water.
Answer: we use 2 tablespoons of water in the liquid.

Find the number of tablespoons of hydrogen peroxide.
You use ________ times as much hydrogen peroxide as water.
Answer: we use 4 times as much Hydrogen peroxide as water.

2 × 4 = ______
Answer: 2 × 4 = 8. 8 is 2 times as many as 4.

So, you use _______ tablespoons of hydrogen peroxide.
Answer: we use 8 tablespoons of Hydrogen peroxide.

Show and Grow

Question 15.
A bicycle-sharing station on Main Street has 5 times as many bicycles as a station on Park Avenue. There are 24 bicycles at the two stations. How many bicycles are at the Main Street station?
Answer: 20

Explanation: Let us say ,Main street as M and Park Avenue as P,
Given that main street has 5 times as many bicycles as a station on Park Avenue so we get, M = 5P
And there are 24 bicycles at the two stations from that we have M + P = 24,
by replacing the M value in the equation M + P = 24 we get,
5P + P = 24
6P = 24
P = 24/6
P = 4. replace this P value in the above equation M = 5P to get M value, then
M = 5(4)
M = 5 × 4
M = 20.
so we have M = 20 and P = 4, in total
20 + 4 = 24. the given equation is balanced.
Finally, we have 20 bicycles at the Main street station and 4 bicycles at the Park Avenue station.

Question 16.
In the 2016 Olympics, Brazil won 6 silver medals. France won 3 times as many silver medals as Brazil. How many silver medals did France win?
Answer: 18 silver medals.

Explanation:
Brazil = 6 silver medals,
France = 3 times as many as brazil ,
so 3 × 6 = 18.
France has won 18 silver medals.

Question 17.
Of all the national flags in the world, there are 3 times as many red, white, and blue flags as there are red, white, and green flags. There are 40 flags with these color combinations. How many more flags are red, white, and blue than red, white, and green?
Answer:

Understand Multiplicative Comparisons Homework & Practice 3.1

Write two comparison sentences for the equation.
Question 1.
24 = 8 × 3
Answer: 24 is 8 times as many as 3. or
24 is 3 times as many as 8.

Explanation: we should multiply the numbers 8 and 3 ,
to get the multiplication result as 24.

Question 2.
14 = 7 × 2
Answer: 14 is 7 times as many as 2. or
14 is 2 times as many as 7.

Explanation: we should multiply the numbers 7 and 2 ,
to get the multiplication result as 14.

Write an equation for the comparison sentence.
Question 3.
30 is 6 times as many as 5.
Answer: 30 = 6 × 5

Explanation: we should multiply the numbers 6 and 5,
to get the multiplication result as 30.

Question 4.
27 is 3 times as many as 9.
Answer: 27 = 3 × 9

Explanation: we should multiply the numbers 3 and 9,
to get the multiplication result as 27.

Question 5.
12 is 7 more than 5.
Answer: 12 = 7 + 5

Explanation: we should add the numbers 7 and 5,
to get the Addition result as 12.

Question 6.
10 is 2 times as many as 5.
Answer: 10 = 2 × 5

Explanation: we should multiply the numbers 2 and 5,
to get the multiplication result as 10.

Question 7.
The House of Representatives has 335 more members than the Senate. The Senate has 100 members. How many members does the House of Representatives have?
Answer: 435 members.

Explanation:
senate has 100 members so, 335 + 100 = 435 .
House of representatives have 435 members.

Question 8.
A lion’s roar can be heard 5 miles away. The vibrations from an elephant’s stomp can be felt 4 times as many miles away as the lion’s roar can be heard. How many miles away can the vibrations be felt?
Answer: 20 miles.

Explanation:
a lion’s roar can be heard from 5 miles away
Elephant’s stomp can be felt 4 times as many miles away from the lion’s roar ,
that is 4 × 5 = 20.

 

Question 9.
Reasoning
Newton says the equation 270 = 30 × 9 means 270 is 30 times as many as 9. Descartes says it means 270 is 9 times as many as 30. Explain how you know they are both correct.

Answer: Multiplication is used to find as much or how, many times.
Since the two numbers 30 and 9 are the  factors of  multiplication they remain constant,
Even their position in the factors place changes,
The result will be the same product , that is 270.

Question 10.
Precision Compare the door’s height to the desk’s height using multiplication and addition.

Answer:  Multiplication comparison:
Door’s height is 4 times the height of the desk’s height,
4 × 2 = 8.
Explanation: by multiplying the 4 times the height of the desk  and 2 we get door’s height 8 feet.
Addition comparison:
Door’s height is 6 feet more than height of the desk,
6 + 2 = 8.
Explanation: by adding the 6 feet height of the desk and 2 we get the door’s height 8 feet.

 

Question 11.
Open-Ended
Write a comparison statement for a sum of 28.
Answer:  28 is  20 more than 8.
Explanation: 28  = 20 + 8 . The term sum here refers to addition of two number to get the total number.

Question 12.
Modeling Real Life
There are 12 shepherds and retrievers in all at a dog park. There are 2 times as many shepherds as retrievers. How many retrievers are there?
Answer: 8 retrievers.

Explanation:  Total count of dogs at a dog park is 12.
Let us say that shepherds as S and retrievers as R , Given that, S + R = 12.
There are 2 times as many shepherds as retrievers.
we have S = 2R,  by replacing this value in equation S + R = 12 we get,
2R + R = 12
3R = 12
R = 12/3
R = 4.  To get S value substitute this value in S = 2R then
S = 2(4)
S = 2 × 4
S = 8.
Finally we have 8 shepherds and 4 retrievers in the dog park.

Question 13.
Modeling Real Life
Pythons sleep 6 times as long as horses. Horses sleep3 hours each day. How many hours do pythons sleep each day?
Answer: 18 hours.

Explanation: Horses sleep 3 hours  and
Python sleep 6 times as long as horses
, so 6 x 3 =  18.
therefore python sleeps 18 hours each day.

Question 14.
DIG DEEPER!
You have 8 times as many dimes as nickels. You have18 dimes and nickels altogether. How much money do you have in all?
Answer: I have 16 dimes and 2 nickels.

Explanation: You have 8 times as many dimes as nickels
You have18 dimes and nickels altogether
Let us say, dimes as D and nickels as N then we have, D = 8N,
so, D + N = 18
8N + N = 18
9N = 18
N = 18/9
N = 2. replace the value in D = 8N , we get  D = 8 × 2 = 16 .

So, I have 16 dimes and 2 nickels.

Review & Refresh

Find the missing factor.
Question 15.
7 × _____ = 280
Answer: 40

Explanation:    280 is 7 times as many as 40 ,
7 × 40 = 280.
we should multiply the numbers 7 and 40,
to get the multiplication result as 280.

Question 16.
____ × 30 = 270
Answer: 9

Explanation: 270 is 9 times as many as 30 ,
9 × 30 = 270.
we should multiply the numbers 30 and 9,
to get the multiplication result as 270.

Question 17.
8 × ____ = 640
Answer: 80

Explanation: 640 is 8 times as many as 80 ,
8 × 80 = 640.
we should multiply the numbers 8 and 80,
to get the multiplication result as 640.

Question 18.
____ × 90 = 540
Answer: 6

Explanation: 540 is 6 times as many as 90 ,
6 × 90 = 540.
we should multiply the numbers 6 and 90,
to get the multiplication result as 540.

Question 19.
2 × _____ = 40
Answer: 20

Explanation: 40 is 2 times as many as 20,
2 × 20 = 40.
we should multiply the numbers 2 and 20,
to get the multiplication result as 40.

Question 20.
_____ × 50 = 350
Answer:  7

Explanation: 350 is 7 times as many as 50 ,
7 × 50 = 350.
we should multiply the numbers 7 and 50,
to get the multiplication result as 350.

Lesson 3.2 Multiply Tens, Hundreds, and Thousands

Explore and Grow

Use models to find each product. Draw your models.

What pattern do you notice?
Answer:   Model

The Place- value method. From the above pattern, we can conclude that the result has different place-values of 12

Explanation:
The position of 3 is different in the given 4 multiplications.
So,
4 × 3 = 12
4 × 30 = 120
4 × 300 = 1200
4 × 3000 = 12,000
From the above pattern, we can conclude that the result has different place-values of 12.

Repeated Reasoning
How does 3 × 7 help you to find 3 × 7,000? Explain.
Answer: 3× 7 = 21
3 × 7000 = 21000

Explanation:   When multiplying 3 by 7 we get 21 , same as that when we multiply 3 by 7000 we get 21000.
the number moves to the left and a zero is put in, ,
the number moves three places to the left and zeros are put in,
so the number would become 21000.

Think and Grow: Multiply Tens, Hundreds, and Thousands

You can use place value to multiply by tens, hundreds, or thousands.
Example
Find each product.
7 × 200 = 7 × _____ hundreds
= _____ hundreds
= ______
So, 7 × 200 = _____.

Answer: 1400

Explanation: Using the Place-value method,
7 × 200 = 7 × 2 hundreds
= 14 hundreds
= 1400
So, 7 × 200 = 1400.

 

3 × 4,000 = 3 × ______ thousands
= _____ thousands
= _____
So, 3 × 4,000 = ______.

Answer: 12000.

Explanation: Using the Place-value method,
3 × 4,000 = 3 × 4 thousands
= 12 thousands
= 12000
So, 3 × 4,000 = 12000.

 

Example
Find each product.
8 × 5 = 40 Multiplication fact
8 × 50 = 400 Find 8 × 5; write 1 zero to show tens.
Answer: 8 × 5 = 40 .

Explanation: By using the place-value method,
8 × 50 = 8 × 5 tens
= 40 × tens
= 400
So,
8 × 50 = 400.

8 × 500 = _____ Find 8 × 5; write 2 zeros to show hundreds.
Answer: 8 × 5 = 40 , 8 × 500 = 4000.

Explanation: By using the place-value method,
8 × 500 = 8 × 50 tens
= 8 × 5 × tens ×  tens
= 40 × tens × tens
= 4000
So,
8 × 500 = 4000.

8 × 5,000= ______ Find 8 × 5; write 3 zeros to show thousands.
Answer : 8 × 5 = 40, 8 × 5,000 = 40,000.

Explanation: By using the place-value method,
8 × 5000 = 8 × 500 tens
= 8 × 5 × tens ×  tens ×tens
= 40 × tens × tens × tens
= 40,000
So,
8 × 5000 = 40,000.
Notice the Pattern: Write the multiplication fact and the same number of zeros that are in the second factor.

Show and Grow

Find each product.
Question 1.
6 × 9 = ____
6 × 90 = ____
6 × 900 = ______
6 × 9,000 = _____
Answer: 6 × 9 = 54
6 × 90 = 540
6 × 900 =5400
6 × 9,000 = 54,000

Explanation:  6 × 9 = 54

By using the place-value method,
6 × 90 = 6× 9 tens
= 6 × 9 × tens
= 54 × tens
= 540
So,
6 × 90= 540.

By using the place-value method,
6 × 900 = 6× 90 tens
= 6 × 9 × tens × tens
= 54 × tens × tens
= 5400
So,
6 × 900= 5400.

By using the place-value method,
6 × 9,000 = 6× 900 tens
= 6 × 90 × tens × tens
= 6 × 9 × tens ×  tens × tens
= 54 × tens × tens × tens
= 54,000
So,
6 × 9,000 = 54,000.

Question 2.
5 × 2 = ____
5 × 20 = ____
5 × 200 = ______
5 × 2,000 = _____
Answer: 5 × 2 = 10
5 × 20 = 100
5 × 200 = 1000
5 × 2,000 = 10,000.

Explanation: 5 × 2 = 10.

By using the place-value method,
5 × 20 = 5 × 2 tens
= 10 × tens
= 100
So,
5 × 20 = 100.

By using the place-value method,
5 × 200 = 5 × 20 tens
= 5 × 2 × tens × tens
= 10 ×  tens × tens
= 1000
So,
5 × 200 = 1000.

By using the place-value method,
5 × 2,000 = 5× 200 tens
= 5 × 20 × tens × tens
= 5 × 2 × tens ×  tens × tens
= 10× tens × tens × tens
= 10,000
So,
5 × 2,000 = 10,000.

Question 3.
2 × 3 = _____
2 × 300 = _____
Answer: 2 × 3 = 6
2 × 300 = 600

Explanation:  we should multiply 2 by 3 to get 6.

By using the place-value method,
2 × 300 = 2 × 30 tens
= 2 × 3 × tens × tens
= 6 ×  tens × tens
= 600
So,
2 × 300 = 600.

Question 4.
9 × 8 = _____
9 × 8,000 = _____
Answer: 9 × 8 = 72
9 × 8,000 = 72,000

Explanation: we should multiply 9 by 8 to get 72.

By using the place-value method,
9 × 8,000 = 9 × 800 tens
= 9 × 80 × tens × tens
= 9 × 8 × tens ×  tens × tens
= 72× tens × tens × tens
= 72,000
So,
9 × 8,000 = 72,000.

 

Apply and Grow: Practice

Find the product
Question 5.
7 × 700 = _____
Answer: 4900

Explanation:

By using the place-value method,
7 × 700 = 7 × 70 tens
= 7 × 7 × tens × tens
= 49 ×  tens × tens
= 4900
So,
7 × 700 = 4900.

 

Question 6.
3 × 100 = _____
Answer: 300

Explanation: By using the place-value method,
3 × 100 = 3 × 10 tens
= 3 × tens × tens
= 30× tens
= 300
So,
3 × 100 = 300.

 

Question 7.
8,000 × 3 = _____
Answer: 24,000

Explanation : By using the place-value method,
3 × 8,000 = 9 × 800 tens
= 3 × 80 × tens × tens
= 3 × 8 × tens ×  tens × tens
= 24× tens × tens × tens
= 24,000
So,
3 × 8,000 = 24,000.

Question 8.
60 × 2 = _____
Answer: 120

Explanation: By using the place-value method,
2 × 60 = 2 × 6 tens
= 12 × tens
= 120
So,
2 × 60 = 120.

Question 9.
4 × 4,000 = ______
Answer: 16,000.

Explanation : By using the place-value method,
4 × 4,000 = 4 × 400 tens
= 4 × 40 × tens × tens
= 4 × 4 × tens ×  tens × tens
= 16 × tens × tens × tens
= 16,000
So,
4 × 4,000 = 16,000.

Question 10.
700 × 5 = _____
Answer: 3500

Explanation: By using the place-value method,
5 × 700 = 5 ×70 tens
= 5 ×7 × tens × tens
= 350× tens
= 3500
So,
5 × 700 = 3500.

Question 11.
900 × 7 = _____
Answer: 6300

Explanation: By using the place-value method,
7 × 900 = 7 × 90 tens
= 7 × 9 × tens × tens
= 63 × tens × tens
= 6300
So,
7  × 900 = 6300.

Question 12.
50 × 3 = _____
Answer: 150

Explanation: By using the place-value method,
3 × 50 = 3 × 5 × tens
= 15 × tens
= 150
So,
3 × 50 = 150.

Question 13.
1,000 × 8 = ______
Answer: 8000

Explanation : By using the place-value method,
8 × 1,000 = 8 × 100 tens
= 8 × 10 × tens × tens
= 8 × tens ×  tens × tens
= 8,000
So,
8 × 1,000 = 8,000.

Find the missing factor
Question 14.
_____ × 6,000 = 24,000
Answer:  4 × 6,000 = 24,000

Explanation :Let the missing number be X
So, X × 6,000 = 24,000
X = 24,000 / 6,000 = 4
Hence, the value of X is: 4.

Question 15.
9 × ____ = 450
Answer: 9 × 50 = 450

Explanation :
Let the missing number be X
So, 9 × X = 450
X = 450 / 9 = 50
Hence, the value of X is: 50.

Question 16.
_____ × 30 = 210
Answer: 7 × 30 = 210

Explanation:
Let the missing number be X
So, X × 30 = 210
X = 210 / 30 = 7
Hence, the value of X is: 7

Question 17.
2 × _____ = 800
Answer: 2 × 400 = 800

Explanation:
Let the missing number be X
So, 2 × X = 800
X = 800 / 2 = 400
Hence, the value of X is: 400.

Question 18.
_____ × 1,000 = 9,000
Answer: 9 × 1,000 = 9,000

Explanation:
Let the missing number be X
So, X × 1,000 = 9,000
X = 9,000 / 1,000 = 9
Hence, the value of X is: 9.

Question 19.
8 × ____ = 640
Answer: 8 × 80 = 640.

Explanation :
Let the missing number be X
So, 8 × X = 640
X = 640 / 8 = 80
Hence, the value of X is: 80

Compare
Question 20.

Answer: 7 × 60 = 420

Explanation:
7 × 60 = 420
Given numbers are: 420 and 400
By comparing 2 values, we can conclude that 420 is greater than 400

Question 21.

Answer: 500 × 4 = 2,000.

Explanation:
500 × 4 = 2,000
Given numbers are: 2,000 and 2,000
By comparing 2 values, we can conclude that 2,000 is equal to 2,000

Question 22.

Answer: 3 × 9,000 = 27,000.

Explanation :
3 × 9,000 = 27,000
Given numbers are: 27,000 and 39,000
By comparing 2 values, we can conclude that 27,000 is less than 39,000.

Question 23.
The North Canadian River is 800 miles long. The Amazon River is 5 times longer than the North Canadian River. How many miles long is the Amazon River?

Answer: 4,000 miles

Explanation: Let us say North Canadian river as N and Amazon river as A ,
Given that Amazon river is 5 times longer than North Canadian river so A = 5N
N = 800 miles, get this value into above equation then,
A = 5(800)
= 5 × 800
= 4,000.
so Amazon river is 4,000 miles long.

Question 24.
One reusable bag can prevent the use of 600 plastic bags. Six reusable bags can prevent the use of how many plastic bags?
Answer: 3600 plastic bags

Explanation:
Given that , One reusable bag can prevent the use of 600 plastic bags.
if one bag = 600 plastic bags then six bags  = ?
let us multiply the bags then we have 6 × 600 = 3600.
so, answer is 3600 plastic bags.

Question 25.
YOU BE THE TEACHER
Your friend says the product of 6 and 500 will have2 zeros. Is your friend correct? Explain.
Answer: NO

Explanation:  6 × 500 = 3,000.
they mentioned that result will have 2 zero’s,
but the actual product gives 3 zero’s.

Think and Grow: Modeling Real Life

Example
An aquarium has 7 bottle nose dolphins. Each dolphin eats 60 pounds of fish each day. The aquarium has510 pounds of fish. Does the aquarium have enough fish to feed the dolphins?

Answer: YES

Explanation: Let us say Aquarium as A and Dolphins as D
Given that A = 7D, Aquarium has 510 pounds of fish
Each D = 60 pounds, we get
A = 7 (60)
= 7 × 60
= 420.
All the dolphins will eat 420 pounds of fish and the Aquarium has 510 pounds.510 is greater than 420

Think: What do you know? What do you need to find? How will you solve?
Step 1: How many pounds of fish do all of the dolphins eat?
7 × 60 = _____
Answer: 7 × 60 = 420.
All of the dolphins eat _______ pounds of fish.
Answer: 420 pounds.
Step 2: Compare the number of pounds of fish all of the dolphin seat to the number of pounds of fish the aquarium has.
Answer: Aquarium has 510 pounds of fish and all the dolphins will eat 420 pounds of fish
so, 510 is greater than 420
The aquarium _______ have enough fish to feed the dolphins.
Answer: 510 pounds of fish.

Show and Grow

Question 26.
Students want to make 400 dream catchers for a craft fair. Each dream catcher needs 8 feathers. The students have3,100 feathers. Do the students have enough feathers for all of the dream catchers?

Answer: NO

Explanation: Let us say Dream catcher as D and Feathers as F, total Dream catchers we want are 400 and
Each Dream catcher needs 8 Feathers  so D = 8F, students have 3,100 Feathers
Then , by calculating , 3,100 /8 = 387.5.
to get up to the total count of required 400 Dream catchers Feathers are not sufficient as 400 is greater than 387.

Question 27.
A principal has 3 rolls of 800 raffle tickets each and 5 rolls of 9,000 raffle tickets each. How many raffle tickets does the principal have?
Answer: 47,400 raffle tickets are there with the principal.

Explanation: A principal has 3 rolls of 800 raffle tickets each and 5 rolls of 9,000 raffle tickets each.
Then we have 3 × 800 and 5 × 9,000
= 2,400 and 45,000
Totally , 45,000 + 2,400 = 47,400 .

So, 47,400 raffle tickets are there with the principal.

Question 28.
You have 2 sheets of 4 stickers each. Your friend has 20 times as many stickers as you. Your teacher has 700 times as many stickers as you. How many stickers do the three of you have in all?
Answer: Three of us have  5,768 stickers in all.

Explanation: You have 2 sheets of 4 stickers each. that is 2 × 4 = 8 stickers.
Your friend has 20 times as many stickers as you. that is 20 × 8 = 160 stickers.
Your teacher has 700 times as many stickers as you. that is 700 × 8 = 5,600 stickers.
Then 5,600 + 160 + 8 = 5,768 stickers.

So, Three of us have  5,768 stickers in all.

Multiply Tens, Hundreds, and Thousands Homework & Practice 3.2

Find each product
Question 1.
3 × 3 = ____
3 × 30 = ____
3× 300 = ______
3 × 3,000 = _____
Answer: 3 × 3 = 9
3 × 30 = 90
3× 300 = 900
3 × 3,000 = 9,000

Explanation:  3 × 3 = 9

By using the place-value method,
3 × 30 = 3× 3 × tens
= 9 × tens
= 90
So,
3 × 30= 90.

By using the place-value method,
3 × 300 = 3× 30 tens
= 3 × 3 × tens × tens
= 9 × tens × tens
= 900
So,
3 × 300= 900.

By using the place-value method,
3 × 3,000 = 3 × 300 tens
= 3 × 30 × tens × tens
= 3 × 3 × tens ×  tens × tens
= 9 × tens × tens × tens
= 9,000
So,
3 × 3,000 = 9,000.

Question 2.
8 × 7 = ____
8 × 70 = ____
8 × 700 = ______
8 × 7,000 = _____
Answer: 8 × 7 = 56
8 × 70 = 560
8 × 700 = 5600
8 × 7,000 = 56,000

Explanation:  8 × 7 = 56

By using the place-value method,
8 × 70 = 8 × 7 tens
= 8 × 7 × tens
= 56 × tens
= 560
So,
8 × 70= 560.

By using the place-value method,
8 × 700 = 8 × 70 tens
= 8 × 7 × tens × tens
= 56 × tens × tens
= 5600
So,
8 × 700= 5600.

By using the place-value method,
8 × 7,000 = 8 × 700 tens
= 8 × 70 × tens × tens
= 8 × 7 × tens ×  tens × tens
= 56 × tens × tens × tens
= 56,000
So,
8 × 7,000 = 56,000.

Find the product.
Question 3.
9 × 90 = ____
Answer: 9 × 90 = 810

Explanation:  we should multiply 9 by 9 to get 81 that implies 9 × 90 = 810.
By using the place-value method,
9 × 90 = 9 × 9 tens
= 81 × tens
= 810
So,
9 × 90 = 810.

Question 4.
6,000 × 1 = _____
Answer: 6,000 × 1 = 6,000.

Explanation : By using the place-value method,
1 × 6,000 = 1× 600 tens
= 1 × 60 × tens × tens
= 1 × 6 × tens ×  tens × tens
= 6 × tens × tens × tens
= 6,000
So,
1 × 6,000 = 6,000.

Question 5.
8 × 200 = _____
Answer: 8 × 200 = 1600.

Explanation: By using the place-value method,
8 × 200 = 8 ×20 tens
= 8 ×2 × tens × tens
= 160× tens
= 1600
So,
8 × 200 = 1600.

Question 6.
3,000 × 6 = _____
Answer: 3,000 × 6 = 18,000.

Explanation : By using the place-value method,
6 × 3,000 = 6 × 300 tens
= 6 × 30 × tens × tens
= 6 × 3 × tens ×  tens × tens
= 18 × tens × tens × tens
= 18,000
So,
6 × 3,000 = 18,000.

Question 7.
5 × 500 = _____
Answer: 5 × 500 = 2500.

Explanation: By using the place-value method,
5 × 500 = 5 ×50 tens
= 5 ×5 × tens × tens
= 250× tens
= 2500
So,
5 × 500 = 2500.

Question 8.
2 × 90 = ______
Answer: 2 × 90 = 180

Explanation: By using the place-value method,
2 × 90 = 2 × 9 × tens
= 18 × tens
= 180
So,
2 × 90 = 180.

Question 9.
7 × 300 = ______
Answer: 7 × 300 = 2100

Explanation: By using the place-value method,
7 × 300 = 7 × 30 tens
= 7 × 3 × tens × tens
= 21 × tens × tens
= 2100
So,
7  × 300 = 2100.

Question 10.
20 × 2 = _____
Answer:  20 × 2 = 40

Explanation: By using the place-value method,
2 × 20 = 2 × 2 × tens
= 4 × tens
= 40
So,
2 × 20 = 40.

Question 11.
6,000 × 5 = _____
Answer:  6,000 × 5 = 30,000.

Explanation : By using the place-value method,
5 × 6,000 = 5 × 600 tens
= 5 × 60 × tens × tens
= 5 × tens ×  tens × tens
= 30,000
So,
5 × 6,000 = 30,000.

Find the missing factor.
Question 12.
_____ × 400 = 3,200
Answer:  8 × 400 = 3,200

Explanation:
Let the missing number be X
So, X × 400 = 3,200
X = 3,200 / 300 = 8
Hence, the value of X is: 8

Question 13.
1 × _____ = 500
Answer: 1 × 500 = 500.

Explanation:
Let the missing number be X
So, 1 × X = 500
X = 500 / 1 = 500
Hence, the value of X is: 500

Question 14.
_____ × 300 = 1,200
Answer:  4 × 300 = 1,200

Explanation:
Let the missing number be X
So, X × 300 = 1,200
X = 1,200 / 300 = 4
Hence, the value of X is: 4

Question 15.
8 × ____ = 720
Answer: 8 × 90 = 720

Explanation:
Let the missing number be X
So, 8 × X = 720
X = 720 / 8 = 90
Hence, the value of X is: 90

Question 16.
_____ × 1,000 = 7,000
Answer:   7 × 1,000 = 7,000

Explanation :
Let the missing number be X
So, X × 1,000 = 7,000
X = 7,000 / 1,000 = 7
Hence, the value of X is: 7

Question 17.
8 × _____ = 4,800
Answer: 8 × 600 = 4,800

Explanation:
Let the missing number be X
So, 8 × X = 4,800
X = 4,800 / 8 = 600
Hence, the value of X is: 600

Compare.
Question 18.

Answer: 9 × 400 is equal to 3,600

Explanation:
9 × 400 = 3,600
Given numbers are: 3,600 and 3,600
By comparing 2 values, we can conclude that 3,600 is equal to 3,600

Question 19.

Answer: 360 is less than 660

Explanation:
6 × 60 = 360
Given numbers are: 360 and 660
By comparing 2 values, we can conclude that 360 is less than 660.

Question 20.

Answer: 16,000 is greater than 10,000

Explanation:
8,000 × 2 = 16,000
Given numbers are: 16,000 and 10,000
By comparing 2 values, we can conclude that 16,000 is greater than 10,000

Question 21.
Patterns
Describe and complete the pattern.
1 × 4 = _____
2 × 40 = _____
3 × 400 = ____
_____ × _____ = _____
_____ × _____ = ______
Answer:  1 × 4 = 4
2 × 40 = 80
3 × 400 = 1200
4 × 4,000 = 16,000
5 × 40,000 = 200,000
Explanation :  we should multiply the numbers 1 and 4 to get 4 .
By using the place-value method,
2 × 40 = 2 × 4 tens
= 8 × tens
= 80
So,
2 × 40 = 80.

By using the place-value method,
3 × 400 = 3 × 40 tens
= 3 × 4 × tens × tens
= 12 ×  tens × tens
= 120× tens
= 1200
So,
3 × 400 = 1200.

By using the place-value method,
4 × 4,000 = 4 × 4,000 tens
= 4 × 40 × tens × tens
= 4 × 4 × tens ×  tens × tens
= 16× tens × tens × tens
= 16,000
So,
4 × 4,000 = 16,000.

By using the place-value method,
5 × 40,000 = 5 × 4,000 tens
= 5 × 400 × tens × tens
= 5 × 40 × tens ×  tens × tens
= 5 × 4 × tens × tens × tens × tens
= 200,000
So,
5 × 40,000 = 200,000.

Question 22.
DIG DEEPER!
Without calculating, tell whether the product of 5 and 700 or the product of 5 and 7,000 is greater. Explain how you know.
Answer: The place value method.
This is a common method used to know how to multiply by ten: When multiplying 5 by ten, the number moves to the left and a zero is put in, so the number becomes 50. If you multiply 5 by 100, the number moves two places to the left and zeros are put in, so the number would become 500.

Explanation:
This shows that to know the greater value of the product of 5 and 700 or the product of 5 and 7,000 is determined by the place of the zero’s in the number.
So, product of 5 and 7,000 is greater than the product of 5 and 700.

Question 23.
Writing
How does 3 × 6 help you to find 3,000 × 6? Explain.
Answer: 3 × 6 = 18   so 3,000 × 6 = 18000.

Explanation: we should multiply the 3 and 6 to get 18 ,like wise
the zero’s value increases when they start to move left side adding another zero until  the thousands place is filled so we have
3,000 × 6 = 18,000.

Question 24.
Open-Ended
Write a multiplication equation with a product of 120.
Answer: 4 × 30 = 120 or 3 × 40 = 120

Explanation: the product of the two numbers 4 and 3 is 12 that implies
if the zero, is in tens place in any one of the number in multiplying
we get the same number as a product so, 4 × 30 = 120 or 3 × 40 = 120.

Question 25.
Modeling Real Life
A second grade student answers about 2,000 math problems each month. A fourth grade student answers about 7,000 math problems each month. How many total problems would 8 second grade students and 9 fourth grade students answer in one month?
Answer: Totally they both answer 79,000 math problems in one month.

Explanation: A second grade student answers about 2,000 math problems each month.
For 8 students to solve in 1 month we have, 8 × 2,000 = 16,000.

A fourth grade student answers about 7,000 math problems each month.
For 9 students to solve in 1 month we have, 9 × 7,000 = 63,000.

They both have answered, 63,000 + 16,000 = 79,000.
Totally they both answer 79,000 math problems in one month.

Question 26.
Modeling Real Life
A teacher has 5 boxes with 50 pencils in each box. He also has 3 boxes with 20 pens in each box. A student donates 25 pencils. How many pens and pencils does the teacher have now?

Answer: He has 275 pencils and 60 pens.

Explanation: A teacher has 5 boxes with 50 pencils in each box.  so, 5 × 50 = 250 pencils
He also has 3 boxes with 20 pens in each box. so, 3 × 20 = 60 pens.
A student donates 25 pencils. so 250 + 25 = 275 pencils.

So, He has 275 pencils and 60 pens.

Review & Refresh

Complete the table.
Question 27.

Answer:

Question 28.

Answer:

Lesson 3.3 Estimate Products of Rounding

Explore and Grow

There are 92 marbles in each jar.

Estimate the total number of marbles in two ways.
Answer:  540 and 600
Explanation: 1 jar has 92 marbles, then 6 jars has  6 × 92 marbles
Round to the nearest tens.
92 is close to 90;
6 x 90 = 540.
and
Round to the nearest tens.
92 is close to 100;
6 x 100 = 600.
So the total number of marbles count is in between 540 and 600.

Round 92 to the nearest ten.
Answer:  90
Explanation: Round the nearest tens , so 92 is close to 90.

Round 92 to the nearest hundred.
Answer: 100
Explanation: Round the nearest tens , so 92 is close to 100.

Which estimate do you think is closer to the total number of marbles? Explain.
Answer:   540. The one 92 is closer to 90.
Because marbles in all jars are calculated as 552 by multiplying the 6 and 992.

Explanation: 1 jar has 92 marbles, then 6 jars has  6 × 92 marbles
Round to the nearest tens.
92 is close to 90;
6 x 90 = 540.
and
Round to the nearest tens.
92 is close to 100;
6 x 100 = 600.
So the total number of marbles count is in between 540 and 600.

Repeated Reasoning
Would your answer to the question above change if there were 192 marbles in each jar? Explain
Answer: Yes. 1,140 and 1200.

Explanation: 1 jar has 192 marbles, then 6 jars has  6 × 192 marbles
Round to the nearest tens.
192 is close to 190;
6 x 190 = 1,140.
and
Round to the nearest tens.
192 is close to 200;
6 x 200 = 1200.
So the total number of marbles count is in between 1,140 and 1200.

Think and Grow: Estimate Products

You can estimate a product by rounding.

Example
Estimate 7 × 491
Round 491 to the nearest hundred. Then multiply.

So, 7 × 491 is about ______.
Answer: 3,430
Explanation:
Round to the nearest tens.
491 is close to 490; ;
7 x 490 = 3,430

When solving multiplication problems, you can check whether an answer is reasonable by finding two estimates that a product is between.
Example
Find two estimates that the product of 4 × 76 is between.
Think: 76 is between 70 and 80.

So, the product is between _______ and ____.
Answer: 280 and 320.

Explanation:
Round to the nearest tens.
76 is close to 70;
4 x 70 = 280.
or
Explanation:
Round to the nearest tens.
76 is close to 80;
80 x 4 = 320

Show and Grow

Estimate the product.
Question 1.
3 × 89
Answer: 270

Explanation:
Round to the nearest tens.
89 is close to 90;
3 x 90 = 270

Question 2.
8 × 721
Answer: 5,760

Explanation:
Round to the nearest tens.
721 is close to 720;
8 x 720 = 5,760

Question 3.
5 × 7,938
Answer: 39,700

Explanation:
Round to the nearest tens.
7,938 is close to 7,940;
5  x  7,940 = 39,700

Find two estimates that the product is between
Question 4.
9 × 44
Answer: 360 and 450

Explanation:
Round to the nearest tens.
44 is close to 40;
9 x 40 = 360
and
Explanation:
Round to the nearest tens.
44 is close to 50;
9 x 50 = 450
The product is in between 360 and 450.

Question 5.
2 × 657
Answer:  1,300 and  1,320

Explanation:
Round to the nearest tens.
657 is close to 650;
2 x 650 = 1,300
and
Explanation:
Round to the nearest tens.
657 is close to 660;
2 x 660 = 1,320
The product is in between 1,300 and  1,320

Question 6.
6 × 4,243
Answer:  25,440 and 25,500

Explanation:
Round to the nearest tens.
4,243 is close to 4,240;
6 x 4,240 = 25,440
and
Explanation:
Round to the nearest tens.
4,243 is close to 4,250;
6 x 4,250 = 25,500
The product is in between 25,440 and 25,500

Apply and Grow: Practice

Estimate the product.
Question 7.
4 × 65
Answer: 240

Explanation:
Round to the nearest tens.
65 is close to 60;
4 x 60 = 240

Question 8.
248 × 7
Answer:  1,750

Explanation:
Round to the nearest tens.
248 is close to 250;
250 x 7 = 1,750

Question 9.
3 × 9,032
Answer: 27,090

Explanation:
Round to the nearest tens.
9,032 is close to 9,030;
3 x 9,030 = 27,090

Find two estimates that the product is between.
Question 10.
32 × 9
Answer:  270 and  360

Explanation:
Round to the nearest tens.
32 is close to 30;
9 x 30 = 270
and
Explanation:
Round to the nearest tens.
32 is close to 40;
9 x 40 = 360
The product is in between 270 and 360

Question 11.
970 × 5
Answer: 4,850

Explanation:
There is no need to Round to the nearest tens. Because the number 970 is already has its tens value
970 x 5 = 4,850 .

Question 12.
6 × 5,328
Answer:  31,920 and 31,968

Explanation:
Round to the nearest tens.
5,328 is close to 5,320;
6 x 5,320 = 31,920

Explanation:
Round to the nearest tens.
5,328 is close to 5,330;
6 x 5,330 = 31,968

Question 13.
A sales representative sells 4 smart watches for $199 each. To determine whether the sales representative collects at least $1,000, can you use an estimate, or is an exact number required? Explain.

Answer: $800

Explanation: if each watch cost $199, 4 × 199
then 4 watches costs $ 800
Round to the nearest tens.
199 is close to 200;
4 x 200 = 800
The estimated cost is $800.

YOU BE THE TEACHER
A student finds the product. Is her answer reasonable? Estimate to check.
Question 14.

Answer:  The answer is not reasonable

Explanation: The given number is already at its tens place so,
There is no need to Round to the nearest tens.
480 x 8 = 3,840

Question 15.

Answer: 7,600
The product is equal to the result but the estimated result is different

Explanation:
Round to the nearest tens.
1,904 is close to 1,900;
1,900 x 4 = 7,600

 

Number Sense
Use two of the numbers below to write an expression for a product that can be estimated as shown. You may use the numbers more than once.

Question 16.
Estimate: 40,000
____ × _____
Answer:  5 and 8,932  are used here.

Explanation:
Round to the nearest tens.
8,932 is close to 8,930;
5 x 8,930 = 44,650
The product is estimated as the nearest to 40,000

Question 17.
Estimate: 9,000
_____ × ____
Answer:  3 and  2,842 are used here.
Explanation:
Round to the nearest tens.
2,842 is close to 2,850;
3 x 2,850 = 8,550
The product is estimated as the nearest to 9,000

Question 18.
Estimate: 72,000
_____ × ______
Answer: 8 and 8,932  are used here.

Explanation:
Round to the nearest tens.
8,932 is close to 8,930;
8 x 8,930 = 71,440
The product is estimated as the nearest to 72,000.

Question 19.
Estimate: 45,000
_____ × ______
Answer:   5 and 8,932  are used here.

Explanation:
Round to the nearest tens.
8,932 is close to 8,930;
5 x 8,930 = 44,650
The product is estimated as the nearest to 45,000.

Think and Grow: Modeling Real Life

Example
About how much more money was earned from new release rentals than from top pick rentals?
Estimate the amount of money earned from new release rentals.


So, about $ ______ more was earned from new release rentals.
Answer: Estimation of the amount of money earned from new release rentals is  $15,250,
Estimation of the amount of money earned from top pick rentals is $11,970,
Money was earned from new release rentals than from top pick rentals is $3,280

Explanation:
Estimation of the amount of money earned from new release rentals is
Round to the nearest tens.
3,047 is close to 3,050;
$5 x 3,050 = $15,250

Estimation of the amount of money earned from top pick rentals is
Round to the nearest tens.
3,986 is close to 3,990;
$3 x 3,990 = $11,970

Money was earned from new release rentals than from top pick rentals is
subtract  $11,970 from $15,250 then we get  $3,280..

Show and Grow

Use the graph above.
Question 20.
About how much more money was earned from top pick rentals than from television episode rentals?
Answer:  Estimation of the amount of money earned from top pick rentals is  $11,970
Estimation of the amount of money earned from to television episode rentals is $1,700
Money was earned from top pick rentals than from television episode rentals is $10,270

Explanation:   Estimation of the amount of money earned from top pick rentals is
Round to the nearest tens.
3,986 is close to 3,990;
$3 x 3,990 = $11,970

Estimation of the amount of money earned from to television episode rentals is
Round to the nearest tens.
849 is close to 850;
$2 x 850 = $1,700

Money was earned from top pick rentals than from television episode rentals is
subtract $1,700 from $11,970 we get  $10,270.

Question 21.
Your friend says that the number of top picks rented is about 5 times as many as the number of television episodes rented. Is your friend correct? Explain.
Answer: yes. The number of top picks rented is about 5 times as many as the number of television episodes rented.

Explanation:   Estimation of the amount of money earned from top pick rentals is
Round to the nearest tens.
3,986 is close to 3,990;
$3 x 3,990 = $11,970

Estimation of the amount of money earned from to television episode rentals is
Round to the nearest tens.
849 is close to 850;
$2 x 850 = $1,700

Money was earned from top pick rentals than from television episode rentals is
subtract $1,700 from $11,970 we get  $10,270.

The number of top picks rented is about 5 times as many as the number of television episodes rented.
5 × $1,700 = $8,500

Question 22.
An accountant says that the total amount of money earned from new release rentals is $11,958. Check whether the accountant’s answer is reasonable by finding two estimates that the total is between.

Answer: The total amount of money earned from new release rentals is $11,958 is not reasonable .

Explanation:
Estimation of the amount of money earned from new release rentals is
Round to the nearest tens.
3,047 is close to 3,040;
$5 x 3,040 = $15,200
Estimation of the amount of money earned from new release rentals is
Round to the nearest tens.
3,047 is close to 3,050;
$5 x 3,050 =  $15,250
The estimated product should be in between  $15,200 and $15,250

Estimate Products of Rounding Homework & Practice 3.3

Estimate the product
Question 1.
7 × 42
Answer: 280

Explanation:
Round to the nearest tens.
42 is close to 40;
7 x 40 = 280

Question 2.
85 × 4
Answer: 320

Explanation:
Round to the nearest tens.
85 is close to 80;
4 x 80 = 320

Question 3.
2 × 698
Answer: 1,400

Explanation:
Round to the nearest tens.
698 is close to 700;
2 x 700 = 1,400

Question 4.
6 × 705
Answer: 4,200

Explanation:
Round to the nearest tens.
705 is close to 700;
6 x 700 = 4,200

Question 5.
1,834 × 9
Answer: 16,470

Explanation:
Round to the nearest tens.
1,834 is close to 1,830;
9 x 1,830 = 16,470

Question 6.
7,923 × 8
Answer: 63,360

Explanation:
Round to the nearest tens.
7,923 is close to 7,920;
8 x 7,920 = 63,360

Find two estimates that the product is between
Question 7.
3 × 95
Answer: 270 and 300

Explanation:
Round to the nearest tens.
95 is close to 90;
3 x 90 = 270
and
Explanation:
Round to the nearest tens.
95 is close to 100;
3 x 100 = 300
The product is in between 270 and 300

Question 8.
23 × 5
Answer: 100 and 150

Explanation:
Round to the nearest tens.
23 is close to 20;
20 x 5 = 100
and
Explanation:
Round to the nearest tens.
23 is close to 30;
5 x 30 = 150
The product is in between 100 and 150

Question 9.
537 × 6
Answer: 3,240 and 3,180

Explanation:
Round to the nearest tens.
537 is close to 530;
6 x 530 = 3,180
and
Explanation:
Round to the nearest tens.
537 is close to 540;
6 x 540 = 3,240
The product is in between 3,240 and 3,180

Question 10.
8 × 309
Answer: 2,400 and 2,4800

Explanation:
Round to the nearest tens.
309 is close to 300;
8 x 300 = 2,400
and
Explanation:
Round to the nearest tens.
309 is close to 310;
8 x 310 = 2,480
The product is in between 2,400 and 2,4800

Question 11.
1,649 × 7
Answer:

Explanation:
Round to the nearest tens.
1,649 is close to 1,640;
1,640 x 7 = 11,480
and
Explanation:
Round to the nearest tens.
1,649 is close to 1,650;
1,650 x 7 = 11,550
The product is in between 11,480  and  11,550.

Question 12.
4 × 6,203
Answer: 24,800  and  24,840.

Explanation:
Round to the nearest tens.
6,203 is close to 6,200;
4 x 6,200 = 24,800

Explanation:
Round to the nearest tens.
6,203 is close to 6,210;
4 x 6,210 = 24,840

The product is in between  24,800  and  24,840.

Question 13.
Construction workers build 54 feet of a bridge each day for 9 days. To determine whether the bridge is at least 490 feet long, can you use an estimate, or is an exact answer required? Explain.
Answer: exact answer is 9 × 54 = 486 feet long .which is close to the target of 490 feet long

Explanation: if 54 feet of a bridge  each day for 9 days, we get 9 × 54
Round to the nearest tens.
54 is close to 500;
9 x 50 = 450.
The estimated product 450 feet is close to given target of 490 feet long and an exact answer can also satisfies.

Question 14.
YOU BE THE TEACHER
A student finds the product. Is his answer reasonable? Estimate to check.

Answer: Yes .The answer is reasonable  and The estimated product is 17,070.
Explanation:   5,692 × 3
Round to the nearest tens.
5,692 is close to 5,690;
3 x 5,690 = 17,070
The estimated product is 17,070

Question 15.
DIG DEEPER!
An astronaut earns $5,428 each month. You estimate that she will earn $35,000 in 7 months. Is the amount she earns in 7 months greater than or less than your estimate? Explain.
Answer: The estimated product  $38,010 is greater than the given $35,000.

Explanation:
if astronaut earns $5,428 each month then ,
Estimation of the amount of money earned for 7 months is $5,428 × 7
Round to the nearest tens.
5,428 is close to 5,430;
$5,430 x 7 = $38,010
The estimated product  $38,010 is greater than the given $35,000.

Question 16.
Writing
Explain how estimating by rounding can be helpful as a check when finding a product.
Answer:  sTo find the actual product of the two factors of numbers are quite hard for larger digit numbers so by rounding the higher digit numbers which have nearest to 10’s or 100’s. We will get the estimated product easily and fast . To get the actual product we have to use the normal method .

Modeling Real Life
Use the graph to answer the question.

Question 17.
Which animal has a heart rate that is about 2 times as fast as a whale’s?
Answer: Horse

Explanation: whale has heart rate of 20 Bpm
Horse has around 40 Bpm as per the given details, so
Horse = 2(20) Bpm
Horse has a heart rate that is about 2 times as fast as a whale’s

Question 18.
About how many times does a giraffe’s heart beat in 5 minutes? Find two estimates that the answer is between.
Answer:  300 and 350 times
Explanation:
Giraffe’s heart beat is in between 60  and 70 as per the details so
to find for 5 min we have to multiply 5 × 60  and 5 × 70
so, 300 and 350 is the estimation times.

Review & Refresh

Use the Distributive Property to find the product.
Question 19.
9 × 6 = ____ × (5 + ____)
= (9 × 5) + (____ + _____)
= ____ + ____
= _____
Answer: 54

Explanation: by using Distributive property
9 × 6 = 9 × (5 + 1)
= (9 × 5) + (9 × 1)
= 45 + 9
= 54

Question 20.
7 × 7 = 7 × (5 + ____)
= (_____ × 5) + (7 × _____)
= _____ + _____
= _____
Answer: 49

Explanation: by using Distributive property
7 × 7 = 7 × (5 + 2)
= (7 × 5) + (7 × 2)
= 35 + 14
= 49

Lesson 3.4 Use the Distributive Property to Multiply

Explore and Grow

Use base ten blocks to model 4 × 16. Draw your model. Then find the area of the model.

4 × 16 = _____
Break apart 16 to show two smaller models. Find the area of each model. What do you notice about the sum of the areas?
Area = _____ Area = ______
Answer:  64
Explanation: 4 × 16 = 4× (10 + 6)
= (4 × 10) + (4 × 6)
= 40 + 24
= 64

Reasoning
How does this strategy relate to the Distributive Property? Explain.
Answer: Because of the evenly distributed  numbering among the place – values of the numbers. This property helps to find the product in a easy way.

Think and Grow: Use the Distributive Property to Multiply

Think: One way to multiply a two-digit number is to first break apart the number. Then use the Distributive Property.


Answer: 108

Show and Grow

Draw an area model. Then find the product.
Question 1.
7 × 11 = ____

Answer: 77
Explanation:
by using Distributive property
7 × 11 = 7 × (7 + 4)
= (7 × 7) + (7 × 4)
= 49 + 28
= 77

Question 2.
2 × 15 = _____

Answer: 30

Explanation:
by using Distributive property
2 × 15 = 2 × 10 + 5)
= (2 × 10) + (2 × 5)
= 20 + 10
= 30

Apply and Grow: Practice

Draw an area model. Then find the product.
Question 3.
4 × 16 = ______

Answer: 64

Explanation:
by using distributive property
4 × 16 = 4× (10 + 6)
= (4 × 10) + (4 × 6)
= 40 + 24
= 64

Question 4.
9 × 18 = _____

Answer: 162
Explanation:
by using distributive property
9 × 18 = 9× (10 + 8)
= (9 × 10) + (9 × 8)
= 90 + 72
= 162

Question 5.
6 × 27 = _____

Answer: 162

Explanation:
by using distributive property
6 × 27 = 6× (20 + 7)
= (6 × 20) + (6 × 7)
= 120 + 42
= 162

Find the product.
Question 6.
3 × 46 = _____
Answer: 129
Explanation: by using distributive property
3 × 43 = 3 × (40 + 3)
= (3 × 40) + (3 × 3)
= 120 + 9
= 129

Question 7.
8 × 35 = ____
Answer: 280

Explanation: by using distributive property
8 × 35 = 8× (30 + 5)
= (8 × 30) + (8 × 5)
= 240 + 40
= 280

Question 8.
5 × 72 = _____
Answer: 360

Explanation: by using distributive property
5 × 72 = 5 × (70 + 2)
= (5 × 70) + (5 × 2)
= 350 + 10
= 360

Question 9.
Number Sense
Use the area model to complete the equation

5 × ____ = _____
Answer: 5 × 34 = 170
Explanation: by using distributive property
5 × 34 = 5 × (30 + 4)
= (5 × 30) + (5 × 4)
= 150 + 20
= 170

Think and Grow: Modeling Real Life

Example
A parking garage has 9 floors. Each floor has 78 parking spaces. 705 cars are trying to park in the garage. Are there enough parking spaces? Explain.
Multiply the number of floors by the number of parking spaces on each floor.

Answer:   Only 702 cars are able to park in the garage

Explanation: by using distributive property
9 × 78 = 9× (70 + 8)
= (9 × 70) + (9 × 8)
= 630 + 72
= 702
Compare the number of spaces to the number of cars trying to park in the garage.
There _____ enough parking spaces.
Explain:
Answer:  Total number of spaces in the garage are 702 but as per the given details 705 are trying to park in the garage .
There are not enough parking spaces.

 

Show and Grow

Question 10.
Your piano teacher wants you to practice playing the piano 160 minutes this month. So far, you have practiced 5 minutes each day for 24 days. Have you reached the goal your teacher set for you? Explain.
Answer: 150 minutes
Explanation : practice for 160 minutes in a month ,say that 30 days
So far, you have practiced 5 minutes each day for 24 days. then, 5 × 24, By using distributive property
= 5 × (20+ 4)
= (5 × 20) + (5 × 4)
= 100 + 20
= 120
Remaining 6 days have to be practiced so ,5 × 6, By using distributive property
= 5 × (4 + 2)
= (5 × 4) + (5 × 2)
= 20 +10
= 30
Totally we have 120 + 30
= 150

Question 11.
Newton has $150. He wants to buy a bicycle that costs 4 times as much as the helmet. Does he have enough money to buy the bicycle and the helmet? Explain.

Answer: $112
Explanation: Helmet = $ 28 ,bicycle that costs 4 times as much as the helmet. so
Bicycle = 4 × $28, By using distributive property
= 4 × (20 + 8 )
= (4 × 20) + (4 × 8)
= 80 + 32
= 112. so ,The bicycle costs $112 and he has $150 so he can buy the Bicycle.

Question 12.
A baby orangutan weighs 3 pounds. The mother orangutan weighs 27 times as much as the baby. The father orangutan weighs 63 times as much as the baby. What are the weights of the mother and the father?

Answer: The mother orangutan weighs  81 pounds
The father orangutan weighs 189 pounds

Explanation: Baby orangutan  = 3 pounds
Mother orangutan weighs 27 times as much as the baby , 27 × 3, By using distributive property
3 × 27 = 3 × (20 + 7)
= (3 × 20) + ( 3 × 7)
= 60+21
= 81.
father orangutan weighs 63 times as much as the baby, 63 × 3, By using distributive property
63 × 3 = 3 × (60 + 3)
= (3 × 60) + (3 × 3)
= 180 + 9
= 189.

Use the Distributive Property to Multiply Homework & Practice 3.4

Draw an model. Then find the product
Question 1.
3 × 12 = _____

Answer: 36
Explanation:
3×12= 3 ×(10+2)
=(3×10)+ (3×2)
=30+6
=36

Question 2.
5 × 16 = _____

Answer: 80

Explanation:
5×16= 5 ×(10+6)
=(5×10)+ (5×6)
=50+30
=80

Question 3.
4 × 34 = _____

Answer: 136

Explanation:
4 ×34 = 4 ×(30+4)
=(4×30)+ (4×4)
=120+ 16
=136

Find the product.
Question 4.
9 × 56 = ____
Answer: 504

Explanation: 9 × 56= 9 ×(50+6)
=(9×50)+ (9×6)
=450 + 54
=504

Question 5.
71 × 2 = _____
Answer:  142

Explanation:  71 × 2 = 2 × (70+ 1)
=(2×70)+ (2×1)
= 140+2
=142

Question 6.
3 × 77 = ____
Answer:  231

Explanation:  77 × 3 = 3 × (70+ 7)
=(3×70)+ (3×7)
= 210 + 21
=231

Question 7.
Writing
Explain how you can use the Distributive Property to find a product.
Answer:
The Distributive Property states that when you multiply the sum of two or more addends by a factor, the product  is the same as if you multiplied each addend by the factor and then added the partial products .

Question 8.
Structure
Use the Distributive Property to find 6 × 18 two different ways.
Answer: 108
Explanation: 6 ×18 = 6 ×(10+8)
=(6×10)+ (6×8)
=60 + 48
= 108
or
Explanation: 6 ×18= 6 ×(9 + 9)
=(6×9)+ (6 ×9)
=54 +54
= 108

Question 9.
Reasoning
To find 4 × 22, would you rather break apart the factor 22 as 20 + 2 or as 11 + 11? Explain.
Answer:  88. I would rather prefer to break apart the factor 22 as 20 + 2 . as it has zero’s in the tens place it would be easier to calculate and both of them are correct and anyone can use it on their
Explanation: 4× 22 = 4 ×(20+2)
= (4×20)+ (4×2)
= 80 + 8
= 88
or
Explanation: 4× 22 = 4 ×(11+11)
= (4×11)+ (4×11)
= 44 + 44
= 88

Question 10.
Modeling Real Life
The diameter of a firework burst, in feet, is 45 times the height of the shell in inches. You have an 8-inch firework shell. Will the shell produce a firework burst that has a diameter greater than 375 feet? Explain.

Answer: NO

Explanation: The diameter of a firework burst, in feet, is 45 times the height of the shell in inches
You have an 8-inch firework shell converting into feet e get 0.66 feet
so diameter = 45 × 0.66
=  29.7 feet
The shell  does not produce a firework burst that has a diameter greater than 375 feet

Question 11.
Modeling Real Life
A juvenile bearded dragon should eat 48 crickets each day. You have150 crickets. Do you have enough crickets to feed 3 juvenile bearded dragons? Explain.?

Answer: yes

Explanation:  juvenile bearded dragon should eat 48 crickets each day
To feed 3 juvenile bearded dragons , 3 × 48
= 3 × (40 + 8)
= ( 3 × 40) + (3 × 8)
= 120 + 24
= 144.
You have150 crickets , we can feed them by 144 crickets

Review & Refresh

Plot the fraction on a number line.
Question 12.
\(\ frac{7}{4}\)

Answer:  7/4
Explanation:
To mark 7/4; move seven parts on the right-side of zero.

Question 13.
\(\frac{4}{3}\)

Answer: 4/3

Explanation:
To mark 4/3 ; move four parts on the right-side of zero

Lesson 3.5 Use Expanded Form to Multiply

Explore and Grow

Write 128 in expanded form.
128 = _____ + _____ + _____
Answer: 100 + 20 + 8
Use expanded form to label the area model for 5 × 128. Find the area of each part.

Answer: 640

Explanation: 5 × 128, by expanding we get
= 5 × (100 + 20 + 8), by using distributive property, we get
= (5 × 100) + ( 5 × 20) + ( 5 × 8)
Area model = 500 + 100 + 40
so the result of the partial products is 640
What is the sum of all of the parts? How does the sum relate to the product of 5 and 128?
Answer:  the sum of all of the parts is 640  the sum relate to the product of 5 and 128 because of their partial products.

Repeated Reasoning
Explain to your partner how you can use expanded form to find 364 × 8.
Answer:  2,912
Explanation: 364 × 8 , by expanding we get
= (300 + 60 + 4) × 8, by using distributive property, we get
= (8 × 300) + ( 8 × 60) + ( 8× 4),
Area model = 2400 + 360  + 32
The sum of the partial products is 2,912.

Think and Grow: Use Expanded Form to Multiply

You can use expanded form and the Distributive Property to multiply.
Example
Find 8 × 74.

Answer:  592
Explanation: 8 × 74 , by expanding
= 8 × (70 + 4), by using distributive property,
= (8 × 70) + ( 8 × 4)
= 560 + 32
= 592.

Example
Find 2 × 5,607.

Answer:  11,214
Explanation: 2 × 5,607, by expanding
= 2 × (5,000 + 600 + 7) by using distributive property we get,
= (2 × 5,000) + ( 2 × 600) +(2 × 7)
= 10,000 + 1200 + 14
= 11,214

Show and Grow

Find the product.
Question 1.
4 × 306 = 4 × (_____ + _____)
= (4 × _____) + (4 × _____)
= ____ + _____
= ______
Answer: 1,224
Explanation :  4 × 306 = 4 × ( 300 + 6 )
= (4 × 300 ) + (4 × 6 ) by using distributive property we get
= 1200 + 24
= 1,224

Question 2.
7 × 549
Answer:  3,843

Explanation :  7 × 549 = 4 × ( 500 + 40 + 9 )
= (7 × 500 ) + (7 × 40 ) + (7 × 9)  by using distributive property we get
= 3,500 + 280 + 63
= 3,843

Apply and Grow: Practice

Find the product.
Question 3.
6 × 85 = _____

Answer: 510

Explanation :  6 × 85 = 6 × ( 80 + 5 )
= (6 × 80 ) + (6 × 5 ) by using distributive property we get
= 480 + 30
= 510

Question 4.
2 × 932 = _____

Answer: 1,864

Explanation :  2 × 932 = 2 × ( 900 + 30 + 2 )
= (2 × 900 ) + (2 × 30 ) + ( 2 ×2)  by using distributive property we get
= 1800 + 60 + 4
= 1,864

Question 5.
4 × 690 = 4 × (_____ + ______)
= (4 × _____) + (4 × ____)
= _____ + _____
= _____
Answer: 2,760

Explanation: 4 × 690 = 4 × (600 + 90)
= (4 × 600) + (4 × 90)
= 2,400 + 360
= 2,760

Question 6.
1,027 × 9 = (_____ + _____ + ______) × 9
= (____ × 9) + (_____ × 9) + (_____ × 9)
= _____ + ______ + _____
= _____
Answer: 9,243

Explanation: 1,027 × 9 = (1000+ 20 + 7) × 9
= (1000 × 9) + (20 × 9) + (7 × 9)
= 9000 + 180 + 63
= 9,243

Question 7.
487 × 5 = _____
Answer: 2,435

Explanation :  5 × 487 = 5 × ( 400 + 80 + 7 )
= (5 × 400 ) + (5 × 80 ) + ( 5 ×7 )  by using distributive property we get
= 2000 + 400 + 35
= 2,435

Question 8.
8 × 2,483 = ______
Answer: 19,864

Explanation :  8 × 2,483 = 8 × (2000 + 400 + 80 + 3 )
= (8 × 2000 ) + (8 × 400 ) + ( 8 ×80) + (8 ×3)   by using distributive property we get
= 16,000 + 3200 + 640 +24
= 19,864

Question 9.
A basketball player made 269 three-point shots in a season. How many points did he score from three-point shots?
Answer:

Question 10.
Your cousin runs 6 miles each week. There are 5,280 feet in a mile. How many feet does your cousin run each week?
Answer: 6 × 5,280 = 31,680

Explanation :  cousin runs 6 miles each week. There are 5,280 feet in a mile.so
we have 6 × 5,280
6 × 5,280 = 6 × (5000 + 200 + 80 )
= (6 × 5000 ) + (6 × 200 ) + ( 6 ×80)  by using distributive property we get
= 30,000 + 1200 + 480
= 31,680.

Question 11.
YOU BE THE TEACHER
Your friend finds 744 × 3. Is your friend correct? Explain.
744 × 3 = (700 + 40 + 4) × 3
= (700 × 3) + (40 × 3) + (4 × 3)
= 2,100 + 120 + 12
= 2,232
Answer: Yes

Explanation: By using expanding form and distributive property we get,
744 × 3 = (700 + 40 + 4) × 3
= (700 × 3) + (40 × 3) + (4 × 3)
= 2,100 + 120 + 12
= 2,232

 

Question 12.
DIG DEEPER!
What is the greatest possible product of a two-digit number and a one-digit number? Explain.
Answer:  891
The greatest two digit number is 99 and the greatest single digit number is 9. The products when the two are multiplied. Multiply 99 by 9 to get 891.

Explanation : 99 × 9 = 891

Think and Grow: Modeling Real Life

Example
A baby hippo is fed 77 fluid ounces of milk 5 times each day. There are 128 fluid ounces in 1 gallon. Are2 gallons of milk enough to feed the baby hippo for 1 day?

Answer: 2 gallons of milk is not enough for baby hippo to feed
Explanation:
baby hippo is fed 77 fluid ounces of milk 5 times each day that is, 5 × 77
= 5 × (70 + 7 )
= (5 ×70) + (5 ×7)
= 350 + 35
= 385
There are 128 fluid ounces in 1 gallon , then for 2 gallons it will 256 fluid ounces

Find the number of fluid ounces of milk the baby hippo drinks in 1 day.
5 × 77 = 5 × (70 + 7)
= (5 × 70) + (5 × 7)
= _____ +____
= _____

Answer : 385
Explanation:
baby hippo is fed 77 fluid ounces of milk 5 times each day that is, 5 × 77
= 5 × (70 + 7 )
= (5 ×70) + (5 ×7)
= 350 + 35
= 385

Find the number of fluid ounces in 2 gallons
2 × 128 = 2 × (100 + 20 + 8)
= (2 × 100) + (2 × 20) + (2 × 8)
= ___ + _____ + _____
= _____
Answer: 256
Explanation: 2 × 128 = 2 × (100 + 20 + 8)
= (2 × 100) + (2 × 20) + (2 × 8)
= 200 +40 + 16
= 256.
Compare the products.
Answer: Baby hippo is fed 77 fluid ounces of milk 5 times each day that is, 5 × 77 is 385 ounces
There are 128 fluid ounces in 1 gallon , then for 2 gallons it will 256 fluid ounces
So, 385 ounces is greater than 256 ounces.

So, 2 gallons of milk ______ enough to feed the baby hippo for 1 day.
Answer: 2 gallons of milk is not enough to feed the baby hippo for 1 day.s

Show and Grow

Question 13.
A school with 6 grades goes on a field trip. There are 64 students in each grade. One bus holds 48 students. Will 8 buses hold all of the students?
Answer: YES , there are 384 students in total.

Explanation : A school with 6 grades goes on a field trip.
There are 64 students in each grade. so we have 6 × 64
= 6 × ( 60 + 4 )
= (6 × 60) + ( 6 × 4)
= 360 + 24
= 384.
There are 8 buses in total
One bus holds 48 students.so 8 × 48
= 8 × ( 40 + 8 )
= (8 × 40) + ( 8 × 8)
= 320 + 64
= 384.

So, 8 buses are enough to carry the 384 students of all 6 grades.

Question 14.
The average life span of a firefly is 61 days. The average life span of a Monarch butterfly is 4 times as long as that of a firefly. How many days longer is the average lifespan of a Monarch butterfly than a firefly?

Answer: 183 days longer.

Explanation: The average life span of a firefly is 61 days.
The average life span of a Monarch butterfly is 4 times as long as that of a firefly. so, we have 4 × 61
= 4 × ( 60 + 1 )
= (4 × 60) + ( 4 × 1)
= 240 + 4
= 244.
Then 244 – 61 (The average life span of a firefly) is 183.
So 183 days longer is the average lifespan of a Monarch butterfly than a firefly.

Question 15.
A tourist is in Denver. His car can travel 337 miles using 1 tank of gasoline. He wants to travel to a city using no more than 3 tanks of gasoline. To which cities could the tourist travel?

Answer: Los Angeles 1,016 miles

Explanation: His car can travel 337 miles using 1 tank of gasoline.
He wants to travel to a city using no more than 3 tanks of gasoline. we have 3 × 337
= 3 × (300 + 30 + 7)
= (3 × 300) + (3 × 30) + (3 × 7)
= 900 +90 + 21
= 1,011.
From the given details Los Angeles is the nearest one which he can drive with 3 tanks of gasoline  that is 1,016 miles.

Use Expanded Form to Multiply Homework & Practice 3.5

Find the product
Question 1.
7 × 803 = _____

Answer:  5,621

Explanation: 7 × 803 = 7 × (800 + 3)
= (7 × 800) + (7 × 3)
= 5600 + 21
= 5,621.

Question 2.
9 × 1,024 = _____

Answer:9,216

Explanation: 9 × 1,024 = 9 × (1000 + 20 + 4)
= (9 × 1000) + (9 × 20) + (9 × 4)
= 9000 +180 + 36
= 9,216.

Question 3.
43 × 8 = (40 + 3) × 8
= (____ × 8) + (____ × 8)
= ____ + ____
= _____
Answer: 344

Explanation: 8 × 43 = (40 + 3) × 8
= (40 × 8) + (3 × 8)
= 320 + 24
= 344.

Question 4.
4 × 742 = 4 × (____ + ____ + _____)
= (4 × ____) + (4 × _____) + (4 × _____)
= ____ + _____ + _____
= _____
Answer: 2,968

Explanation: 4 × 742 = 4 × (700 + 40 + 2)
= (4 × 700) + (4 × 40) + (4 × 2)
= 2800 +160 + 8
= 2,968.

Question 5.
3 × 482 = _____
Answer: 1,446

Explanation: 3 × 482 = 2 × (400 + 80 + 2)
= (3 × 400) + (3 × 80) + (3 × 2)
= 1200 + 240 + 6
= 1,446.

Question 6.
4,591 × 6 = _____
Answer: 27,546

Explanation: 6 × 4,591
= 6 × (4,000 + 500 + 90 + 1)
= (6 × 4,000) + ( 6 × 500) +(6 × 90) + ((6 × 1)
= 24,000 + 3,000 + 540 + 6
= 27,546.

Question 7.
Each lobe on a Venus flytrap has 6 trigger hairs that sense and capture insects. A Venus flytrap has 128 lobes. How many trigger hairs does it have?

Answer: 768 trigger hairs.

Explanation: A Venus flytrap has 128 lobes
Each lobe on a Venus flytrap has 6 trigger hairs, we have 6 × 128
= 6 × (120 + 8)
= (6 × 120) + (6 × 8)
= 720 + 48
= 768
So, it has 768 trigger hairs.

Question 8.
A human skeleton has 206 bones. How many bones do 4 skeletons have in all?
Answer: 824

Explanation:  A human skeleton has 206 bones
so 4 × 206 = 4 × ( 200 + 6)
= (4 × 200) + ( 4 × 6)
= 800 + 24
=824.
So, 4 skeletons have 824 bones.

Question 9.
Writing
Explain how you can find 5 × 7,303 using expanded form.
Answer: 36,515

Explanation: 5 × 7,303
= 5 × (7,000 + 300 + 3)
= (5 × 7,000) + ( 5 × 300) + (5 × 3)
= 35,000 + 15,000 + 15
= 36,515.

Question 10.
Structure
Rewrite the expression as a product of two factors.
(6,000 × 3) + (70 × 3) + (4 × 3)
Answer: 18,222

Explanation:
(6,000 × 3) + (70 × 3) + (4 × 3)
= (6,000 + 70 + 4) × 3
= 3 × 6,074
= 18,222.

Question 11.
Modeling Real Life
A summer camp has 8 different groups of students. There are 35 students in each group. There are 24 shirts in a box. Will 9 boxes be enough for each student to get one shirt?

Answer: No, 9 boxes are not enough.

Explanation:  A summer camp has 8 different groups of students.
There are 35 students in each group, we have 8 × 35
= 8 × (30 + 5)
= (8 × 30) + ( 8 × 5)
= 240 + 40
=280.
There are 24 shirts in a box
for 9 boxes we have , 9 × 24
= 9 × (20 + 4)
= (9 × 20) + ( 9 × 4)
= 180 + 36
=216.
So, there are not enough shirts for all the students.

Question 12.
Modeling Real Life
Firefighters respond to 65 calls in 1 week. Police officers respond to 8 times as many calls as firefighters in the same week. How many more calls do police officers respond to than firefighters in that week?
Answer: 455 calls.

Explanation: Firefighters respond to 65 calls in 1 week.
Police officers respond to 8 times as many calls as firefighters in the same week.
we have, 8 × 65
= 8 × (60 + 5)
= (8 × 60) + ( 8 × 5)
= 480 + 40
=520.
Then 520 – 65 (Firefighters respond to 65 calls in 1 week.) = 455.
So, 455 calls do police officers respond to than firefighters in that week.

Review & Refresh

Write the total mass shown.
Question 13.

Answer:  913 g

Explanation: Total mass = 500 +100 + 100 + 100 +100 +1 + 1 + 1 +10
=500 + 400 + 3 + 10
=913g

Question 14.

Answer:  1,346

Explanation: 1 kg = 1000g
Total mass = 1000 + 5 + 100 +100 + 100 + 10 + 10 +10 + 1g
=1000 + 300 + 40 + 6
=1,346g

Lesson 3.6 Use Partial Products to Multiply

Explore and Grow

Use the area model to find 263 × 4.

Answer:  1,052

Explanation:
The sum of the partial products = 800 + 240 +12 = 1,052

Repeated Reasoning
Does the product change if you multiply the ones first, then the tens and hundreds? Explain.

Answer: product does not changes.

Explanation:  The product of the two numbers does not change as long as the factor numbers are same in the multiplication products that is (3 × 4 ) + ( 60 × 4) + ( 200 × 4)
= 12 + 240 + 800
= 1,052.

Think and Grow: Practice

Partial products are found by breaking apart a factor into ones, tens, hundreds, and so on, and multiplying each of these by the other factor.

Answer: 194 × 3 = 582.

Explanation:
partial products = (3 × 100) + ( 3 × 90) + (3 × 4)
the sum of the partial products = 300 + 270 + 12 = 582
so, 194 × 3 = 582

Example 2
Answer: 3,190 × 2 = 6,380

Explanation:
The sum of the partial products = 6000 + 200 + 180 + 0 = 6,380
So, 3,190 × 2 = 6,380

Show and Grow

Find the product.
Question 1.

Answer: 430

Explanation:
the sum of the partial products = 400 + 30 = 430
So, 86 × 5 = 430

Question 2.

Answer: 3,514

Explanation:
The sum of the partial products = 3500 + 14 = 3,514
So, 502 × 7 = 3,514

Question 3.

Answer: 21,468

Explanation:
The sum of the partial products is 20,000 + 1200 + 240 + 28 = 21,468
So, 5,367 × 4 = 21,468

Apply and Grow: Practice

Find the product.
Question 4.

Answer: 162

Explanation:
The sum of the partial products are 120 + 42 = 162
So, 27 × 6 = 162

Question 5.

Answer: 4,437

Explanation:
The sum of the partial products are  3600 + 810 + 27 = 4,437
So, 493 × 9 = 4,437

Question 6.

Answer: 55,856

Explanation:
The sum of the partial products are 48000 + 7200 + 640 + 16 = 55,856
So, 6,982 × 8 = 55,856

Question 7.

Answer: 138

Explanation:
The sum of the partial products are 120 + 18 = 138
So, 69 × 2 = 138

Question 8.

Answer: 2,451

Explanation:
The sum of the partial products are 2400 + 30 + 21 = 2,451
So, 817 × 3 = 2,451

Question 9.

Answer: 19,810

Explanation:
The sum of the partial products are 15000 + 4500 + 300 + 10 = 19,810
So, 3,962 × 5 = 19,810

Question 10.

Answer: 5,096

Explanation:
The sum of the partial products are 4900 + 140 + 56 = 5,096
So, 728 × 7 = 5,096

Question 11.

Answer: 9,324

Explanation:
The sum of the partial products are 9000 + 2700 + 54 = 9,324
So, 1,036 × 9 = 9,324

Question 12.

Answer: 39,804

Explanation:
The sum of the partial products are 36,000 + 3600 + 200 + 4  = 39,804
So, 9,951 × 4 = 39,804

Question 13.
Descartes finds 472 × 3. Is he correct? Explain.

Answer: 472 × 3 = 1,416

Explanation:
The sum of the partial products are 1200 + 210 + 6 = 1,416
So, 472 × 3 = 1,416

Question 14.
DIG DEEPER!
Write the multiplication equation shown by the model.

Answer: The given model is not according to the distributive property there is no proper place for tens, hundreds in the zeros place . so, it should be modified in order to get the correct answer.

Question 15.
Writing
Write a multiplication word problem using the numbers 506 and 8. Then solve.
Answer: 506 × 8 = 4,048

Explanation:
The sum of the partial products are 4000 + 48 = 4,048
So, 506 × 8 = 4,048

Think and Grow: Modeling Real Life

Example
The Grand Prismatic Spring at Yellowstone National Park is 121 feet deep. Its width is 7 feet more than3 times its depth. What is the width of the spring?


So, the width of the Grand Prismatic Spring is ______ feet.
Answer: The depth of the Grand Prismatic Spring  is 363 feet and width of the Grand Prismatic Spring  is 370 feet.

Explanation:
The sum of the partial products are 360 + 3 = 363. The depth of the Grand Prismatic Spring  is 363 feet
So, the width of the Grand Prismatic Spring  is 363 + 7 = 370 feet.

Show and Grow

Question 16.
The mass of a giant squid is 202 kilograms. The mass of a beluga whale is 78 kilograms less than 6 times the mass of the giant squid. What is the mass of beluga whale?

Answer: the mass of beluga whale is 1,134 kilograms.

Explanation : 6 times the mass of the giant squid. is 6 × 202  = 1,212
The sum of the partial products are 1200 + 12 = 1,212.
The mass of a beluga whale is 78 kilograms less than 6 times the mass of the giant squid = 1,212 – 78 = 1,134 kg
The mass of beluga whale is 1,134 kilograms

Question 17.
Newton replaces all 4 tires on his car and pays $159 for each tire. Descartes replaces all 4 tires on his truck and pays $227 for each tire. How much more does Descartes pay to replace his tires?
Answer:  $272 more has to pay to replace the tires.

Explanation: Newton replaces all 4 tires on his car and pays $159 for each tire
so, $159 × 4 =  $636
The sum of the partial products are 400 + 200 + 36 =  $636.

Descartes replaces all 4 tires on his truck and pays $227 for each tire.
so, $227 × 4 =  $908
The sum of the partial products are 800 + 80 + 28 = $908.
Then $908  – $636 = $272.
So,$272 more has to pay to replace the tires.

Question 18.
There are 52 weeks and 1 day in a year. There are 52 weeks and 2 days in a leap year. How many weeks are there in 6 years if one of the years is a leap year?
Answer: There are 313 weeks are there for 6 years .

Explanation: Apart from the leap year there are 5 years with 52 weeks and 1 day
then, 52 × 5 = 260 weeks and 5 × 1 = 5 days ( 5 years)
In a leap year we have 52 weeks and 2 days adding this to the total count above we have,
260 weeks + 52 weeks = 312 weeks and by adding days we have 5 + 2 = 7 that is  a week
Totally 312 weeks + 1 week = 313 weeks.
There are 313 weeks are there for 6 years .

Use Partial Products to Multiply Homework & Practice 3.6

Find the product
Question 1.

Answer: 185

Explanation:
The sum of the partial products are 150 + 35 = 185
So, 37 × 5 = 185

Question 2.

Answer: 7,029

Explanation :
The sum of the partial products are 6300 + 720 + 9 = 7,029
So, 781 × 9 = 7,029

Question 3.

Answer: 8,811

Explanation:
The sum of the partial products are 6000 + 2700 + 90 + 21 = 8,811
S0, 2,937 × 3 = 8,811

Question 4.

Answer: 4,692

Explanation:
The sum of the partial products are 4200 + 480 12 = 4,962
So, 782 × 6 = 4,962

Question 5.

Answer: 680

Explanation:
The sum of the partial products are 640 + 40 = 680
So, 85 × 8 = 680

Question 6.

Answer: 16,052

Explanation:
The sum of the partial products are 16,000 + 40 + 12 = 16,052
So, 8,026 × 2 = 16,052

Find the product
Question 7.

Answer: 2,793

Explanation:
The sum of the partial products are 2100 + 630 + 63 = 2,793
So, 399 × 7 = 2,793

Question 8.

Answer: 29,560

Explanation:
The sum of the partial products are 28,000 + 1200 + 360 = 29,560
So, 7,390 × 4 = 29,560

Question 9.

Answer: 46,430

Explanation:
The sum of the partial products are 45,000 + 1000 + 400 + 30 = 46,430
So, 9,286 × 5 = 46,430

Question 10.
Number Sense
Which four numbers are the partial products that you add to find the product of 3,472 and 6?

Answer: 18,000 ; 24,00 ; 420 ; 12.

Explanation:
The sum of the partial products are 18,000 + 2400 + 420 + 12 = 20,832.
So, The partial products are 18,000 ; 24,00 ; 420 ; 12.

Question 11.
DIG DEEPER!
The sum of a four-digit number and a one-digit number is 7,231. The product of the numbers is 28,908. What are the numbers?
Answer: 7227 and 4

Explanation:
The sum of a four-digit number and a one-digit number is 7,231. The product of the numbers is 28,908. so we have 7,227 + 4 = 7,231 and product is 7,227 × 4 = 28,908.
So, The numbers are 7,227 and 4

Question 12.
Modeling Real Life
The height of the Eiffel Tower is 38 feet more than 3 times the height of Big Ben. What is the height of the Eiffel Tower?

Answer: The height of the Eiffel Tower is 986 feet.

Explanation:
The height of the Eiffel Tower is 38 feet more than 3 times the height of Big Ben.
so, 316 × 3 = 948 feet,
The height of the Eiffel Tower is 38 feet more we get 948 + 38 = 986 feet.
So, The height of the Eiffel Tower is 986 feet.

Question 13.
Modeling Real Life
An animal shelter owner has 9 dogs and 4 cats ready for adoption. How much money will the owner collect when all of the animals are adopted?

Answer:  $12,150 money is collect from the adoption.

Explanation:
Each cat is $450 , Then for 4 cats we get 4 × 450 = 1,800
Each dog is $1,150 ,Then for 9 dogs we get 9 × 1,150 = 10,350.

Total amount is $1800 + $10,350 = $ 12,150.
So, $12,150 money is collect from the adoption.

Review & Refresh

Write all of the names for the quadrilateral.
Question 14.

Answer: Parallelogram, Square, Rectangle, Rhombus, Trapezoid.

Explanation: In geometry, a quadrilateral can be defined as a closed, two dimensional shape which has four straight sides.
We can find the shape of quadrilaterals in various things around us, like in a chess board, a deck of cards, a kite, a tub of popcorn, a sign board and in an arrow.

Question 15.

Answer: square.

Explanation:
All sides are equal.
All angles are equal and measure 90°.

Lesson 3.7 Multiply Two-Digit Numbers by One-Digit Numbers

Explore and Grow

Model 24 × 3. Draw to show your model.

24 × 3 = _____
How did you use regrouping to find the product?
Answer: 72

Precision
Use a model to find 15 × 8.
Answer: 120

Think and Grow: Use Regrouping to Multiply

Example
Find 32 × 6.
Estimate: 30 × 6 = _____

Check: Because ____ is close to the estimate, _____, the answer is reasonable.
Answer: 32 × 6 = 192 , Then 30 × 6 = 180.

Explanation: Because 192  is close to the estimate, 180 , the answer is reasonable.

Show and Grow

Question 1.
Use the model to find the product.

Answer: 132.

Explanation:

Find the product. Check whether your answer is reasonable.
Question 2.
Estimate: ______

Answer: 96, the Estimate product is 80

Explanation:
8 × 2 ones = 16 ones
regroup 16 ones as
1 ten and 6 ones
Then 8 × 1 ten = 8 tens
8 tens + 1 ten = 9 tens , then we have 96

And the Estimate product is 80,
because 12 is close to 10 , we have 10 × 8 = 80
So, 12 × 8 = 96

Question 3.
Estimate: _____

Answer: 235, the estimated product is 250

Explanation:

5 × 7 ones = 35 ones
regrouping 35 ones as 3 tens and 5 ones
5 × 4 tens = 20 tens
20 tens + 3 tens = 23 tens  or  2 hundreds and 3 tens ,Then we get 235

And the estimated product is 250
because 47 is close to 50,then 50 × 5 = 250
So, 47 × 5 = 235

Question 4.
Estimate: _____

Answer:  138, he estimated product is 120

Explanation:
6 × 3 ones = 18 ones
regrouping 18 ones as 1 ten and 8 ones
6 × 2tens = 12 tens
12 tens + 1 ten = 13 tens or 1 hundred and 3 tens ,Then we have 138

And the estimated product is 120
because 23 is close to 20 , so 20 × 6 = 120
So, 23 × 6 = 138

 

Apply and Grow: Practice

Find the product. Check whether your answer is reasonable.
Question 5.
Estimate: ______

Answer: 175 , the Estimate product is 140

Explanation:
7 × 5 ones = 35 ones
regrouping 35 ones as 3 tens and 5 ones
7 × 2tens = 14 tens
14 tens + 3 tens = 17 tens or 1 hundred and 7 tens ,Then we have 175

And the estimated product is 140
because 25 is close to 20 , so 20 × 7 = 120
So, 25 × 7 = 175

Question 6.
Estimate: ______

Answer:  68 , the Estimate product is 60

Explanation:
2 × 4 ones = 8 ones
2 × 3tens = 6 tens
Then we have 68

And the estimated product is 60
because 34 is close to 30 , so 30 × 2 = 60
So, 34 × 2 = 68

 

Question 7.
Estimate: ______

Answer: 504, the Estimate product is 540

Explanation:
9 × 6 = 54 ones
regrouping 54 ones as 5 tens and 4 ones
9 × 5 tens = 45 tens
45 tens + 5 tens = 50 tens or 5 hundreds ,Then we have 504

And the estimated product is 540
because 56 is close to 60 , so 60 × 9 = 540
So, 56 × 9 = 504

Question 8.
Estimate: ______

Answer:  328, the Estimate product is 320

Explanation:   
8 × 1 ones = 8 ones
8 × 4 tens = 32 tens
Then we have 328

And the estimated product is 320
because 41 is close to 40 , so 40 × 8 = 320
So, 41 × 8 = 328

Question 9.
Estimate: ______

Answer: 292, the Estimate product is 280

Explanation:
4 × 3 ones = 12 ones
regrouping 12 ones as 1 ten and 2 ones
4 × 7 tens = 28  tens
28 tens + 1 ten  = 29 tens or 2 hundred and 9 tens ,Then we have 292

And the estimated product is 280
because 73 is close to 70 , so 70 × 4 = 280
So, 73 × 4 = 292

Question 10.
Estimate: ______

Answer:  534, the Estimate product is 540

Explanation:
6 × 9 ones = 54 ones
regrouping 54 ones as 5 tens and 4 ones
6 × 8 tens = 48 tens
48 tens + 5 tens = 53 tens or 5 hundred and 3 tens ,Then we have 534

And the estimated product is 540
because 89 is close to 90 , so 90 × 6 = 540
So, 89 × 6 = 534

Question 11.
Estimate: ______
65 × 7 = ______
Answer:  455, the Estimate product is 490

Explanation:
7 × 5 ones = 35 ones
regrouping 35 ones as 3 tens and 5 ones
7 × 2tens = 14 tens
14 tens + 3 tens = 17 tens or 1 hundred and 7 tens ,Then we have 175

And the estimated product is 140
because 25 is close to 20 , so 20 × 7 = 120
So, 25 × 7 = 175

Question 12.
Estimate: ____
3 × 92 = _____
Answer: 276, the Estimate product is 270

Explanation:

3 × 2 ones = 6 ones
3 × 9 tens = 27 tens
Then we get 235

And the estimated product is 270
because 92 is close to 90, then 90 × 3 = 270
So, 92 × 3 = 276

Question 4.

Question 13.
Estimate: _____
47 × 5 = ______
Answer:  235, the Estimate product is 250

Explanation:

5 × 7 ones = 35 ones
regrouping 35 ones as 3 tens and 5 ones
5 × 4 tens = 20 tens
20 tens + 3 tens = 23 tens  or  2 hundreds and 3 tens ,Then we get 235

And the estimated product is 250
because 47 is close to 50,then 50 × 5 = 250
So, 47 × 5 = 235

Question 14.
There are 8 questions during the first round of a game show. Each question is worth 15 points. What is the greatest number of points that a contestant can earn during the first round?
Answer: 120 is the greatest number of points that a contestant can earn during the first round

Explanation:

8 × 5 ones = 40 ones
regrouping 40 ones as 4 tens and 0 ones
1 × 8 tens = 8 tens
8 tens + 4 tens = 12 tens  or  1 hundred  and 2 tens ,Then we get 120
So, 120 is the greatest number of points that a contestant can earn during the first round.

Question 15.
DIG DEEPER!
Find the missing digits.

Answer:  1 , 7 and 1 , 9 and 7.

Explanation:

5 × 3 ones = 15 ones
regrouping 15 ones as 1 ten and 5 ones
5 × 1 ten = 5 tens
5 tens + 1 ten = 6 tens  ,Then we get  65

Explanation:

2 × 7 ones = 14 ones
regrouping 14 ones as 1 ten and 4 ones
2 × 3 tens = 6 tens
6 tens + 1 ten = 7 tens  Then we get  74

Explanation:

9 × 8 ones = 72 ones
regrouping 72 ones as 7 tens and 2 ones
9 × 8 tens = 72 tens
72 tens + 7 tens = 79 tens  or  7 hundreds and 9 tens ,Then we get  792
Think and Grow: Modeling Real Life

Example
The fastest human on Earth can run up to 27 miles per hour. A pronghorn antelope can run up to 2 times as fast as the fastest human. A cheetah can run up to 61 miles per hour. Which animal can run faster?

Find how fast a pronghorn antelope can run.

Answer: 54 miles per hour

Explanation:
2 × 7 ones = 14 ones
regrouping 14 ones as 1 ten and 4 ones
2 × 2 tens = 4 tens
4 tens + 1 ten = 5 tens  Then we get  54.

A pronghorn antelope can run up to _____ miles per hour.
Answer: A pronghorn antelope can run up to 54 miles per hour.

Compare the fastest speeds of a pronghorn antelope and a cheetah.
So, a _______ can run faster.
Answer: A cheetah can run faster.

Show and Grow

Question 16.
A band’s goal is to produce 100 songs. The band has produced 6 albums with 13 songs on each album. Has the band reached its goal?
Answer: NO , 78 songs

Explanation:

6 × 3 ones = 18 ones
regrouping 18 ones as 1 ten and 8 ones
6 × 1 ten = 6 tens
6 tens + 1 ten = 7 ten Then we get  78
A band’s goal is to produce 100 songs but they had only 78 songs so they did not get to their goal.

Question 17.
A teenager must practice driving with an adult for 50 hours before taking a driver’s license test. A teenager practices driving with an adult 4 hours each week for14 weeks. Has the teenager practiced long enough to take the test?

Answer: YES , 56 hours

Explanation:

4 × 4 ones = 16 ones
regrouping 16 ones as 1 ten and 6 ones
4 × 1 ten = 4 tens
4 tens + 1 ten = 5 tens Then we get  56
A teenager must practice driving with an adult for 50 hours So she has practiced enough to get a license.

Question 18.
How much more does it cost to rent a personal water craft for 3 hours than a motor boat for 3 hours?

Answer: It cost $60 more to rent a personal water craft for 3 hours than a motor boat for 3 hours.

Explanation:
A Personal water craft cost $89 per hour then per 3 hours, $89 × 3 ,so
3 × 9 ones = 27 ones
regrouping 27 ones as 2 tens and 7 ones
3 × 8 tens = 24 tens
24 tens + 2 tens = 26 tens  or  2 hundreds and 6 tens ,Then we get  $267.

Explanation:
A Motor boat costs $69 per hour then for 3 hours,$69 × 3 ,so
3 × 9 ones = 27 ones
regrouping 27 ones as 2 tens and 7 ones
3 × 6 tens = 18 tens
18 tens + 2 tens = 20 tens or  2 hundreds ,Then we get  $207

Finally $267 – $ 207 = $60
It cost $60 more to rent a personal water craft for 3 hours than a motor boat for 3 hours.

Multiply Two-Digit Numbers by One-Digit Numbers Homework & Practice 3.7

Question 1.
Use the model to find the product.

Answer: 106

Find the product. Check whether your answer is reasonable.
Question 2.
Estimate: _____

Answer: 72 , the Estimate product is 60,

Explanation: 
3 × 4 ones = 12 ones
regroup 12 ones as 1 ten and 2 ones
Then 3 × 2 tens = 6 tens
6 tens + 1 ten = 7 tens , then we have 72

And the Estimate product is 60,
because 24 is close to 20 , we have 20 × 3 = 60
So, 24 × 3 = 72.

Question 3.
Estimate: _____

Answer: 126 ,the Estimate product is 140,

Explanation: 
7 × 8 ones = 56 ones
regroup 56 ones as
5 tens and 6 ones
Then 7 × 1 ten = 7 tens
7 tens + 5 tens = 12  tens , then we have 126

And the Estimate product is 140,
because 18 is close to 20 , we have 20 × 7 = 140
So, 18 × 7 = 126

Question 4.
Estimate: ______

Answer: 198, the Estimate product is 180.

Explanation: 
6 × 3 ones = 18 ones
regroup 18 ones as 1 ten and 8 ones
Then 6 × 3 tens = 18 tens
18 tens + 1 ten = 19 tens , then we have  198

And the Estimate product is 180,
because 33 is close to 30 , we have 30 × 6 = 180
So, 33 × 6 = 198

Question 5.
Estimate: ______

Answer: 423, the Estimate product is 450.

Explanation: 
9 × 7 ones = 63 ones
regroup 63 ones as 6 tens and  3 ones
Then 9 × 4 tens = 36 tens
36 tens + 6 ten = 42 tens , then we have 423

And the Estimate product is 450,
because 47 is close to 50 , we have 50 × 9 = 450
So, 47 × 9 = 423

Question 6.
Estimate: ______

Answer: 260,the Estimate product is 240.

Explanation: 
4 × 5 ones = 20 ones
regroup 20 ones as 2 tens and 0 ones
Then 4 × 6 tens = 24 tens
24 tens + 2 tens = 26 tens , then we have 260

And the Estimate product is 240,
because 65 is close to 60 , we have 60 × 4 = 240
So, 65 × 4 = 260

Question 7.
Estimate: ______

Answer:  576 ,the Estimate product is 560.

Explanation:
8 × 2 ones = 16 ones
regroup 16 ones as
1 ten and 6 ones
Then 8 × 7 tens = 56 tens
56 tens + 1 ten = 57 tens , then we have 576

And the Estimate product is 560,
because 72 is close to 70 , we have 70 × 8 = 560
So, 72 × 8 = 576.

Find the product. Check whether your answer is reasonable.
Question 8.
Estimate: ______
49 × 5 = _____
Answer: 245 , the Estimate product is 250,

Explanation: 
5 × 9 ones = 45 ones
regroup 45 ones as 4 tens and 5 ones
Then 5 × 4 tens = 20 tens
20 tens + 4 tens = 24 tens , then we have  245

And the Estimate product is 250,
because 49 is close to 50 , we have 50 × 5 = 250
So, 49 × 5 = 250.

Question 9.
Estimate: _______
7 × 86 = _______
Answer: 602, the Estimate product is 630,

Explanation: 
7 × 6 ones = 42 ones
regroup 42 ones as 4 tens and  2 ones
Then 7 × 8 tens = 56 tens
56 tens + 4 tens = 60 tens , then we have 602

And the Estimate product is 630,
because 86 is close to 90 , we have 90 × 7 = 630
So, 86 × 7 = 602.

Question 10.
Estimate: ______
93 × 3 = ______
Answer: 279 , the Estimate product is 270

Explanation: 
3 × 3 ones = 9 ones
Then 3 × 9 tens = 27 tens
then we have 279

And the Estimate product is 270,
because 93 is close to 90 , we have 90 × 3 = 270
So, 93 × 3 = 270.

Question 11.
You read 56 pages each week. How many pages do you read in 8 weeks?
Answer: 448 pages.

Explanation:
8 × 6 ones = 48 ones
regroup 48 ones as 4 tens and 8 ones
Then 8 × 5 tens = 40 tens
40 tens + 4 tens = 44 tens , then we have  448.
So, 448 pages for 8 weeks.

Question 12.
Number Sense
The sum of two numbers is 20. The product of the two numbers is 51. What are the two numbers?
Answer: 17 and 3  are the two numbers.

Explanation:
3 × 7 ones = 21 ones
regroup 21 ones as 2 tens and 1 ones
Then 3 × 1 ten = 3 tens
3 tens + 2 tens = 5 tens , then we have  51

Question 13.
Reasoning
Your friend multiplies 58 by6 and says that the product is 3,048. Is your friend’s answer reasonable? Explain.
Answer:  Correct answer is 348

Explanation:

6 × 8 ones = 48 ones
regroup 48 ones as 4 tens and 2 ones
Then 6 × 5 tens = 30 tens
30 tens + 4 tens = 34 tens , then we have  348
So, Correct answer is 348 .

Question 14.
DIG DEEPER!
How much greater is 4 × 26 than 3 × 26? Explain how you know without multiplying.
Answer:  4 × 26 is greater than 3 × 26 .
Explanation: The two factors of the given multiplication are 4 and 3 as 4 is greater than 3 , So , any same number multiplied by these two numbers individually will give the greatest number as a result for the number 4 .
So, 4 × 26 is greater than 3 × 26.

Question 15.
Modeling Real Life
A self-balancing scooter travels 12 miles per hour. An all-terrain vehicle can travel 6 times as fast as the scooter. A go-kart can travel 67 miles per hour. Which vehicle can travel the fastest?

Answer: An all-terrain vehicle can travel faster that is 72 miles per hour

Explanation:
6 × 2 ones = 12 ones
regroup 12 ones as 1 ten and 2 ones
Then 6 × 1 tens = 6 tens
6 tens + 1 tens = 7 tens , then we have  72.
Given , A go-kart can travel 67 miles per hour.
An all-terrain vehicle can travel faster that is 72 miles per hour

Question 16.
Modeling Real Life
It takes a spaceship 3 days to reach the moon from Earth. It takes a spaceship 14 times as many days to reach Mars from Earth. How long would it take the spaceship to travel from Earth to Mars and back?

Answer: It takes 84 days for the spaceship to travel from Earth to Mars and back.

Explanation:
3 × 4 ones = 12 ones
regroup 12 ones as 1 ten and 2 ones
Then 3 × 1 ten = 3 tens
3 tens + 1 ten = 4 tens , then we have  42.
It takes 42 days to reach Mars from Earth.
And IT take 42 + 42  = 84 days
It takes 84 days to travel from Earth to Mars and back.

Review & Refresh

Question 17.
You spend 21 fewer minutes riding the bus to school than getting ready in the morning. You take 36 minutes to get ready. How long is the bus ride?
Answer: 15 minutes to ride the bus

Explanation:  You take 36 minutes to get ready.
You spend 21 fewer minutes riding the bus to school than getting ready in the morning
So, 36 – 21 = 15
It takes 15 minutes to ride the bus.

Lesson 3.8 Multiply Three-and Four-Digit Numbers by One-Digit Numbers

Explore and Grow

Use any strategy to find each product.
7 × 39 = _____
7 × 439 = ______
Answer: 273 and 3,073

Explanation: by using distributive property
7 × 39 = 7 × ( 30 + 9)
= (7 × 30)+ (7 ×9)
=210  + 63
= 273.

Explanation:
7 × 439 = 7 × ( 400 + 30 + 9)
= ( 7 × 400) +(7 × 30)+ (7 ×9)
= 2800  + 210 + 63
= 3,073.

Structure
How are the equations the same? How are they different?
Answer: The equations are same when the numbers in the equation are in their standard position as ones, tens, hundreds etc. And at the same time the factors of the products must not change .
The equation is different when the two factor numbers are changing their position irrespective of their places.

Think and Grow: Use Regrouping to Multiply

Example
Find 795 × 4.

Example
Find 6,084 × 2.

Check: Because ____ is close to the estimate, _____, the answer is reasonable.
Answer: 6,000 × 2 = 12,000 and 6,084 × 2 = 12,168.

Explanation:
Step 1:  Multiply ones and regroup
2 × 4 ones = 8 ones

Step 2: Multiply tens and regroup tens
2 × 8 tens = 16 tens
regroup as 1 ten and 6 ones

Step 3: Multiply hundreds and regroup hundreds
2 × 0 = 0

Step 4: Multiply thousands and regroup thousands
2 × 6 thousands = 12 thousands
Finally we have , 12,168.
So, 6,000 × 2 = 12,000 and 6,084 × 2 = 12,168.
Because 6,000  is close to the estimate, 12,000, the answer is reasonable.

Show and Grow

Find the product. Check whether your answer is reasonable.
Question 1.
Estimate: _____

Answer: 615, the estimated product is 600

Explanation:

Step 1:  Multiply ones and regroup
5 × 3 ones = 15 ones
Regroup as 1 ten and 5 ones

Step 2: Multiply tens and regroup tens
5 × 2 tens = 10 tens
10 tens + 1 ten = 11 tens
regroup 11 tens  as 1 hundreds and 1 ten

Step 3: Multiply hundreds and regroup hundreds
5 × 1 hundreds  = 5 hundreds
5 hundreds + 1 hundred = 6 hundreds

By holding all places together we get 615 .

And the estimated product is 600
because 123 is close to 120, 120 × 5 = 600
So, 123 × 5 = 615 .

Question 2.
Estimate: _____

Answer:  2,721 , the estimated product is 2,700

Explanation:

Step 1:  Multiply ones and regroup
3 × 7 ones = 21 ones
Regroup 21 ones  as 2 tens and 1 ones

Step 2: Multiply tens and regroup tens
3 × 0 tens = 0 tens
0 tens + 2 tens = 2 tens

 Step 3: Multiply hundreds and regroup hundreds
3 × 9 hundreds  = 27 hundreds
Regroup 27 hundreds  as 2 thousands and 7 hundreds

By holding all places together we get 2,721 .

And the estimated product is 2,700
because 907 is close to 900, 900 × 3 = 2,700
So, 907 × 3 = 2,721 .

Question 3.
Estimate: ______

Answer:  43,890, the estimated product is 43,800

Explanation:

Step 1:  Multiply ones and regroup
6 × 5 ones = 30 ones
regroup 30 ones as 3 tens and 0 ones

Step 2: Multiply tens and regroup tens
6 × 1 tens = 6 tens
6 tens + 3 tens = 9 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 3 hundreds  = 18 hundreds
regroup as 1 thousand and 8 hundreds

Step 4: Multiply thousands and regroup thousands
6 × 7 thousands = 42 thousands
42 thousands + 1 thousand = 43 thousands

By holding all the places together we get 43,890

And the estimated product is 43,800
because 7,315 is close to 7300, 7300 × 6 = 43,800
So, 7,315 × 6  = 43,890 .

Apply and Grow: Practice

Find the product. Check whether your answer is reasonable.
Question 4.
Estimate: ______

Answer:  840, the estimated product is also 840

Explanation:

Step 1:  Multiply ones and regroup
7 × 0 ones = 0 ones

Step 2: Multiply tens and regroup tens
7 × 2 tens = 14 tens
regroup 14 tens  as 1 hundred and 4 tens

Step 3: Multiply hundreds and regroup hundreds
7 × 1 hundreds  = 7 hundreds
7 hundreds + 1 hundred = 8 hundreds

By holding all places together we get 840.

And the estimated product is 840
because 120 has a zero’s in ones place ,
so it is already close to the nearest 0’s so we don’t have to  estimate separately.

Question 5.
Estimate: ______

Answer:  10,136, the estimated product is 10,000

Explanation:

Step 1:  Multiply ones and regroup
4 × 4 ones = 16 ones
regroup as 1 ten and 6 ones

Step 2: Multiply tens and regroup tens
4 × 3 tens = 12 tens
12 tens + 1 ten = 13 tens
regroup as 1 hundred and 3 tens

Step 3: Multiply hundreds and regroup hundreds
4 × 5 hundreds  = 20 hundreds
20 hundreds +  1 hundred = 21 hundreds
regroup as 2 thousands and 1 hundred

Step 4: Multiply thousands and regroup thousands
4 × 2 thousands = 8 thousands
8 thousands + 2 thousands = 10 thousands

By holding all the places together we get 10,136

And the estimated product is 10,000
because 2,534 is close to 2500, 2500 × 4 = 10,000
So, 2,534 × 4 = 10,136 .

Question 6.
Estimate: ______

Answer: 15,234, the estimated product is 15,200

Explanation:

Step 1:  Multiply ones and regroup
2 × 7 ones = 14 ones
regroup as 1 ten and 4 ones

Step 2: Multiply tens and regroup tens
2 × 1 ten = 2 tens
2 tens + 1 ten = 3 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 6 hundreds  = 12 hundreds
regroup as 1 thousand  and 2 hundreds

Step 4: Multiply thousands and regroup thousands
2 × 7 thousands = 14 thousands
14 thousands + 1 thousand = 15 thousands

By holding all the places together we get 15,234

And the estimated product is 15,200
because 7,617 is close to 7600, 7,600 × 2 = 15,200
So, 7,617 × 2 = 15,234 .

Question 7.
Estimate: _____

Answer:  3,664, the estimated product is 3,680

Explanation:

Step 1:  Multiply ones and regroup
8 × 8 ones = 64 ones
regroup as 6 tens and 4 ones

Step 2: Multiply tens and regroup tens
8 × 5 tens = 40 tens
40 tens + 6 tens = 46 tens
regroup 46 tens  as 4 hundreds  and 6 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 4 hundreds  = 32 hundreds
32 hundreds + 4 hundreds = 36 hundreds

By holding all places together we get 3,664.

And the estimated product is 3,680
because 458 is close to 460, 460 × 8 = 36,80
So, 458 × 8 = 3,664 .

Question 8.
Estimate: _____

Answer: 79,407, the estimated product is 79,200

Explanation:

Step 1:  Multiply ones and regroup
9 × 3 ones = 27 ones
regroup 27 ones as 2 tens and 7 ones

Step 2: Multiply tens and regroup tens
9 × 2 tens = 18 tens
18 tens + 2 tens = 20 tens
regroup as 2 hundreds and 0 tens

Step 3: Multiply hundreds and regroup hundreds
9 × 8 hundreds  = 72 hundreds
72 hundreds + 2 hundreds = 74 hundreds
regroup as 7 thousands and 4 hundreds

Step 4: Multiply thousands and regroup thousands
9 × 8 thousands = 72 thousands
72 thousands + 7 thousand = 79 thousands

By holding all the places together we get 79,407.

And the estimated product is 79,200
because 8,823 is close to 8,800, 8,800 × 9 = 79,200
So, 8,823 × 9  = 79,407 .

Question 9.
Estimate: _____

Answer:  5,850, the estimated product is 5,820

Explanation:

Step 1:  Multiply ones and regroup
6 × 5 ones = 30 ones
regroup as 3 tens and 0 ones

Step 2: Multiply tens and regroup tens
6 × 7 tens = 42 tens
42 tens + 3 tens = 45 tens
regroup 45 tens  as 4 hundreds  and 5 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 9 hundreds  = 54 hundreds
54 hundreds + 4 hundreds = 58 hundreds

By holding all places together we get 5,850.

And the estimated product is 5,820
because 975 is close to 970, 970 × 6 = 5,820
So, 975 × 6 = 5,850 .

Question 10.
Estimate: _____
1,762 × 3 = ______
Answer:  5,286, the estimated product is 5,100

Explanation:

Step 1:  Multiply ones and regroup
3 × 2 ones = 6 ones

Step 2: Multiply tens and regroup tens
3 × 6 tens = 18 tens
regroup as 1 hundred and 8 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 7 hundreds  = 21 hundreds
21 hundreds + 1 hundred = 22 hundreds
regroup as 2 thousands and 2 hundreds

Step 4: Multiply thousands and regroup thousands
3 × 1 thousands = 3 thousands
3 thousands + 2 thousands = 5 thousands

By holding all the places together we get 5,286.

And the estimated product is 5,100
because 1,762 is close to 1,700, 1,700 × 3  = 5,100
So, 1,762 × 3  = 5,286 .

Question 11.
Estimate: ______
5 × 5,492 = ______
Answer:  27,460, the estimated product is 27,000

Explanation:

Step 1:  Multiply ones and regroup
5 × 2 ones = 10 ones
regroup 10 ones as 1 ten and 0 ones

Step 2: Multiply tens and regroup tens
5 × 9 tens = 45 tens
45 tens + 1 ten = 46 tens
regroup as 4 hundreds and 6 tens

Step 3: Multiply hundreds and regroup hundreds
5 × 4 hundreds  = 20 hundreds
20 hundreds + 4 hundreds = 24 hundreds
regroup as 2 thousands and 4 hundreds

Step 4: Multiply thousands and regroup thousands
5 × 5 thousands = 25 thousands
25 thousands + 2 thousand = 27 thousands

By holding all the places together we get 27,460.

And the estimated product is 27,000
because 5,492 is close to 5,400, 5,400 × 5 = 27,000
So, 5,492  × 5  =  27,460.

Question 12.
Estimate: ______
347 × 7 = _____
Answer:   2,429, the estimated product is 2,380

Explanation:

Step 1:  Multiply ones and regroup
7 × 7 ones = 49 ones
regroup as 4 tens and 9 ones

Step 2: Multiply tens and regroup tens
7 × 4 tens = 28 tens
28 tens + 4 tens = 32 tens
regroup 32 tens  as 3 hundreds  and 2 tens

Step 3: Multiply hundreds and regroup hundreds
7 × 3 hundreds  = 21 hundreds
21 hundreds + 3 hundreds = 24 hundreds

By holding all places together we get 2,429.

And the estimated product is 2,380
because 347 is close to 340, 340 × 7 = 2,380
So, 347 × 7 = 2,429 .

Compare
Question 13.

Answer: 6 × 2,843 is greater than 8 × 1,645
17,058  is greater than 13,160

Explanation:

Step 1:  Multiply ones and regroup
6 × 3 ones = 18 ones
regroup 18 ones as 1 ten and 8 ones

Step 2: Multiply tens and regroup tens
6 × 4 tens = 24 tens
24 tens + 1 ten = 25 tens
regroup as 2 hundreds and 5 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 8 hundreds  = 48 hundreds
48 hundreds + 2 hundreds = 50 hundreds
regroup as 5 thousands and 0 hundreds

Step 4: Multiply thousands and regroup thousands
6 × 2 thousands = 12 thousands
12 thousands + 5 thousands = 17 thousands

By holding all the places together we get 17,058
So, 2,843 × 6 = 17,058

Step 1:  Multiply ones and regroup
8 × 5 ones = 40 ones
regroup 40 ones as 4 tens and 0 ones

Step 2: Multiply tens and regroup tens
8 × 4 tens = 32 tens
32 tens + 4 tens = 36 tens
regroup as 3 hundreds and 6 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 6 hundreds  = 48 hundreds
48 hundreds + 3 hundreds = 51 hundreds
regroup as 5 thousands and 1 hundred

Step 4: Multiply thousands and regroup thousands
8 × 1 thousand = 8 thousands
8 thousands + 5 thousands = 13 thousands

By holding all the places together we get 13,160
So, 1,645 × 8 = 13,160

Finally , 17,058  is greater than 13,160.

Question 14.

Answer: 6,582 × 3 is greater than 2,394 × 8
19,746  is greater than 19,152

Explanation:

Step 1:  Multiply ones and regroup
3 × 2 ones = 6 ones

Step 2: Multiply tens and regroup tens
3 × 8 tens = 24 tens
regroup as 2 hundreds and 4 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 5 hundreds  = 15 hundreds
15 hundreds + 2 hundreds = 17 hundreds
regroup as 1 thousand and 7 hundreds

Step 4: Multiply thousands and regroup thousands
3 × 6 thousands = 18 thousands
18 thousands + 1 thousand = 19 thousands

By holding all the places together we get 19,746.
So, 6,582 × 3 = 19,746.

Step 1:  Multiply ones and regroup
8 × 4 ones = 32 ones
regroup 32 ones as 3 tens and 2 ones

Step 2: Multiply tens and regroup tens
8 × 9 tens = 72 tens
72 tens + 3 tens = 75 tens
regroup as 7 hundreds and 5 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 3 hundreds  = 24 hundreds
24 hundreds + 7 hundreds = 31 hundreds
regroup as 3 thousands and 1 hundred

Step 4: Multiply thousands and regroup thousands
8 × 2 thousands = 16 thousands
16 thousands + 3 thousands = 19 thousands

By holding all the places together we get 19,152
So, 2,394 × 8 =  19,152

Finally , 19,746  is greater than 19,152.

Question 15.
A roller coaster is twice as tall as a Ferris wheel that is 228 feet tall. How tall is the roller coaster?
Answer:  The roller coaster is 456 feet tall.

Explanation:
Given, A roller coaster is twice as tall as a Ferris wheel that is 228 feet tall.so , 228 × 2

Step 1:  Multiply ones and regroup
2 × 8 ones = 16 ones
Regroup as 1 ten and 6 ones

Step 2: Multiply tens and regroup tens
2 × 2 tens = 4 tens
4 tens + 1 ten = 5 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 2 hundreds  = 4 hundreds

By holding all places together we get 456 .
The roller coaster is 456 feet tall.

Question 16.
DIG DEEPER!
How can you estimate the number of digits in the product of 7 and 8,348?

Answer:  58,436 , There are 7 digits i the product of the 7 and 8,348

Explanation:

Step 1:  Multiply ones and regroup
7 × 8 ones = 56  ones
regroup 56 ones as 5 tens and 6 ones

Step 2: Multiply tens and regroup tens
7 × 4 tens = 28 tens
28 tens +  5 tens = 33 tens
regroup as 3 hundreds and 3 tens

Step 3: Multiply hundreds and regroup hundreds
7 × 3 hundreds  = 21 hundreds
21 hundreds + 3 hundreds = 24 hundreds
regroup as 2 thousands and 4 hundreds

Step 4: Multiply thousands and regroup thousands
7 × 8 thousand = 56 thousands
56 thousands + 2 thousands = 58 thousands

By holding all the places together we get 58,436.
So, 8,348 × 7 = 58,436

Think and Grow: Modeling Real Life

Example
The lengths of time that a penguin and an elephant seal can hold their breaths are shown. A sea turtle can hold its breath 7 times as long as a penguin. Which animal can hold its breath the longest?

Multiply to find how long a sea turtle can hold its breath.

Answer : The sea turtle can hold its breath up to 84,00 seconds

Explanation:

Step 1:  Multiply ones and regroup
0

Step 2: Multiply tens and regroup tens
0

Step 3: Multiply hundreds and regroup hundreds
7 × 2 hundreds  = 14 hundreds
regroup as 1 thousand and 4 hundreds

Step 4:  Multiply thousands and regroup
7 × 1 thousands = 7 thousands
7 thousands + 1 thousand  =  8 thousands

By holding all places together we get 8,400 .

Compare the lengths of time that each animal can hold its breath.
Answer:     Penguin can hold its breath by 1,200 seconds.
Elephant seal can hold its breath by 7,200 seconds.
Sea turtle can hold it breath by 8,400 seconds.

The _______ can hold its breath the longest.
Answer: Sea turtle can hold its breath the longest

Show and Grow

Question 17.
You have 796 baseball cards and 284 hockey cards. You have 3 times as many football cards as hockey cards. Which type of card do you have the greatest number of?

Answer: 852 football cards are the greatest number of cards.

Explanation:

Step 1:  Multiply ones and regroup
3 × 4 ones = 12 ones
Regroup as 1 ten and 2 ones

Step 2: Multiply tens and regroup tens
3 × 8 tens = 24 tens
24 tens + 1 ten = 25 tens
regroup 25 tens  as 2 hundreds and 5 ten

Step 3: Multiply hundreds and regroup hundreds
3 × 2 hundreds  = 6 hundreds
6 hundreds + 2 hundreds = 8 hundreds

By holding all places together we get 852.

So,852 football cards are the greatest number of cards.

Question 18.
A principal has $2,000 to spend on updating some of your school’s tablets. She buys 4 tablets that each cost $299. How much money does the principal have left?
Answer: $299 × 4 = $1,196 , the principal have left $804.

Explanation:
She buys 4 tablets that each cost $299

Step 1:  Multiply ones and regroup
4 × 9 ones = 36 ones
Regroup as 3 tens and 6 ones

Step 2: Multiply tens and regroup tens
4 × 9 tens = 36 tens
36 tens + 3 tens = 39 tens
regroup 39 tens  as 3 hundreds and 9 tens

Step 3: Multiply hundreds and regroup hundreds
4 × 2 hundreds  = 8 hundreds
8 hundreds + 3 hundreds = 11 hundreds
regroup as 1 thousand and 1 hundred.

By holding all places together we get 1,196
So, $299 × 4 = $1,196 ,
the principal have left $804.

Question 19.
A train ticket from New York City to Miami costs $152. A train ticket from New York City to Orlando costs $144. A group of 8 friends is in New York City. How much money can the group save by going to Orlando instead of going to Miami?
Answer: They can save $64.

Explanation:
A group of 8 friends is in New York City,
A train ticket from New York City to Miami costs $152.

Step 1:  Multiply ones and regroup
8 × 2 ones = 16 ones
Regroup as 1 ten and 6 ones

Step 2: Multiply tens and regroup tens
8 × 5 tens = 40 tens
40 tens + 1 ten = 41 tens
regroup 41 tens  as 4 hundreds and 1 ten

Step 3: Multiply hundreds and regroup hundreds
8 × 1 hundreds  = 8 hundreds
8 hundreds + 4 hundred = 12 hundreds
regroup as 1 thousand and 2 hundreds

By holding all places together we get 1,216
So, $152 × 8 = $1,216.

A train ticket from New York City to Orlando costs $144,

Step 1:  Multiply ones and regroup
8 × 4 ones = 32 ones
Regroup as 3 tens and 2 ones

Step 2: Multiply tens and regroup tens
8 × 4 tens = 32 tens
32 tens + 3 tens = 35 tens
regroup 35 tens  as 3 hundreds and 5 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 1 hundreds  = 8 hundreds
8 hundreds + 3 hundred = 11 hundreds
regroup as 1 thousand and 1 hundred

By holding all places together we get 1,152
So, $144 × 8 = $1,152.

Finally , $1,216  – $1,152 = $ 64
They can save $64.

Multiply Three-and Four-Digit Numbers by One-Digit Numbers Homework & Practice 3.8

Find the product. Check whether your answer is reasonable.
Question 1.
Estimate: ______

Answer:  2,736, The estimated product is  2,700

Explanation:

Step 1:  Multiply ones and regroup
9 × 4 ones = 36 ones
Regroup 36 ones  as 3 tens and 6 ones

Step 2: Multiply tens and regroup tens
9 × 0 tens = 0 tens
0 tens + 3 tens = 3 tens

Step 3: Multiply hundreds and regroup hundreds
9 × 3 hundreds  = 27 hundreds
Regroup 27 hundreds  as 2 thousands and 7 hundreds

By holding all places together we get 2,736 .

And the estimated product is 2,700
because 304 is close to 300, 300 × 9 = 2,700
So, 304 × 9 = 2,736 .

Question 2.
Estimate: _____

Answer:  4,936, The estimated product is 4,880

Explanation:

Step 1:  Multiply ones and regroup
8 × 7 ones = 56 ones
Regroup 56 ones  as 5 tens and 6 ones

Step 2: Multiply tens and regroup tens
8 × 1 ten = 8 tens
8 tens + 5 tens = 10 tens
regroup as 1 hundred and 0 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 6 hundreds  = 48 hundreds
48 hundreds + 1 hundred = 49 hundreds
Regroup 49 hundreds  as 4 thousands and 9 hundreds

By holding all places together we get 4,936 .

And the estimated product is 4,880
because 617 is close to 610, 610 × 8 = 4,880
So, 617 × 8 = 4,936 .

Question 3.
Estimate: _____

Answer: 23,756, The estimated product is  23,760

Explanation:

Step 1:  Multiply ones and regroup
4 × 9 ones = 36 ones
regroup 36 ones as 3 tens and 6 ones

Step 2: Multiply tens and regroup tens
4 × 3 tens = 12 tens
12 tens + 3 tens = 15 tens
regroup as 1 hundred and 5 tens

Step 3: Multiply hundreds and regroup hundreds
4 × 9 hundreds  = 36 hundreds
36 hundreds + 1 hundred = 37 hundreds
regroup as 3 thousands and 7 hundreds

Step 4: Multiply thousands and regroup thousands
4 × 5 thousands = 20 thousands
20 thousands + 3 thousands = 23 thousands

By holding all the places together we get 23,756.

And the estimated product is 23,760
because 5,939 is close to 5,940, 5,940 × 4 = 23,760
So, 5,939 × 4  = 23,756 .

Question 4.
Estimate: ______

Answer:  8,262, The estimated product is  8,250.

Explanation:

Step 1:  Multiply ones and regroup
3 × 4 ones = 12 ones
regroup 12 ones as 1 ten and 2 ones

Step 2: Multiply tens and regroup tens
3 × 5 tens = 15 tens
15 tens + 1 ten = 16 tens
regroup as 1 hundred and 6 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 7 hundreds  = 21 hundreds
21 hundreds + 1 hundred = 22 hundreds
regroup as 2 thousands and 2 hundreds

Step 4: Multiply thousands and regroup thousands
3 × 2 thousands = 6 thousands
6 thousands + 2 thousands = 8 thousands

By holding all the places together we get 8,262

And the estimated product is 8,250
because 2,754 is close to 2,750, 2,750 × 3 = 8,250
So, 2,754 × 3  = 8,262 .

Question 5.
Estimate: ______

Answer: 50,790, The estimated product is  50,760

Explanation:

Step 1:  Multiply ones and regroup
6 × 5 ones = 30 ones
regroup 30 ones as 3 tens and 0 ones

Step 2: Multiply tens and regroup tens
6 × 6 tens = 36 tens
36 tens + 3 tens = 39 tens
regroup as 3 hundreds and 9 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 4 hundreds  = 24 hundreds
24 hundreds + 3 hundreds = 27 hundreds
regroup as 2 thousands and 7 hundreds

Step 4: Multiply thousands and regroup thousands
6 × 8 thousands = 48 thousands
48 thousands + 2 thousands = 50 thousands

By holding all the places together we get 50,790.

And the estimated product is 50,760,
because 8,465 is close to 8,460, 8,460 × 6 = 50,760
So, 8,465 × 6  = 50,790.

Question 6.
Estimate: ______

Answer: 5,754 ,The estimated product is  5,740

Explanation:

Step 1:  Multiply ones and regroup
7 × 2 ones = 14 ones
Regroup 14 ones  as 1 tens and 4 ones

Step 2: Multiply tens and regroup tens
7 × 2 tens = 14 tens
14 tens + 1 tens = 15 tens
regroup as 1 hundred and 5 tens

Step 3: Multiply hundreds and regroup hundreds
7 × 8 hundreds  = 56 hundreds
56 hundreds + 1 hundred = 57 hundreds
Regroup 57 hundreds  as 5 thousands and 7 hundreds

By holding all places together we get 5,754 .

And the estimated product is 5,740
because 822 is close to 820, 820 × 7 = 5,740
So, 822 × 7 = 5,754 .

Find the product. Check whether your answer is reasonable.
Question 7.
Estimate: _______
629 × 5 = ______
Answer: 3,145 , The estimated product is 3,150

Explanation:

Step 1:  Multiply ones and regroup
5 × 9 ones = 45 ones
Regroup 45 ones  as 4 tens and 5 ones

Step 2: Multiply tens and regroup tens
5 × 2 tens = 10 tens
10 tens + 4 tens = 14 tens
regroup as 1 hundred and 4 tens

Step 3: Multiply hundreds and regroup hundreds
5 × 6 hundreds  = 30 hundreds
30 hundreds + 1 hundred = 31 hundreds
Regroup 31 hundreds  as 3 thousands and 1 hundred

By holding all places together we get 3,145.

And the estimated product is 3,150.
because 629 is close to 630, 630 × 5 = 3,150.
So, 629 × 5 = 3,145 .

Question 8.
Estimate: _______
7 × 1,836 = ______
Answer:  12,852 , The estimated product is  12,880

Explanation:

Step 1:  Multiply ones and regroup
7 × 6 ones = 42 ones
regroup 42 ones as 4 tens and 2 ones

Step 2: Multiply tens and regroup tens
7 × 3 tens = 21 tens
21 tens + 4 tens = 25 tens
regroup as 2 hundreds and 5 tens

Step 3: Multiply hundreds and regroup hundreds
7 × 8 hundreds  = 56 hundreds
56 hundreds + 2 hundreds = 58 hundreds
regroup as 5 thousands and 8 hundreds

Step 4: Multiply thousands and regroup thousands
7 × 1 thousands = 7 thousands
7 thousands + 5 thousands = 12 thousands

By holding all the places together we get 12,852

And the estimated product is 12,880
because 1,836 is close to 1,840, 1,840 × 7 = 12,880
So, 1,836 × 7  = 12,852 .

Question 9.
Estimate: ______
453 × 3 = ______
Answer: 1,359 , The estimated product is  1,350

Explanation:

Step 1:  Multiply ones and regroup
3 ×  ones = 9 ones

Step 2: Multiply tens and regroup tens
3 × 5 tens = 15 tens
regroup as 1 hundred and 5 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 4 hundreds  = 12 hundreds
12 hundreds + 1 hundred = 13 hundreds
Regroup 13 hundreds  as 1 thousand and 3 hundreds

By holding all places together we get 1,359 .

And the estimated product is 1,350
because 453 is close to 450, 450 × 3 = 1,350.
So, 453 × 3 = 1,359 .

Compare
Question 10.

Answer: 20,346 is greater than 6,782.

Explanation:

Step 1:  Multiply ones and regroup
3 × 2 ones = 6 ones

Step 2: Multiply tens and regroup tens
3 × 8 tens = 24 tens
regroup as 2 hundreds and 4 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 7 hundreds  = 21 hundreds
21 hundreds + 2 hundreds = 23 hundreds
regroup as 2 thousands and 3 hundreds

Step 4: Multiply thousands and regroup thousands
3 × 6 thousands = 18 thousands
18 thousands + 2 thousands = 20 thousands

By holding all the places together we get 20,346

Step 1:  Multiply ones and regroup
2 × 1 ones = 2 ones

Step 2: Multiply tens and regroup tens
2 × 9 tens = 18 tens
regroup as 1 hundred and 8 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 3 hundreds  = 6 hundreds
6 hundreds + 1 hundred = 7 hundreds

Step 4: Multiply thousands and regroup thousands
2 × 3 thousands = 6 thousands

By holding all the places together we get 6,782.
So, 20,346 is greater than 6,782.

Question 11.

Answer: 12,528 is greater than 9,972.

Explanation:

Step 1:  Multiply ones and regroup
9 × 2 ones = 18 ones
regroup 18 ones as 1 tens and 8 ones

Step 2: Multiply tens and regroup tens
9 × 9 tens = 81 tens
81 tens + 1 tens = 82 tens
regroup as 8 hundreds and 2 tens

Step 3: Multiply hundreds and regroup hundreds
9 × 3 hundreds  = 27 hundreds
27 hundreds + 8 hundreds = 35 hundreds
regroup as 3 thousands and 5 hundreds

Step 4: Multiply thousands and regroup thousands
9 × 1 thousands = 9 thousands
9 thousands + 3 thousands = 12 thousands

By holding all the places together we get 12,528.

Step 1:  Multiply ones and regroup
4 × 3 ones = 12 ones
regroup 12 ones as 1 tens and 2 ones

Step 2: Multiply tens and regroup tens
4 × 9 tens = 36 tens
36 tens + 1 tens = 37 tens
regroup as 3 hundreds and 7 tens

Step 3: Multiply hundreds and regroup hundreds
4 × 4 hundreds  = 16 hundreds
16 hundreds + 3 hundreds = 19 thousands
regroup as 1 thousand and 9 hundreds

Step 4: Multiply thousands and regroup thousands
4 × 2 thousands = 8 thousands
8 thousands + 1 thousand = 9 thousands

By holding all the places together we get 9,972.
So, 12,528 is greater than 9,972.

Question 12.
A backpacker hikes the Buckeye Trail 5 times. The trail is 1,444 miles long. How many miles has he hiked on the Buckeye Trail in all?
Answer: 7,220 miles has hiked on the Buckeye Trail.

Explanation:
The trail is 1,444 miles long,
A backpacker hikes the Buckeye Trail 5 times. Then 1,444 × 5 ,

Step 1:  Multiply ones and regroup
5 × 4 ones = 20 ones
regroup 20 ones as 2 tens and 0 ones

Step 2: Multiply tens and regroup tens
5 × 4 tens = 20 tens
20 tens + 2 tens = 22 tens
regroup as 2 hundreds and 2 tens

Step 3: Multiply hundreds and regroup hundreds
5 × 4 hundreds  = 20 hundreds
20 hundreds + 2 hundreds = 22 hundreds
regroup as 2 thousands and 2 hundreds

Step 4: Multiply thousands and regroup thousands
5 × 1 thousands = 5 thousands
5 thousands + 2 thousands  = 7 thousands

By holding all the places together we get 7,220.

Question 13.
Number Sense
What number is 980 more than the product of 6,029 and 8?
Answer: 49,212, The product of 6,029 and 8 is 48,232

Explanation:

Step 1:  Multiply ones and regroup
8 × 9 ones = 72 ones
regroup 72 ones as 7 tens and 2 ones

Step 2: Multiply tens and regroup tens
8 × 2 tens = 16 tens
16 tens + 7 tens = 23 tens
regroup as 2 hundreds and 3 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 0 hundreds  = 0 hundreds
0 hundreds + 2 hundreds = 2 hundreds

Step 4: Multiply thousands and regroup thousands
8 × 6 thousands = 48 thousands

By holding all the places together we get  48,232.

Given, The required number is 980 more than  48,232,
Then 48,232 + 980 = 49,212.
Finally the wanted number is 49,212.

Question 14.
YOU BE THE TEACHER
Newton says that the product of a three-digit number and a one-digit number is always a three-digit number. Is Newton correct? Explain.
Answer: NO

Explanation: The Assumption that the product of a three-digit number and a one-digit number is always a three-digit number is wrong because,
the the factor numbers of a product might vary in places .
The highest one digit number is multiplied by the highest 3 digit number will not be a 3 digit number as a output product .
So, Newton is wrong about his statement.

Question 15.
Modeling Real Life
The numbers of songs Newton and Descartes’s download are shown. You download 2 times as many songs as Descartes. Who downloads the most songs?

Answer: I downloaded 334 songs and Newton has the most songs that is 351 songs .

Explanation:

Step 1:  Multiply ones and regroup
2 × 7 ones = 14 ones
Regroup as 1 ten and 4 ones

Step 2: Multiply tens and regroup tens
2 × 6 tens = 12 tens
12 tens + 1 ten = 13 tens
regroup 11 tens  as 1 hundreds and 3 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 1 hundreds  = 2 hundreds
2 hundreds + 1 hundred = 3 hundreds

By holding all places together we get 334 .
The songs that i downloaded are 334
Newton has 351 songs and Descartes has 167 songs and compared to all of us ,
Newton has more songs that is 315 songs.

Review & Refresh

Find the sum or difference. Use the inverse operation to check.
Question 16.

Answer: 685.

Explanation:
By subtracting 847 – 162 , we get 685
To check inverse operation :
Add 685 + 162  = 847.

So 847 – 162 = 685 .
Question 17.

Answer:  901

Explanation:
By adding 612 + 289 , we get 901
To check inverse operation:
Subtract 901 – 289 ,we get  612

So, 612 + 289 = 901 .

Question 18.

Answer:  149

Explanation:

By subtracting 500 – 351 , we get 149
To check inverse operation :
Add 315 + 149  = 500 .

So 500 – 351 = 149 .

Lesson 3.9 Use Properties to Multiply

Explore and Grow

Use any strategy to find each product. Explain the strategy you used to find each product.

Answer:  700 , 790, 45,009 , 44,991.

Explanation:  7 × 25 × 4
By using commutative property of multiplication
= 7 × ( 25 × 4)
=7 × 100
= 700
So, 7 × 25 × 4 = 700 .

5 × 79 × 2
By using commutative property of multiplication
= 79 × ( 5 × 2)
=79 × 10
= 790
So, 5 × 79 × 2 = 790 .

9 × 5,001
By using Distributive property of multiplication
= 9 × (5,000 + 1)
= ( 9 × 5,000) + ( 9 × 1)
= 45,000 + 9
= 45,009
So,  9 × 5,001 = 45,009.

9 × 4,999
By using Distributive property of multiplication
= 9 × (5,000 – 1)
= ( 9 × 5,000) – ( 9 × 1)
= 45,000 – 9
= 44,991
So,  9 × 4,999 = 44,991.

Construct Arguments
Compare your strategies with your partner’s. How are they alike? How are they different?
Answer:  All the strategies are correct that are used by my friend and me.
Using any method in multiplication is frequently used to find the product as fast as possible
some methods are long and some of them are really fast and easy .
Every method is correct unless you did not do any mistakes, so , all are alike.

Think and Grow: Use Properties to Multiply

You can use properties to multiply.

Answer :  2,000 , 3,490 , 900

Explanation:
8 × 250
By using associative property of multiplication
=( 4 × 2 ) × 250
= 4 × ( 2 × 250)
= 4 × 500
= 2,000.
So, 8 × 250 = 2,000.

5 × 698
By using Distributive property of multiplication
= 5 × ( 700 – 2)
= (5 × 700) – ( 5 × 2)
= 3,500 – 10
= 3,490
So,  5 × 698 = 3,490.

4 × 9 × 25
By using commutative property of multiplication
= 9 × ( 4 × 25)
= 9 × 100
= 900.
So, 4 × 9 × 25 = 900.

Show and Grow

Use properties to find the product. Explain your reasoning.
Question 1.
6 × 150
Answer: 900

Explanation:
By using Distributive property of multiplication
= 6 × ( 100 + 50)
= (6 × 100) + ( 6 × 50)
= 600 + 300
= 900.
So, 6 × 150 = 900.

Question 2.
3 × 494
Answer: 1,482

Explanation:
By using Distributive property of multiplication
= 3 × ( 500 – 6)
= (3 × 500) – ( 3 × 6)
= 1500 – 18
= 1,482.
So, 3 × 494 = 1,482.

Question 3.
25 × 7 × 4
Answer:  700

Explanation:
By using commutative property of multiplication
= 7 × ( 25 × 4 )
= 7 × 100
= 700.
So, 25 × 7 × 4 = 700.

Apply and Grow: Practice

Use properties to find the product. Explain your reasoning.
Question 4.
7 × 798
Answer: 5,586

Explanation:
By using Distributive property of multiplication
= 7 × ( 800 – 2)
= (7 × 800) – ( 7 × 2)
= 5,600 – 14
= 5,586
So,  7 × 798 = 5,586.

Question 5.
350 × 6
Answer:  2,100

Explanation:
By using Distributive property of multiplication
= 6 × ( 300 + 50)
= (6 × 300) + ( 6 × 50)
= 1,800 + 300
= 2,100.
So, 6 × 350 = 2,100.

Question 6.
106 × 5
Answer: 530

Explanation:
By using Distributive property of multiplication
= 5 × ( 100 + 6)
= (5 × 100) + ( 5 × 6)
= 500 + 30
= 530.
So, 5 × 106 = 530.

Question 7.
4 × 625
Answer:  2,500

Explanation:
By using Distributive property of multiplication
= 4 × ( 600 + 25)
= (4 × 600) + ( 4 × 25)
= 2400 + 100
= 2,500.
So, 4 × 625 = 2,500.

Question 8.
395 × 8
Answer: 3,160.

Explanation:
By using Distributive property of multiplication
= 8 × ( 400 – 5)
= (8 × 400) – ( 8 × 5)
= 3,200 – 40
= 3,160.
So, 8 × 395 = 3,160.

Question 9.
2 × 7 × 15
Answer: 210

Explanation:
By using commutative property of multiplication
= 7 × ( 15 × 2 )
= 7 × 30
= 210.
So, 2 × 7 × 15 = 210.

Question 10.
430 × 2
Answer:  860

Explanation:
By using Distributive property of multiplication
= 2 × ( 400 + 30)
= (2 × 400) + ( 2 × 30)
= 800 + 60
= 860.
So, 2 × 430 = 860.

Question 11.
8 × 150
Answer: 1,200

Explanation:
By using Distributive property of multiplication
= 8 × ( 100 + 50)
= (8 × 100) + ( 8 × 50)
= 800 + 400
= 1,200.
So, 8 × 150 = 1,200.

Question 12.
3 × 1,997
Answer: 5,991

Explanation:
By using Distributive property of multiplication
= 3 × ( 2000 – 3)
= (3 × 2000) – ( 3 × 3)
= 6000 – 9
= 5,991.
So, 3 × 1,997 = 5,991.

Question 13.
25 × 9 × 2
Answer: 450

Explanation:
By using commutative property of multiplication
= 9 × ( 25 × 2 )
= 9 × 50
= 450.
So, 25 × 9 × 2 = 450.

Question 14.
404 × 6
Answer:  2,424

Explanation:
By using Distributive property of multiplication
= 6 × ( 400 + 4)
= (6 × 400) + ( 6 × 4)
= 2400 + 24
= 2,424.
So, 6 × 404 = 2,424.

Question 15.
4 × 2,004
Answer:  8,016

Explanation:
By using Distributive property of multiplication
= 4 × ( 2000 + 4)
= (4 × 2000) + ( 4 × 4)
= 8000 + 16
= 8,016.
So, 4 × 2,004 = 8,016.

Question 16.
Which One Doesn’t Belong?
Which expression does not belong with the other three?
(3 × 30) + (3 × 7), (3 × 40) – (3 × 3)
3 × (30 + 7), 3 × 3 × 7
Answer: 3 × 3 × 7

Explanation:
(3 × 30) + (3 × 7)
(3 × 40) – (3 × 3)
3 × (30 + 7) ,
These 3 expressions are related to the distributive property of multiplication.
So, 3 × 3 × 7 this expression does not belong to the distributive property of multiplication.

Question 17.
Number Sense
Use properties to find each product
9 × 80 = 720, so 18 × 40 = _____.
5 × 70 = 350, so 5 × 72 = _____.
Answer: 720, 360.

Explanation:
By using associative property of multiplication
=( 6 × 3 ) × 40
= 6 × ( 3 × 40)
= 6 × 120
= 720.
So, 18 × 40 = 720.

5 × 72
By using Distributive property of multiplication
= 5 × ( 70 + 2)
= (5 × 70) + ( 5 × 2)
= 350 + 10
= 360.
So, 5 × 72 = 360.

Think and Grow: Modeling Real Life

Example
The fastest recorded speed of a dragster car in the United States was 31 miles per hour less than 3 times the top speed of the roller coaster. What was the speed of the car?

Multiply to find 3 times the top speed of the roller coaster.

Answer: 360 miles per hour.

Explanation:
3 × 120
By using Distributive property of multiplication
= 3 × ( 100 + 20)
= (3 × 100) + ( 3 × 20)
= 300 + 60
= 360.
So, 3 × 120 = 360.

Subtract to find 31 miles per hour less
_____ – 31 = _______ miles per hour
Answer:  329 miles per hour

Explanation:
360 – 31 = 329 .

So, the speed of the car was ______ miles per hour.
Answer: the speed of the car was 329 miles per hour.

Show and Grow

Question 18.
In 2016, a theme park used 300 drones for a holiday show. In 2017, China used 200 fewer than 4 times as many drones for a lantern festival. How many drones did China use?

Answer: China used  1,000 drones.

Explanation:
A theme park used 300 drones for a holiday show
China used 4 times as many drones for a lantern festival.
So, 4 × 300 = 1,200
Given, China used 200 fewer than 4 times as many drones for a lantern festival.
So 1,200 – 200 = 1,000.

China used 1,000 drones.

Question 19.
A subway train has 8 cars. Each car can hold 198 passengers. How many passengers can two subway trains hold?

Answer: 3,168 passengers.

Explanation:
A subway train has 8 cars
Each car can hold 198 passengers.
So, 8 × 198
By using Distributive property of multiplication
= 8 × ( 200 – 2)
= (8 × 200) – ( 8 × 2)
= 1600 – 16
= 1,584.
So, 8 × 198 = 1,584. 1,584 passengers for a subway train
For 2 subway trains , 1,584 + 1,584 = 3,168.
Two subway trains can hold 3,168 passengers.

Question 20.
You plant cucumbers, green beans, squash, and corn in a community garden. You plant 3 rows of each vegetable with 24 seeds in each row. How many seeds do you plant?
Answer: 288 seeds

Explanation:
plant 3 rows of each vegetable with 24 seeds in each row.
So, 3 × 24
By using Distributive property of multiplication
= 3 × ( 20 + 4)
= (3 × 20) + ( 3 × 4)
= 60 + 12
= 72.
So, 3 × 24 = 72.

You plant cucumbers, green beans, squash, and corn in a community garden.
Total 4 various vegetables are planted
So, 4 × 72
By using Distributive property of multiplication
= 4 × ( 70 + 2)
= (4 × 70) + ( 4 × 2)
= 280 + 8
= 288.
So, 4 × 72 = 288.

We have to plant 288 seeds.

Use Properties to Multiply Homework & Practice 3.9

Use properties to find the product. Explain your reasoning.
Question 1.
3 × 497
Answer: 1,491

Explanation:
By using Distributive property of multiplication
= 3 × ( 500 – 3)
= (3 × 500) – ( 3 × 3)
= 1500 – 9
= 1,491.
So, 3 × 497 = 1,491.

Question 2.
36 × 9
Answer: 324

Explanation:
By using Distributive property of multiplication
= 9 × ( 30 + 6)
= (9 × 30) + ( 9 × 6)
= 270 + 54
= 324.
So, 9 × 36 = 324.

Question 3.
8 × 350
Answer: 2,800

Explanation:
By using Distributive property of multiplication
= 8 × ( 300 + 50)
= (8 × 300) + ( 8 × 50)
= 2400 + 400
= 2,800.
So, 8 × 350 = 2,800.

Question 4.
25 × 8 × 4
Answer: 800

Explanation:
By using commutative property of multiplication
= 8 × ( 25 × 4 )
= 8 × 100
= 800.
So, 25 × 8 × 4 = 800.

Question 5.
999 × 5
Answer:  4,995

Explanation:
By using Distributive property of multiplication
= 5 × ( 1000 – 1)
= (5 × 1000) – ( 5 × 1)
= 5,000 –  5
= 4,995
So,  5 × 999 = 4,995.

Question 6.
9 × 402
Answer: 3,618

Explanation:
By using Distributive property of multiplication
= 9 × ( 400 + 2)
= (9 × 400) + ( 9 × 2)
= 3600 + 18
= 3,618.
So, 9 × 402 = 3,618.

Question 7.
509 × 4
Answer: 2,036

Explanation:
By using Distributive property of multiplication
= 4 × ( 500 + 9)
= (4 × 500) + ( 4 × 9)
= 2000 + 36
= 2,036 .
So, 4 × 509 = 2,036.

Question 8.
2 × 9 × 15
Answer: 270

Explanation:
By using commutative property of multiplication
= 9 × ( 15 × 2 )
= 9 × 30
= 270.
So, 2 × 9 × 15 = 270.

Question 9.
3,998 × 7
Answer: 27,986

Explanation:
By using Distributive property of multiplication
= 7 × ( 4000 – 2)
= (7 × 4000) – ( 7 × 2)
= 28,000 – 14
= 27,986.
So, 3,998 × 7 = 27,986.

Question 10.
Is Descartes correct? Explain.

Answer: Descartes is not correct, 895 × 4 = 3,580.

Explanation:
By using Distributive property of multiplication
= 4 × ( 900 – 5)
= (4 × 900) – ( 4 × 5)
= 3600 – 20
= 3,620.
So, 895 × 4 = 3,580.

Question 11.
DIG DEEPER!
Complete the square so that the product of each row and each column is 2,400.

Answer:

Explanation: 
By using column 1 , we get 4 × 200 = 800,
To obtain the 2,400 the missing number should be 3 , Then 800 × 3  = 2,400.
By using column 3 , we get 2 × 2 = 4,
To obtain the 2,400 the missing number should be 600 , Then 600 × 4 = 2,400.
By using row 2 , we get 200 × 2 = 400,
To obtain the 2,400 the missing number should be 6 , Then 400 × 6 = 2,400.
By using row 1 , we get 4 × 600 = 2,400,
To obtain the 2,400 the missing number should be 1 , Then 2,400 × 1 = 2,400.
By using row 3 , we get 3 × 2 = 6,
To obtain the 2,400 the missing number should be 400 , Then 6 × 400 = 2,400.

The complete Square is .

Question 12.
Modeling Real Life
The height of the Great Pyramid of Giza is 275 feet shorter than 2 times the height of the Luxor Hotel in Las Vegas. How tall is the Great Pyramid of Giza?

Answer: The height of the Great Pyramid of Giza is 455 feet

Explanation:
The height of the Great Pyramid of Giza is 2 times the height of the Luxor Hotel in Las Vegas
The height of the Luxor Hotel in Las Vegas is 365 feet,
So, 2 × 365
By using Distributive property of multiplication
= 2 × ( 300 + 60 + 5 )
= (2 × 300) + ( 2 × 60) + ( 2 × 5 )
= 600 + 120 + 10
= 730.
So, 2 × 365 = 730.

Given, The height of the Great Pyramid of Giza is 275 feet shorter than 2 times the height of the Luxor Hotel in Las Vegas. So, 730 -275 = 455

The height of the Great Pyramid of Giza is 455 feet.

Question 13.
Modeling Real Life
Some firefighters are testing their equipment. Water from their firetruck hose hits the wall 12 feet higher than 7 times where the spray from a fire extinguisher hits the wall. How many feet high does the wall water from the hose hit the wall?

Answer: 180 feet.

Explanation:
Water from their firetruck hose hits the wall 7 times where the spray from a fire extinguisher hits the wall.
The spray from a fire extinguisher hits the wall at 24 feet,
So, 7 × 24
By using Distributive property of multiplication
= 7 × ( 20 + 4)
= (7 × 20) + ( 7 × 4)
= 140 + 28
= 168.
So, 7 × 24 = 168.
Water from their firetruck hose hits the wall 12 feet higher than 7 times where the spray from a fire extinguisher hits the wall. Then 168 + 12 = 180 feet

The wall water from the hose hit the wall is 180 feet.

Review & Refresh

Question 14.
You want to learn 95 new vocabulary words. You learn 5 words the first week and an equal number of words each week for the next 9 weeks. How many words do you learn in each of the 9 weeks?
Answer: 45 words.

Explanation:
You learn 5 words the first week
An equal number of words each week for the next 9 weeks
So, 5 × 9 = 45

We learn 45 words for 9 weeks.

Lesson 3.10 Problem Solving: Multiplication

Explore and Grow

Make a plan to solve the problem.
A group of sea otters that swim together is called a raft. There are 37 otters in a raft. Each otter eats about 16 pounds of food each day. About how many pounds of food does the raft eat in 1 week?

Answer: A raft eats 4,144 pounds in 1 week.

Explanation:
Each otter eats about 16 pounds of food each day.
There are 37 otters in a raft,
The food eaten by a raft in a day is 37 × 16 = 592 pounds.

The food eaten by a raft in 7 days ( 1 week ) is 592 × 7
By using Distributive property of multiplication
= 7 × ( 500 + 90 + 2 )
= (7 × 500) + ( 7 × 90) + ( 7 × 2 )
= 3500 + 630 + 14
= 4,144.
So, 592 × 7 = 4,144.

A raft eats 4,144 pounds in 1 week.

Critique Reasoning
Compare your plan to your partner’s. How are your plans alike? How are they different?
Answer: All the strategies are correct that are used by my friend and me.
Using any method in multiplication is frequently used to find the product as fast as possible
some methods are long and some of them are really fast and easy .
Every method is correct unless you did not do any mistakes, so , all are alike. in obtaining the correct answer.

Think and Grow: Problem Solving: Multiplication

Example
A coach buys 6 cases of sports drinks and spends $60. Each case has 28 bottles. A team drinks 85 bottles at a tournament. How many bottles are left?
Understand the Problem
What do you know?
• The coach buys 6 cases.
• The coach spends $60.
• Each case has 28 bottles.
• The team drinks 85 bottles.

What do you need to find?
• You need to find how many bottles are left.

Make a Plan

How will you solve?
• Multiply 28 by6 to find the total number of bottles in 6 cases.
• Then subtract 85 bottles from the product to find how many bottles are left.
• The amount of money the coach spends is unnecessary information.

Solve

Answer: 28 × 6 = 168 , k = 168
The total bottles in 6 cases are 168
n = k – 85
= 168 – 85
= 83 .
The total bottles left are 83 .
There are ______ bottles left.
Answer: There are 83 bottles left.

Show and Grow

Question 1.
Explain how you can check whether your answer above is reasonable.
Answer: Calculate the given information to get the result.

Explanation:
The coach buys 6 cases.
Each case has 28 bottles.
The team drinks 85 bottles
We need to find how many bottles are left.

So, by calculating these we obtain the result.

Apply and Grow: Practice

Understand the problem. What do you know? What do you need to find? Explain.
Question 2.
A zookeeper has 4 boxes. There are137 grams of leaves in each box. A koala eats 483 grams of leaves in 1 day. The zookeeper wants to know how many grams of leaves are left.

Answer: 65 grams of leaves are left.

By understanding the problem
We know that,
A zookeeper has 4 boxes.
There are137 grams of leaves in each box.
A koala eats 483 grams of leaves in 1 day

We need to find,
how many grams of leaves are left.

Explanation: A zookeeper has 4 boxes.
There are137 grams of leaves in each box.
So, 137 × 4
By using Distributive property of multiplication
= 4 × ( 100 + 30 + 7 )
= (4 × 100) + ( 4 × 30) + ( 4 × 7 )
= 400 + 120 + 28
= 548.
So, 137 × 4 = 548.

A koala eats 483 grams of leaves in 1 day
The zookeeper wants to know how many grams of leaves are left.
Then 548 – 483 = 65
There are 65 leaves left.

Question 3.
A beekeeper has 2 hives. Hive A produces 14 pounds of honey. Hive B produces 4 times as much honey as Hive A. The beekeeper wants to know how many pounds of honey are produced in all.

Answer: 70 pounds

By understanding the problem
We know that,
A beekeeper has 2 hives.
Hive A produces 14 pounds of honey.
Hive B produces 4 times as much honey as Hive A.

We should know ,
The beekeeper wants to know how many pounds of honey are produced in all.

Explanation: Hive A produces 14 pounds of honey.
Hive B produces 4 times as much honey as Hive A.
So, 4 × 14
By using Distributive property of multiplication
= 4 × ( 10 + 4)
= (4 × 10) + ( 4 × 4)
= 40 + 16
= 56.
So, 4 × 14 = 56.
Hive B produces 56 pounds of honey

The honey produced by the 2 hives is 14  + 56 = 70  pounds.

Understand the problem. Then make a plan. How will you solve? Explain.
Question 4.
A runner completes 12 races each year. He improves his time by 10 seconds each year. Each race is 5 kilometers long. The runner wants to know how many kilometers he runs in races in 3 years.
Answer: 180 kilometers.

Explanation:
By understanding the problem
we know that,
A runner completes 12 races each year.
Each race is 5 kilometers long.

We should know that,
how many kilometers he runs in races in 3 years.

Then, he completes 12 races each year. The races in 3 years. are
12  × 3 = 36 , 36 races in 3 years .

Each race is 5 kilometers long. Then for 36 races , we get
36 × 5 = 180 km.

So, The runner completes 180 km in 3 years.

Question 5.
A volunteer bikes 4 miles in all to travel from her home to a shelter and back. At the shelter, she walks a dog 1 mile. The volunteer wants to know how many miles she travels doing these tasks for 28 days.
Answer: 140 miles

Explanation:
By understanding the problem
we know that,
A volunteer bikes 4 miles in all to travel from her home to a shelter and back.
At the shelter, she walks a dog 1 mile.

We should know that,
how many miles she travels doing these tasks for 28 days.

A volunteer bikes 4 miles in all to travel from her home to a shelter and back. For 28 days, we get
4 × 28 = 112, she bikes for 112 miles.
At the shelter, she walks a dog 1 mile. for 28 days, we get
1 × 28 = 28, she walks dog for 28 miles .

she travels doing these tasks for 28 days by covering  a total of 140 miles.

Question 6.
Cats have 32 muscles in each ear. Humans have 12 ear muscles in all. How many more muscles do cats have in both ears than humans have in both ears?

Answer: 52 more muscles .

Explanation:
By understanding the problem
we know that,
Cats have 32 muscles in each ear.
Humans have 12 ear muscles in all.

We should know that,
How many more muscles do cats have in both ears than humans have in both ears

Cats have 32 muscles in each ear, For 2 ears we get 32 +32 = 64 muscles
Humans have 12 ear muscles in all. Then 64 – 12 = 52 .

Cats have 52 more muscles than humans have in both the ears.

Question 7.
A school has 5 hallways. Each hallway has 124 lockers. 310 lockers are red. 586 lockers are in use. How many lockers not are in use?
Answer: 34 lockers are not in use.

Explanation:
By understanding the problem
we know that,
A school has 5 hallways.
Each hallway has 124 lockers
586 lockers are in use.

We should know that,
How many lockers not are in use.

Each hallway has 124 lockers , then for 5 hallways , we get 124 × 5 = 620.
620 lockers in the whole school
586 lockers are in use, to get the number of lockers that are not in use are 620 – 586 = 34.

34 lockers are not in use.

Think and Grow: Modeling Real Life

Example
A group of scientists has $7,500 to spend on microscopes and balances. They buy 3 microscopes that each cost $1,642 and 6 balances that each cost $236. How much money do the scientists have left to spend?

Think: What do you know? What do you need to find? How will you solve?

Answer : They have $1,160 left.

Explanation:
Step 1 : Scientists buy 3 microscopes and each costs $1,642 , then we have $1,642 × 3  = m,
we get m = $4,926,

Step 2 :
scientists buy 6 balances that each cost $236, then we have $236 × 6 = r, we get  r = $1,416.

Step 3 :
The total money spent is m + r = k ,we have , $4,926 + $1,416 = $ 6,340, we get k = $ 6,340.

Step 4 :
The money left J = $7,500 – $6,340, we get J = $ 1,160.

Totally scientists have $1,160 left.

Show and Grow

Question 8.
City A has 3,296 fourth graders. City B has 3 times as many fourth graders as City A. City C has 5 times as many fourth graders as City B. How many fourth graders are in all three cities?
Answer: Totally There are 62,624 fourth graders in all the three cities.

Explanation: City A has 3,296 fourth graders,

Step 1 :
City B has 3 times as many fourth graders as City A. we get , 3,296 × 3 = 9,888.

Step 2 :
City C has 5 times as many fourth graders as City B. we get , 9,888 × 5 = 49,440.

Step 3 :
Fourth graders in all the three cities are 3,296 + 9,888 + 49,440 = 62,624.

Fourth graders in all the three cities are 62,624.

Problem Solving: Multiplication Homework & Practice 3.10

Understand the problem. Then make a plan. How will you solve? Explain.
Question 1.
Raku is a Japanese-inspired art form. fires An artist, or bakes, a raku pot at1,409 degrees Fahrenheit. The artist fires a porcelain pot at 266 degrees Fahrenheit less than 2 times the temperature at which the raku pot is fired. You want to find the temperature at which the porcelain pot is fired.
Answer: The porcelain pot is fired at the temperature of 2,552 degrees Fahrenheit.

Explanation:
By understanding the problem
we know that,
Raku is a Japanese-inspired art form
bakes, a raku pot at1,409 degrees Fahrenheit.
The artist fires a porcelain pot at 266 degrees Fahrenheit less than 2 times the temperature at which, the raku pot is fired.

We should know that,
find the temperature at which the porcelain pot is fired.

Let us say porcelain pot as p and raku pot as r ,
Now, The artist fires a porcelain pot at 266 degrees Fahrenheit less than 2 times the temperature at which, the raku pot is fired. we have  p =  2r  – 266 degrees Fahrenheit
2r = 2 × 1,409
= 2,818.
Then  p = 2,818 – 266 = 2,552  degrees Fahrenheit

The temperature at which the porcelain pot is fired is  2,552  degrees Fahrenheit.

Question 2.
You practice basketball 6 times each week. Each basketball practice is 55 minutes long. You practice dance for 225 minutes altogether each week. You want to find how many minutes you practice basketball and dance in all each week.
Answer: you practice basketball and dance  in all  for 555 minutes each week.

Explanation:
By understanding the problem
we know that,
practice basketball 6 times each week.
Each basketball practice is 55 minutes long.
practice dance for 225 minutes altogether each week.

We should know that,
find how many minutes you practice basketball and dance in all each week.

So, practice basketball 6 times each week.
Each basketball practice is 55 minutes long. we have 6 × 55 = 330,
Basketball practice for a week is for 330 min.
Dance practice for a week is for 225 minutes. Then 330 + 225 = 555.

The basketball and dance practice for a whole week is for 555 minutes.

Question 3.
You buy 7 books of stamps. There are 35 stamps in each book. You give some away and have124 stamps left. How many stamps did you give away?
Answer: 121 stamps have left.

Explanation:
By understanding the problem
we know that,
You buy 7 books of stamps
There are 35 stamps in each book.
You give some away and have124 stamps left.

We should know that,
How many stamps did you give away?

So, You buy 7 books of stamps
There are 35 stamps in each book. we have 35 × 7 = 245 stamps. 245 stamps are gave away.

You give some away and have124 stamps left. we get , 245 – 124 = 121.

So, Totally 121 stamps have left.

Question 4.
Your neighbor fills his car’s gasoline tank with 9 gallons of gasoline. Each gallon of gasoline allows him to drive 23 miles. Can he drive for 210 miles without filling his gasoline tank? Explain.
Answer: No , He can only drive for 207 miles without filling his gasoline.

Explanation:
By understanding the problem
we know that,
Neighbor fills his car’s gasoline tank with 9 gallons of gasoline.
Each gallon of gasoline allows him to drive 23 miles.

We should know that,
Can he drive for 210 miles without filling his gasoline tank?

So, Neighbor fills his car’s gasoline tank with 9 gallons of gasoline.
Each gallon of gasoline allows him to drive 23 miles. we have 9 × 23 = 207.
He can travel for 207 miles for 9 gallons of gasoline.

So, he cannot drive for 210 miles without filling any extra gasoline.

Question 5.
Writing
Write and solve a two-step word problem that can be solved using multiplication as one step.
Answer:  David has 20 oranges.

Explanation:
Problem- Newton has 10 apples and David  has oranges 2 times as many as Newton’s apples. How many oranges does David have?
Solution:  Newton has 10 apples , David  has oranges 2 times as many as Newton’s apples, then we have
2 × 10 = 20 . So, David has 20 oranges.

Question 6.
Modeling Real Life
There are 1,203 pictures taken for a yearbook. There are 124 student pictures for each of the 6 grades. There are also 7 pictures for each of the 23 school clubs. The rest of the pictures are teacher or candid pictures. How many teacher or candid pictures are there?

Answer: There are 298 pictures of teacher or candid pictures.

Explanation:
By understanding the problem
we know that,
There are 1,203 pictures taken for a yearbook.
There are 124 student pictures for each of the 6 grades.
There are also 7 pictures for each of the 23 school clubs.

We should know that,
How many teacher or candid pictures are there?

So, There are 124 student pictures for each of the 6 grades. , we have 6 × 124 = 744 . There are 744 pictures from all 6 grades students.
There are also 7 pictures for each of the 23 school clubs. we have, 7 × 23 = 161 .There are 161 pictures from 23 school club students.
Total number of students we calculated are 744 + 161 = 905.
Totally a  number of 1,203 pictures taken for a yearbook. Then 1,203 – 905 = 298.

Finally, There are 298 pictures of teacher or candid pictures.

Question 7.
Modeling Real Life
A construction worker earns $19 each hour she works. A supervisor earns $35 each hour she works. How much money do the construction worker and supervisor earn in all after 8 hours of work?
Answer: $432 are earned by the construction worker and supervisor earn in all after 8 hours of work.

Explanation:
By understanding the problem
we know that,
A construction worker earns $19 each hour she works.
A supervisor earns $35 each hour she works.

We should know that,
How much money do the construction worker and supervisor earn in all after 8 hours of work.

So, A construction worker earns $19 each hour she works. Then for 8 hours we get , $19 × 8 = $152.
Also A supervisor earns $35 each hour she works. Then for 8 hours we get , $35 × 8 = $280.
Total pay of both worker and supervisor is $152 + $280 = $432.

So,$432 are earned by the construction worker and supervisor earn in all after 8 hours of work.

Question 8.
Modeling Real Life
A family of 8 has $1,934 to spend on a vacation that is 1,305 miles away. They buy a $217 plane ticket and an $8 shirt for each person. How much money does the family have left?
Answer: The Family  have $134 left.

Explanation:
By understanding the problem
we know that,
A family of 8 has $1,934 to spend.
They buy a $217 plane ticket and an $8 shirt for each person.

We should know that,
How much money does the family have left.

So, They buy a $217 plane ticket for each person, Then for 8 members , we have , $217 × 8 = $1,736.
They buy an $8 shirt for each person ,Then for 8 members , we have , $8 × 8 = $64.
Total cost for plane tickets and shirts are $1,736 + $64 = $1,800.
The money they left is $1,934 – $1,800 = $134.

So, The Family  have $134 left.

Review & Refresh

What fraction of the whole is shaded?
Question 9.

Answer: 5/8  is shaded.

Question 10.

Answer: 2/4 is shaded.

Multiply by One-Digit Numbers Performance Task

Sounds are vibrations that travel as waves through solids, liquids, and gases. Sound waves travel 1,125 feet per second through air.

Question 1.
You see a flash of lightning 5 seconds before you hear the thunder. How far away is the storm?
Answer:  5625 feet

Explanation: As sound waves travel 1,125 feet per second through air. A flash of lightning 5 seconds before you hear the thunder, at a distance of 5625 feet.

Question 2.
Sound waves travel 22,572 feet per second faster through iron than through diamond. The speed of sound through the diamond is 39,370 feet per second.
a. Estimate the speed of sound through iron in feet per second.
b. What is the actual speed of sound through iron in feet per second?
c. Is your estimate close to the exact speed of sound through iron? Explain.
Answer:
a. 61,9412 feet per second
b. 61,9412 feet per second
c. yes

Explanation:
Sound waves travel 22,572 feet per second faster through iron than through diamond. The speed of sound through the diamond is 39,370 feet per second.
speed of iron is 22,572 + 39,370  = 61,9412.
So, we have the same answer for both of the questions because  estimated speed is the actual speed of sound through iron .

Question 3.
Sound waves travel about 4 times faster through water than through air.

a. What is the speed of sound through the water in feet per second?
b. A horn blows underwater. A diver is about 9,000 feet away from the horn. About how many seconds does it take the diver to hear the sound of the horn?
Answer: a. 4,500 feet per second and  b. 8 seconds

Explanation:
a. speed of sound is 1,125 feet per second And sound travels 4 times faster in water than in air
So, 1,125 × 4 =  4,500.
b. We know that, time = distance / speed , Given distance is 9,000 feet and speed is 1,125 feet per second. so
time = 9,000 / 1,125 = 8 seconds.
So, Diver can hear the Horn sound from 8 seconds away.

Question 4.
Do sound waves travel the fastest through solids, liquids, or gases? Explain.
Answer: Yes

Explanation:
As the density of the materials high, sound waves travel faster. That is the reason for high speed in solids then liquids compare to gases.

Multiply by One-Digit Numbers Activity

Multiplication Quest
Directions:
1. Players take turns rolling a die. Players solve problems on their boards to race the knights to their castles.
2. On your turn, solve the next multiplication problem in the row of your roll.
3.The first player to get a knight to a castle wins

Answer:

Multiply by One-Digit Numbers Chapter Test

3.1 Understand Multiplicative Comparisons

Write two comparison sentences for the equation.
Question 1.
72 = 8 × 9
Answer:  72 is 8 times as many as 9. or
72 is 9 times as many as 8.

Explanation: we should multiply the numbers 8 and 9 ,
to get the multiplication result as 72.

Question 2.
60 = 10 × 6
Answer: 60 is 10 times as many as 6. or
60 is 6 times as many as 10.

Explanation: we should multiply the numbers 10 and 6 ,
to get the multiplication result as 60.

Write an equation for the comparison sentence.
Question 3.
28 is 4 times as many as 7.
Answer:  28 = 4 × 7

Explanation:  we should multiply the numbers 4 and 7,
to get the multiplication result as 28.

Question 4.
40 is 8 times as many as 5.
Answer:  40 = 8 × 5

Explanation: we should multiply the numbers 8 and 5,
to get the multiplication result as 40.

Question 5.
Newton saves $40. Descartes saves $25 more than Newton. How much money does Descartes save?
Answer: Descartes saves $65.

Explanation: Newton saves $40.
Descartes saves $25 more than Newton.,
So, $40 + $25 = $65.

Question 6.
Your friend is 10 years old. Your neighbor is 4 times as old as your friend. How old is your neighbor?
Answer: 40 years old.

Explanation: Your friend is 10 years old.
Your neighbor is 4 times as old as your friend.
So, 10 × 4 = 40.

3.2 Multiply Tens, Hundreds, and Thousands

Find the product
Question 7.
6 × 90 = _____
Answer:  540

Explanation: Using the Place-value method,
6 × 90 = 6 × 9 tens
= 54 tens
= 540
So, 6 × 90 = 540.

Question 8.
3,000 × 1 = _____
Answer: 3000

Explanation: Using the Place-value method,
1 × 3,000 = 1 × 3 thousands
= 3 thousands
= 3,000
So, 1 × 3,000 = 3,000.

Question 9.
4 × 200 = _____
Answer: 800

Explanation: Using the Place-value method,
4 × 200 = 4 × 2 hundreds
= 8 hundreds
= 800
So, 4 × 200 = 800.

Question 10.
4,000 × 4 = _____
Answer: 16000

Explanation: Using the Place-value method,
4 × 4,000 = 4 × 4 thousands
= 16 thousands
= 16,000
So, 4 × 4,000 = 16,000.

Question 11.
8 × 700 = _____
Answer: 5,600

Explanation: Using the Place-value method,
8 × 700 = 8 × 7 hundreds
= 56 hundreds
= 5600
So, 8 × 700 = 5,600.

Question 12.
2 × 60 = _____
Answer: 120

Explanation: Using the Place-value method,
2 × 60 = 2 × 6 tens
= 12 tens
= 120
So, 2 × 60 = 120.

Find the missing factor.
Question 13.
____ × 200 = 1,000
Answer: 5

Explanation :Let the missing number be X
So, X × 200 = 1,000
X = 1,000 / 200 = 5
Hence, the value of X is: 5.

Question 14.
5 × _____ = 450
Answer: 90

Explanation :Let the missing number be X
So, 5 × X  = 450
X = 450 / 5 = 90
Hence, the value of X is: 90.

Question 15.
_____ × 800 = 6,400
Answer: 8

Explanation :Let the missing number be X
So, X × 800 = 6,400
X = 6,400 / 800 = 8
Hence, the value of X is: 8.

3.3 Estimate Products by Rounding

Estimate the product
Question 16.
5 × 65
Answer: 350.

Explanation:
Round to the nearest tens.
65 is close to 70;
5 x 70 = 350

Question 17.
2 × 903
Answer: 1,800

Explanation:
Round to the nearest tens.
903 is close to 900;
2 x 900 = 1,800.

Question 18.
7 × 3,592
Answer:  25,130.

Explanation:
Round to the nearest tens.
3,592 is close to 3,590;
7 x 3,590 = 25,130.

Find two estimates that the product is between.
Question 19.
8 × 32
Answer: The product is in between 240 and 320.

Explanation:
Round to the nearest tens.
32 is close to 30;
8 × 30 = 240
and
Explanation:
Round to the nearest tens.
32 is close to 40;
8 × 40 = 320
The product is in between 240 and 320.

Question 20.
4 × 284
Answer:

Explanation:
Round to the nearest tens.
32 is close to 30;
8 × 30 = 240
and
Explanation:
Round to the nearest tens.
32 is close to 40;
8 × 40 = 320
The product is in between 240 and 320.

Question 21.
6 × 5,945
Answer:  The product is in between 35,640 and 35,700.

Explanation:
Round to the nearest tens.
5,945 is close to 5,940;
6 × 5,940 = 35,640
and
Explanation:
Round to the nearest tens.
5,945 is close to 5,950;
6 × 5,950 = 35,700
The product is in between 35,640 and 35,700.

Question 22.
A charity organizer raises $9,154 each month for 6 months. To determine whether the charity raises $50,000, can you use an estimate, or is an exact answer required? Explain.
Answer: estimated raise is $54,900, The charity can raise above $50,000.

Explanation:  organizer raises $9,154 each month for 6 months.
$9,154 is close to $9,150;
So, $9,150 × 6 = $54,900.
The charity can raise above $50,000.

3.4 Use Distributive Property to Multiply

Find the product.
Question 23.
3 × 14 = _____
Answer: 42

Explanation: by using distributive property
3 × 14 = 3× (10 + 4)
= (3 × 10) + (3 × 4)
= 30 + 12
= 42

Question 24.
18 × 9 = _____
Answer: 162

Explanation: by using distributive property
9 × 18 = 9× (10 + 8)
= (9 × 10) + (9 × 8)
= 90 + 72
= 162

Question 25.
36 × 5 = _____
Answer:  180

Explanation: by using distributive property
5 × 36 = 5 × (30 + 6)
= (5 × 30) + (5 × 6)
= 150 + 30
= 180

Question 26.
8 × 56 = _____
Answer: 448

Explanation: by using distributive property
8 × 56 = 8 × (50 + 6)
= (8 × 50) + (8 × 6)
= 400 + 48
= 448

Question 27.
6 × 67 = ______
Answer: 402

Explanation: by using distributive property
6 × 67 = 6 × (60 + 7)
= (6 × 60) + (6 × 7)
= 360 + 42
= 402

Question 28.
83 × 2 = _____
Answer:  166

Explanation: by using distributive property
2 × 83 = 2 × (80 + 3)
= (2 × 80) + (2 × 3)
= 160 + 6
= 166

Question 29.
Structure
Use the Distributive Property to write an equation shown by the model.

Answer: 4 × 23 =  92

Explanation: by using distributive property
4 × 23 = 4× (20 + 3)
= (4 × 20) + (4 × 3)
= 80 + 12
= 92

3.5 Use Expanded Form to Multiply

Find the product
Question 30.
487 × 3 = _____
Answer: 1,461

Explanation: 3 × 487, by expanding we get
= 3 × (400 + 80 + 7), by using distributive property, we get
= (3 × 400) + ( 3 × 80) + ( 3 × 7)
= 1,200 + 240 + 21
Area model = 1,200 + 240 + 21
so the result of the partial products is 1,461

Question 31.
5 × 7,402
Answer:  37,010

Explanation: 5 × 7,402 , by expanding we get
= 5 × (7,000 + 400 + 2), by using distributive property, we get
= (5 × 7,000) + ( 5 × 400) + ( 5 × 2)
= 3,500 + 2,000 + 10
Area model = 3,500 + 2,000 + 10
so the result of the partial products is 37,010.

Question 32.
8,395 × 7 = _____
Answer: 58,765

Explanation :  7 × 8,395 = 7 × (8,000 + 300 + 90 + 5 )
= (7 × 8,000 ) + (7 × 300 ) + ( 7 × 90) + (7 × 5)   by using Expanded form we get
= 56,000 + 2,100 + 630 + 35
Area model = 56,000 + 2,100 + 630 + 35
so the result of the partial products is 58,765

3.6 Use Partial Products of Multiply

Find the product
Question 33.

Answer: 2,394.

Explanation:
The sum of the partial products are 1800 + 540 + 54 = 2,394
So, 266 × 9 = 2,3942.

Question 34.

Answer:  680

Explanation:
The sum of the partial products are 640 + 40 = 680
So, 85 × 8 = 680.

Question 35.

Answer:  28,128

Explanation:
The sum of the partial products are 28,000 + 120 + 8 = 28,128
So, 7,032 × 4 = 28,128.

Question 36.
Number Sense
Which three numbers are the partial products that you add to find the product of 518 and 2?

Answer: 518 × 2 = 1,036 , So,1,000 , 20 , 16 are the partial products .

Explanation:
The sum of the partial products are 1,000 + 20 + 16 = 1,036.
So, 518 × 2 = 1,036.

So, 1,000 , 20 , 16 are the partial products .

3.7 Multiply Two-Digit Numbers by One-Digit Numbers

Find the product. Check whether your answer is reasonable.
Question 37.
Estimate: _____

Answer: 114 ,  the estimated product is 120.

Explanation: 

6 × 9 ones = 54 ones
regrouping 54 ones as 5 tens and 4 ones
6 × 1 tens = 6 tens
6 tens + 5 tens = 11 tens  or  1 hundreds and 1 tens ,Then we get 114

And the estimated product is 120
because 19 is close to 20,then 20 × 6 = 120
So, 19 × 6 = 114.

Question 38.
Estimate: _____

Answer: 511 , the estimated product is 490.

Explanation:

7 × 3 ones = 21 ones
regrouping 21 ones as 2 tens and 1 ones
7 × 7 tens = 49 tens
49 tens + 2 tens = 51 tens  or  5 hundreds and 1 tens ,Then we get 511

And the estimated product is 490
because 73 is close to 70,then 70 × 7 = 490
So, 73 × 7 = 511.

Question 39.
Estimate: _____

Answer:  290, the estimated product is 300.

Explanation:

5 × 8 ones = 40 ones
regrouping 40 ones as 4 tens and 0 ones
5 × 5 tens = 25 tens
25 tens + 4 tens = 29 tens  or  2 hundreds and 9 tens ,Then we get 290.

And the estimated product is 300
because 58 is close to 70,then 60 × 7 = 300
So, 58 × 5 = 290.

3.8 Multiply Three – and Four-Digit Numbers by One-Digit Numbers

Find the product. Check whether your answer is reasonable
Question 40.
Estimate: _____
402 × 3 = _____
Answer: 1,206, And the estimated product is 1,200

Explanation:

Step 1:  Multiply ones and regroup
3 × 2 ones = 6 ones

Step 2: Multiply tens and regroup tens
3 × 0 tens = 0 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 4 hundreds  = 12 hundreds
or 1 thousands and 2 hundreds

By holding all places together we get 1,206 .

And the estimated product is 1,200
because 402 is close to 400, 400 × 3 = 1,200
So, 402 × 3 =  1,206.

Question 41.
Estimate: _____
8 × 3,861 = _____
Answer: 30,888, And the estimated product is 30,880

Explanation:

Step 1:  Multiply ones and regroup
8 × 1 ones = 8 ones

Step 2: Multiply tens and regroup tens
8 × 6 tens = 48 tens
or 4 hundreds and 8 tens

Step 3: Multiply hundreds and regroup hundreds
8 × 8 hundreds  = 64 hundreds
64 hundreds + 4 hundreds = 68 hundreds
regroup as 6 thousands and 8 hundreds

Step 4: Multiply thousands and regroup thousands
8 × 3 thousands = 24 thousands
24 thousands + 6 thousands = 30 thousands

By holding all the places together we get 30,888.

And the estimated product is 30,880
because 3,861 is close to 3,860, 3,860 × 8 = 30,880
So, 3,861 × 8  = 30,888 .

Question 42.
Estimate: ______
977 × 2 = ______
Answer: 1,954 ,And the estimated product is 1,960

Explanation:

Step 1:  Multiply ones and regroup
2 × 7 ones = 14 ones
Regroup 14 ones  as 1 tens and 4 ones

Step 2: Multiply tens and regroup tens
2 × 7 tens = 14 tens
14 tens + 1 tens = 15 tens
regroup as 1 hundred and 5 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 9 hundreds  = 18 hundreds
18 hundreds  + 1 hundreds = 19 hundreds
Regroup  as 1 thousands and 9 hundreds

By holding all places together we get 1,954 .

And the estimated product is 1,960
because 977 is close to 980, 980 × 2 = 1,960
So, 977 × 2 = 1,954 .

Compare.
Question 43

Answer: 306 × 6 = 1,848 is less than 408 × 5 = 2,040.

Explanation:

Step 1:  Multiply ones and regroup
6 × 8 ones = 48 ones
Regroup 48 ones  as 4 tens and 8 ones

Step 2: Multiply tens and regroup tens
6 × 0 tens = 0 tens
0 tens + 4 tens = 4 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 3 hundreds  = 18 hundreds

By holding all places together we get 1,848 .

Step 1:  Multiply ones and regroup
5 × 8 ones = 40 ones
Regroup 40 ones  as 4 tens and 0 ones

Step 2: Multiply tens and regroup tens
5 × 0 tens = 0 tens
0 tens + 4 tens = 4 tens

Step 3: Multiply hundreds and regroup hundreds
5 × 4 hundreds  = 20 hundreds

By holding all places together we get 2,040 .

So, 306 × 6 = 1,848 is less than 408 × 5 = 2,040.

Question 44.

Answer:  6 × 789 = 4,734 is equal to 2 × 2,367 = 4,734.

Explanation:

Step 1:  Multiply ones and regroup
6 × 9 ones = 54 ones
regroup 54 ones as 5 tens and 4 ones

Step 2: Multiply tens and regroup tens
6 × 8 tens = 48 tens
48 tens + 5 tens = 53 tens or regroup as 5 hundreds and 3 tens

Step 3: Multiply hundreds and regroup hundreds
6 × 7 hundreds  = 42 hundreds
42 hundreds + 5 hundreds = 47 hundreds
regroup as 4 thousands and 7 hundreds

By holding all the places together we get 4,734.

Step 1:  Multiply ones and regroup
2 × 7 ones = 14 ones
regroup 14 ones as 1 tens and 4 ones

Step 2: Multiply tens and regroup tens
2 × 6 tens = 12 tens
12 tens + 1 tens = 13 tens
regroup as 1 hundred and 3 tens

Step 3: Multiply hundreds and regroup hundreds
2 × 3 hundreds  = 6 hundreds
6 hundreds + 1 hundred =  7 hundreds

Step 4: Multiply thousands and regroup thousands
2 × 2 thousands = 4 thousands

By holding all the places together we get 4,734.
So, 6 × 789 = 4,734 is equal to 2 × 2,367 = 4,734.

Question 45.

Answer: 454 × 3 = 1,362 is grater than 313 × 4 = 1,252.

Explanation:

Step 1:  Multiply ones and regroup
3 × 4 ones = 12 ones
regroup as 1 tens and 2 ones

Step 2: Multiply tens and regroup tens
3 × 5 tens = 15 tens
15 tens + 1 tens = 16 tens
regroup 16 tens  as 1 hundreds  and 6 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 4 hundreds  = 12 hundreds
12 hundreds + 1 hundreds = 13 hundreds

By holding all places together we get 1,362.

Step 1:  Multiply ones and regroup
4 × 3 ones = 12 ones
regroup as 1 tens and 2 ones

Step 2: Multiply tens and regroup tens
4 × 1 tens = 4 tens
4 tens + 1 tens = 5 tens

Step 3: Multiply hundreds and regroup hundreds
3 × 4 hundreds  = 12 hundreds

By holding all places together we get 1,252.

So, 454 × 3 = 1,362 is grater than 313 × 4 = 1,252.

3.9 Use Properties to Multiply

Use properties to find the product. Explain your reasoning.
Question 46.
25 × 9 × 4
Answer: 900

Explanation:
By using commutative property of multiplication
= 9 × ( 25 × 4)
=9 × 100
= 900
So,  25 × 9 × 4 = 900 .

Question 47.
8 × 250
Answer:  2,000.

Explanation:
By using Distributive property of multiplication
= 8 × ( 200 + 50)
= (8 × 200) + ( 8 × 50)
= 1,600 + 400
= 2,000.
So, 8 × 250 = 2,000.

Question 48.
3 × 497
Answer: 1,491

Explanation:
By using Distributive property of multiplication
= 3 × ( 400 + 90 + 7 )
= (3 × 400) + ( 3 × 90) + (3 × 7)
= 1,200 + 270 + 21
= 1,491.
So, 3 × 497 = 1,491.

Question 49.
2 × 8 × 15
Answer: 240

Explanation:
By using commutative property of multiplication
= 8 × ( 15 × 2)
= 8 × 30
= 240
So,  2 × 8 × 15 = 240 .

Question 50.
699 × 9
Answer: 6,291.

Explanation:
By using Distributive property of multiplication
= 9 × ( 600 + 90 + 9 )
= (9 × 600) + ( 9 × 90) + (9 × 9)
= 5,400 + 810 + 81
= 6,291 .
So, 9 × 699 = 6,291.

Question 51.
1,003 × 6
Answer: 6,018.

Explanation:
By using Distributive property of multiplication
= 6 × ( 1000 + 3)
= (6 × 1000) + (6 × 3)
= 6000 + 18
= 6,018
So, 1,003 × 6 = 6,018.

3.10 Problem Solving: Multiplication

Question 52.
A musical cast sells 1,761 tickets for a big show. The cast needs to complete 36 days of rehearsal. Each rehearsal is 8 hours long. The cast has rehearsed for 102 hours so far. How many hours does the cast have left to rehearse?

Answer: 186 hours left to rehearse .

Explanation:
By understanding the problem
we know that,
A musical cast sells 1,761 tickets for a big show.
The cast needs to complete 36 days of rehearsal.
Each rehearsal is 8 hours long.
The cast has rehearsed for 102 hours so far.

We should know that,
How many hours does the cast have left to rehearse

The total number of hours to practice are 36 × 8 = 288 hours
They practiced for 102 hours so far then , to get the remaining hours 288 – 102 = 186 hours.

So, the cast has 186 hours left to rehearse .

Multiply by One-Digit Numbers Cumulative Practice

Question 1.
Which number is greater than 884,592?
A. 89,621
B. 884,592
C. 805,592
D. 894,592
Answer: D. 894,592

Explanation: of all the given options , D is the greatest number .

Question 2.
What is the difference of 30,501 and 6,874?

Answer:  23,627.

Explanation : To fin the difference between two numbers, we should subtract the lowest number from the largest number, Then, 30,501 – 6,874 = 23,627.

Question 3.
A teenager will send about 37,000 text messages within 1 year. Which numbers could be the exact number of text messages sent?

Answer: 36.834

Explanation: Because this number is the closest number to the 37,000 from the given options.
so, 36,834 could be the exact number.

Question 4.
What is the product of 4,582 and 6?
A. 27,492
B. 24,082
C. 27,432
D. 42,117
Answer: 27,492.

Explanation: by using distributive property we get
6 × 4,582 = 6 × ( 4,000 + 500 + 80 + 2)
= (6 × 4,000) + ( 6 × 500)+ ( 6 × 80 ) + ( 6 × 2)
= 24,000 + 3,000 + 480 + 12
= 27,492.
So, 6 × 4,582 = 27,492.

Question 5.
Which expression is equal to 246,951 + 73,084?
A. 246,951 + 70,000 + 3,000 + 800 + 40
B. 246,951 + 7,000 + 300 + 80 + 4
C. 246,951 + 70,000 + 3,000 + 80 + 4
D. 246,951 + 70,000 + 3,000 + 800 + 4
Answer: C

Explanation:    246,951 + 73,084, 73,084 can be written as,
246,951 + 70,000 + 3,000 + 80 + 4.

Question 6.
Newton reads the number ”four hundred six thousand, twenty-nine” in a book. What is this number written in standard form?
A. 4,629
B. 406,029
C. 460,029
D. 406,290
Answer: B

Explanation: 406,029 can be written as four hundred six thousand, twenty-nine.

Question 7.
Which expressions have a product of 225?

Answer: 9 × 25 , 75 × 3 , 5 × 45.

Explanation: all the above expressions give the product of 225,
9 × 25  = 225
75 × 3  = 225
5 × 45  = 225.

Question 8.
What is the greatest possible number you can make with the number cards below?

Answer: B

Explanation: from the give numbers 6 ,1,8,4,2 , place the numbers in the descending order we get 86,421.

Question 9.
What is the sum of 62,671 and 48,396?
A. 111,067
B. 11,067
C. 100,967
D. 100,067
Answer:  A.1,11,067.

Explanation: Sum means Addition we should add the two numbers in order to get the result,
62,671  +  48,396 = 1,11,067.

Question 10.
Your friend drinks 8 glasses of water each day. He wants to know how many glasses of water he will drink in 1 year. Between which two estimates is the number of glasses he will drink in a year?

A. 2,400 and 3,200
B. 240 and 320
C. 3,200 and 4,000
D. 320 and 400
Answer: He drinks 2, 920 glasses for 1 year , so it is in between 2,400 and 3,200 .

Explanation: He drinks 8 glasses of water each day so for 1 year that is for 365 days , we get 365 × 8 = 2,920.
So, He drinks 2, 920 glasses for 1 year , so it is in between 2,400 and 3,200.

Question 11.
Which statement about the number 420,933 is true?
A. The value of the 2 is 2,000.
B. The 4 is in the ten thousands place.
C. The value of the 3 in the tens place is ten times the value of the 3 in the ones place.
D. There are 9 tens.
Answer: C

Explanation: The number 420,933  has the value of the 3 in the tens place is ten times the value of the 3 in the ones place.

Question 12.
Descartes rounds to the nearest ten thousand and gets an estimate of 360,000. Which expressions could be the problem he estimated?

Answer: 480,012 – 127,465.

Explanation: Because the number 360,000 should be in between the same range of the given numbers that is,
480,012 – 127,465.

Question 13.
There are 8 students in a book club. There are 5 times as many students in a drama club as the book club. How many students are in the drama club?
A. 13 students
B. 45 students
C. 3 students
D. 40 students
Answer: D . 40 students

Explanation: There are 8 students in a book club.
There are 5 times as many students in a drama club as the book club.
Then, 8 × 5 = 40.
So, There are 40 students in the drama club.

Question 14.
The force required to shatter concrete is 3 times the amount of force required to shatter plexiglass. The force required to shatter plexiglass is 72 pounds per square foot. What is the force required to shatter concrete?
A. 216 pounds per square foot
B. 69 pounds per square foot
C. 75 pounds per square foot
D. 2,106 pounds per square foot
Answer: A.216 pounds per square foot

Explanation: The force required to shatter plexiglass is 72 pounds per square foot.
The force required to shatter concrete is 3 times the amount of force required to shatter plexiglass.
Then, 72 × 3 = 216.

So, the force required to shatter concrete is 216 pounds per square foot.

Question 15.
Which number, when rounded to the nearest hundred thousand, is equal to 100,000?
A. 9,802
B. 83,016
C. 152,853
D. 46,921
Answer: B. 83,016

Explanation: it is  only number in the options which is close to 100,000 , All the other options are too far to roundoff .

Question 16.
You want to find 5,193 × 8. Which expressions show how to use the Distributive Property to find the product?

Answer:  8 × (5,000 + 100 + 90 +3)

Explanation: To use distributive property we have to organize the largest number in the expression in equal distributions respectively. so,  5,193 × 8 = 8 × (5,000 + 100 + 90 +3).

Question 17.
Use the table to answer the questions.

Part A Five friends each buy all of the items listed in the table. Use rounding to find about how much money they spend in all.
Part B An Ultimate Package costs $40 and includes all of the items listed in the table. About how much money would the 5 friends have saved in all if they would have bought an Ultimate Package instead of purchasing the items individually? Explain.
Answer:  part A = $250 , part B = $50 can be saved.

Explanation:
Part A: By rounding the money from the table , we have 90 – minute jump pass = $20
water bottle = $10 and T-shirt = $20 ,That total amount will be $50
For five friends we get, 5 × $50 = $250. for all of them.

Part B : An Ultimate Package costs $40 and includes all of the items listed in the table. if each of the 5 members buys the package then , $40  × 5 = $200.
So, from Part A and Part B we get to know that they would have save $50 if they bought the package , that is
$250 – $200 = $50.

Question 18.
A pair of walruses weigh 4,710 pounds together. The female weighs1,942 pounds. How much more does the male weigh than the female?
A. 826 pounds
B. 6,652 pounds
C. 2,768 pounds
D. 2,710 pounds
Answer: C. 2,768 pounds

Explanation: A pair of walruses weigh 4,710 pounds together.
The female weighs1,942 pounds. Then , to get the male walrus weight , we have ,
4,710 – 1,942 = 2,768

So, The weight of male walrus is 2,768 pounds.

Question 19.
You use compensation as shown. What is the final step?

Answer: D. There is no final step . You are finished.

Multiply by One-Digit Numbers STEAM Performance Task

Question 1.
A teacher visits the Golden Gate Bridge in San Francisco, California. 1. Use the figure above to answer the questions.
a. The roadway is 8 times as wide as the width of both walkways combined. Each walkway has the same width. How wide is one of the walkways
b. What is the length of the bridge that is suspended above the water?
c. What is the length of the bridge that is not suspended above the water?
Answer:  a. each walk way is 5 feet,
b. 6,450 feet ,
c. 2,531 feet.

Explanation:
a. Given , in picture the whole roadway and walk way is 90 feet long in total
Considering 2 walkways and assuming walkway as W and roadway as R  we get
R + 2W = 90 feet and The roadway is 8 times as wide as the width of both walkways combined. can say as
R = 8(2W) then we have 8(2W) + 2W  = 90
16W + 2W = 90
18W = 90
W = 90 /8
W = 5 , so, each walk way is 5 feet long.

b. the length of the bridge that is suspended above the water is
from the picture we have 1,125 + 4,200 + 1,125 = 6,450. feet

c. the length of the bridge that is not  suspended above the water is
From the picture we have 8,981 – 6,450 = 2,531 feet.

Question 2.
The teacher wants to estimate the distance between the bridge and the water.

a. He rides in a ferry boat under the bridge. He says that the distance between the bridge and the water is about 5 times the height of the ferry boat. What is the distance between the bridge and the water?
b. The teacher estimates that the height of one tower is about three times the distance between the bridge and the water. What is the teacher’s estimate for the height of the tower?
c. You learn the exact height of the tower is 86 feet taller than the teacher’s estimate. How tall is the tower?
Answer:

a.220 ft.

b.660 ft.

c.746 ft.

Explanation:

a. 44 ft is the height of the boat as per the picture.so the distance between the water and the  bridge is 5 times the height of the ferry boat ,answer is 44 ft multiplied by 5 times i.e.220 ft. so 44 × 5 = 220 ft.

b. As per the teachers estimation height of tower is 3 times the distance between the bridge and the water, so 220ft is multiplied by 3, i.e.  660 ft.  so  220 × 3 = 660

c. learn the exact height of the tower is 86 feet taller than the teacher’s estimate.so 660 + 86 = 746 feet.

The teacher uses his fitness tracker to count the number of steps he walks on the Golden Gate Bridge each day for 1 week.

Question 3.
The table shows the numbers of steps taken on the first 3 days of the week. Use this information to complete the table.
a. He takes one thousand one hundred twenty-two more steps on Wednesday than on Monday.
b. On Thursday, he takes one hundred eighteen steps less than on Sunday.
c. He takes ten thousand fifty steps on Friday.
d. On Saturday, he takes twice as many steps as on Tuesday.
Answer:

a. 11,834 steps on Wednesday
b.  12,260 steps on Thursday
c. 10,050 steps on Friday
d.  15,978 steps on Saturday

Explanation:
a. He walks 1,122 more steps on Wednesday than on Monday = 10,712 + 1,122 = 11,834 steps.
b. He walks 118 steps less on Thursday than on Sunday =  12,378 – 118 = 12,260 steps.
c. He walks 10,050 steps on Friday
d. He walks 2 times as many steps on Saturday than on Tuesday = 2 × 7,989 = 15,978 steps.

Question 4.
The teacher’s goal is to take 11,500 steps each day.

a. What is his goal for the week?
b. Estimate the total number of steps he takes from Sunday to Saturday. Does he meet his goal for the week? Explain.
c. One mile is about two thousand steps. The teacher estimates that he walks 3 miles from one end of the Golden Gate Bridge to the other and back again. About how many steps does he walk?
Answer:
a. 80,500 steps
b. 81,200 steps are estimated. He reached his goal for the week.
c. 6,000 steps

Explanation:
a. He walks 11,500 steps each day , Then for 1 Week we have 7 days , so 7 × 11,500 = 80,500 steps for a week
b. By rounding the nearest numbers of all the steps he walked from Sunday to Saturday , we have,
12,380 + 10,710 + 7,990 + 11,830 + 12,260 + 10,050 + 15,980 = 81,200 steps.
81,200 steps are estimated. He reached his goal for the week.
c. 1 mile = 2,000 steps.
The teacher estimates that he walks 3 miles from one end of the Golden Gate Bridge to the other and back again.
So, 3 × 2,000  = 6,000 steps.
He walked 6,000 steps.

 

Big Ideas Math Book 3rd Grade Answer Key Chapter 15 Find Perimeter and Area answer key are attainable in this chapter. This answer key is useful for students who are preparing for their examinations and can download this pdf for free of cost. Here, each and every question was explained in detail which helps students to understand easily. Big Ideas Math Answers Grade 3 Chapter 15 explains different types of questions on perimeter and area.

Big Ideas Math Book 3rd Grade Answer Key Chapter 15 Find Perimeter and Area

This chapter contains different topics like finding the perimeter of the polygons, finding unknown side lengths and the same perimeter with different areas, etc. Those topics were being set up by the numerical specialists as indicated by the most recent release. Look down this page to get the answers for all the inquiries. Tap the connection to look at the subjects shrouded in this chapter Find Perimeter and Area.

Lesson 1 Understand Perimeter

• Lesson 15.1 Understand Perimeter
• Understand Perimeter Homework & Practice 15.1

Lesson 2 Find Perimeters of Polygons

• Find Perimeters of Polygons 15.2
• Find Perimeters of Polygons Homework & Practice 15.2

Lesson 3 Find Unknown Side Lengths

• Lesson 15.3 Find Unknown Side Lengths
• Find Unknown Side Lengths Homework & Practice 15.3

Lesson 4 Same Perimeter, Different Areas

• Lesson 15.4 Same Perimeter, Different Areas
• Same Perimeter, Different Areas Homework & Practice 15.4

Lesson 5 Same Area, Different Perimeters

• Lesson 15.5 Same Area, Different Perimeters
• Same Area, Different Perimeters Homework & Practice 15.5

Performance Task

• Find Perimeter and Area Performance Task
• Find Perimeter and Area Activity
• Find Perimeter and Area Chapter Practice
• Find Perimeter and Area Cumulative practice 1 – 15
• Find Perimeter and Area Cumulative Steam Performance Task 1 – 15

Lesson 15.1 Understand Perimeter

Explore and Grow

Question 1.
Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. What is the distance around the rectangle?

Answer:
The units around the rectangle are 15 units.

Explanation:

In the above figure, the length of the rectangle is 4 units, and
the breadth of the rectangle is 3 units.
The units around the rectangle are 15 units.

Structure
Change the side lengths of the rectangle on your geoboard. What do you notice about the distance around your rectangle compared to the distance around the rectangle above? Explain

Answer:
The distance around the rectangle is 15 units

Explanation:

In the above figure, we can see the length of the rectangle is 3 units, and
the breadth of the rectangle is 4 units.
There is no change in the distance around the rectangle which is 15 units.

Think and Grow: Understand Perimeter
Perimeter is the distance around a figure. You can measure perimeter using standard units, such as inches, feet, centimeters, and meters.
Example
Find the perimeter of the rectangle.
Choose a unit to begin counting. Count each unit around the rectangle.

Answer:
The perimeter of the rectangle is 20 in.

Explanation:
To find the perimeter of the triangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 3 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+3)
= 2(10)
= 20 in.

Show and Grow

Question 1.
Find the perimeter of the figure.

There are ___ units around the figure.
Answer:
There are 22 units around the figure.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 6 m,
and the breadth is 5 m,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(6+5)
= 2(11)
= 22 m.

Question 2.

There are ___ units around the figure.
So, the perimeter is ____ feet.
Answer:
The perimeter of the figure is 24 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 5,3,2,4,3,7.
and the perimeter of the figure is 5+3+2+4+3+7
= 24 ft.

Question 3.
Draw the figure that has a perimeter of 16 centimeters.

Answer:
The sides of the figure are 4 cm.

Explanation:
As the perimeter of the figure is 16 cm,
So let the sides of the figure be 4 cm,
by that, we can get the perimeter as 16 cm, i.e
4+4+4+4= 16 cm.

Apply and Grow: Practice

Question 4.
Find the perimeter of the figure.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 20 units around the figure.
So, the perimeter is 20 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 3,3,2,2,5,5.
and the perimeter of the figure is 3+3+2+2+5+5
= 20 units,
So the perimeter is 20 cm.

Question 5.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 26 units around the figure.
So, the perimeter is 26 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,5,2,3,1,3,2,5
and the perimeter of the figure is 5+5+2+3+1+3+2+5
= 26 units,
So the perimeter is 26 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the figure is 22 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 4,3,2,3,3,3,1,3
and the perimeter of the figure is 4+3+2+3+3+3+1+3
= 22 units,
So the perimeter is 22 in.

Question 7.

Perimeter = ___
Answer:
The perimeter is 24 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,7,2,2,1,2,1,1,1,2
and the perimeter of the figure is 5+7+2+2+1+2+1+1+1+2
= 24 units,
So the perimeter is 24 m.

Question 8.
Draw a figure that has a perimeter likely measurement for the of 14 centimeters.

Answer:
The length of the rectangle is 2 cm, and
the breadth is 5 cm.

Explanation:

In the above figure, we can see the length of the rectangle is 2 cm
and the breadth of the rectangle is 5 cm
perimeter = 2(length + breadth)
= 2(5+2)
= 2(7)
= 14 cm.
so the perimeter of the rectangle is 14 cm.

Question 9.
Precision Which is the most likely measurement for the perimeter of a photo?
20 inches
100 meters
5 centimeters
2 inches
Answer:
2 inches.

Explanation:
The most likely measurement for the perimeter of a photo is 2 inches.

Question 10.
You be the teacher Your friend counts the units around the figure and says the perimeter is 12 units. Is your friend correct? Explain.

Answer:
No, he is not correct.

Explanation:
No, he is not correct. As there are 2+2+1+1+4+1+1+2= 14 units, but not 12 units. So he is not correct.

Think and Grow: Modeling Real Life
Use a centimeter ruler to find the perimeter of the bookmark.

The perimeter is ___
Answer:
The perimeter of the bookmark is 26 cm.

Explanation:
On measuring, the length of the sides of the bookmark are 4 cm, 9 cm, 2 cm, 2 cm, 9 cm
to find the perimeter of the bookmark, we will add all the length of the sides
so the perimeter of the bookmark is
p = 4 cm+ 9 cm+ 2 cm+ 2 cm+ 9 cm
= 26 cm.
The perimeter of the bookmark is 26 cm.

Show and Grow

Question 11.
Use an inch ruler to find the perimeter of the decal.

Answer:
The perimeter of the decal is 17 in.

Explanation:
On measuring, the length of the sides of the decal is 5in, 2in, 5in, 5in
to find the perimeter of the decal, we will add all the length of the sides
so the perimeter of the decal is
p= 5in+2in+5in+5in
= 17 in
The perimeter of the decal is 17 in.

Question 12.
How much greater is the perimeter of your friend’s desk than the perimeter of your desk?

Answer:
Friend’s perimeter is 22-16= 6 times greater.

Explanation:
As we can see figure 1 is a square and the sides of the square are 4 ft.
So the perimeter of the square is a+a+a+a,
= 4+4+4+4
= 16 ft.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 2,6,5,2,3,4
and the perimeter is 2+6+5+2+3+4= 22 ft.
By this, we can see that the friend’s figure has a greater perimeter,
and friend’s perimeter is 22-16= 6 times greater.

Question 13.
DIG DEEPER!
Explain how you might use a centimeter ruler and string to estimate the perimeter of the photo of the window.

Answer:
Here, we will use the ruler to find the length of the bottom part of the window and the sides which are straight we can find the length using the ruler. And the curve sides we will measure using the string.

Understand Perimeter Homework & Practice 15.1

Question 1.

There are ___ units around the figure.
So, the perimeter is ___ inches.
Answer:
There are 20 units around the figure,
So the perimeter is 20 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
As we can see the above image is a square,
and the perimeter of the square is s+s+s+s
= 5+5+5+5
= 20 in.
As there are 20 units around the figure,
So the perimeter is 20 in.

Question 2.

There are ___ units around the figure.
So, the perimeter is ___ feet.

Answer:
There are 32 units around the figure,
So the perimeter is 32 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,3,4,3,2,5,8,5
and the perimeter of the figure is 2+3+4+3+2+5+8+5
= 32 units.
As there are 32 units around the figure,
So the perimeter is 32 ft.

Question 3.

Perimeter = ___
Answer:
There are 26 units around the figure,
So the perimeter is 26 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,2,2,3,1,3,5,8
and the perimeter of the figure is 2+2+2+3+1+3+5+8
= 26 units.
As there are 26 units around the figure,
So the perimeter is 26 cm.

Question 4.

Perimeter = ___
Answer:
There are 38 units around the figure,
So the perimeter is 38 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,8,2,3,1,3,2,8,2,3,1,3
and the perimeter of the figure is 2+8+2+3+1+3+2+8+2+3+1+3
= 38 units.
As there are 38 units around the figure,
So the perimeter is 38 m.

Question 5.
Draw a figure that has a perimeter of 18 inches.

Answer:
The length of the rectangle is 5 in
and the breadth of the rectangle is 4 in.

Explanation:

Let the length of the rectangle be 5 in
and the breadth of the rectangle be 4 in
so the perimeter of the rectangle is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 in.

Question 6.
Reasoning
Which color represents the perimeter of the rectangle? What does the other color represent?

Answer:
Blue color represents the perimeter of the rectangle and
the other color yellow represents the area of the figure.

Explanation:
In the above figure, the blue color represents the perimeter of the rectangle.
Because the perimeter represents the distance around the edge of the shape.
And the other color yellow represents the area of the figure,
as area represents the amount of space inside a shape.

Question 7.
Modeling Real Life
Use a centimeter ruler tofind the perimeter of the library card.

Answer:
The perimeter of the library card is 24 cm.

Explanation:
On measuring, the length of the sides of the library card is 5cm, 7cm, 5cm, 7cm
to find the perimeter of the library card, we will add all the length of the sides
so the perimeter of the library card is
p= 5cm+7cm+5cm+7cm
= 24 cm
The perimeter of the library card is 24 cm.

Question 8.
Modeling Real Life
How much greater is the perimeter of your piece of fabric than the perimeter of your friend’ spiece of fabric?

Answer:
My fabric is 4 inches greater than my friend’s fabric.

Explanation:
To find the perimeter of the figure 1, we will add all the sides of the figure,
The sides of the above figure is 2,1,1,2,2,2,8,2,2,2,1,1
and the perimeter of the figure is 2+1+1+2+2+2+8+2+2+2+1+1
= 26 in.
So the perimeter of the figure is 26 in.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 1,2,2,4,1,5,4,3
and the perimeter is 1+2+2+4+1+5+4+3
= 22 in.
By this, we can see that my fabric has the highest perimeter than the friend’s fabric
which is 26-22= 4 in greater.

Review & Refresh

Write two equivalent fractions for the whole number.

Question 9.

Answer:
1= 4/4 = 6/6

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 1= 4/4 = 6/6.

Question 10.

Answer:
4= 4/1 = 8/2.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 4= 4/1 = 8/2.

Question 11.

Answer:
6= 24/4 = 36/6.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 6= 24/4 = 36/6.

Find Perimeters of Polygons 15.2

Explore and Grow

Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. Then find the perimeter in more than one way.

Answer:
The perimeter of the rectangle is 14 units.

Explanation:

In  the above figure, we can see the length of the rectangle is 4 units
and the breadth of the rectangle is 3 units,
so the perimeter of the rectangle is
p =2(length + breadth)
= 2(4+3)
= 2(7)
= 14 units.
The perimeter of the rectangle is 14 units.
Critique the Reasoning of Others
Compare your methods of finding the perimeter to your partner’s methods. Explain how they are alike or different.
Answer:
There are two methods to find the perimeter explained below.

Explanation:
There are two ways to find the perimeter.
The first method is
Perimeter = 2(length + breadth)
here, we will add length and breadth, and then we will multiply the result by 2.
and the second method is
Perimeter = a+b+c+d
here, we will add all the sides of the figure.

Think and Grow: Find Perimeter

Example
Find the perimeter of the trapezoid.

You can find the perimeter of a figure by adding all of the side lengths.

Write an equation. The letter P represents the unknown perimeter. Add the side lengths.
So, the perimeter is ___.
Answer:
So, the perimeter is 36 in.

Explanation:
To find the perimeter of the trapezoid, we will add all the sides of the trapezoid,
so the sides of the trapezoid is 6 in,12 in,8 in,10 in
The perimeter is 6+12+8+10
= 36 in.

Example
Find the perimeter of the rectangle

Because a rectangle has two pairs of equal sides, you can also use multiplication to solve.

One Way:

Another Way:

So, the perimeter is ___.

Answer:
Perimeter = 7+9+7+9
= 32 cm.
Perimeter= 2×9 + 2× 7
= 32 cm.
So, the perimeter is 32 cm.

Explanation:
To find the perimeter of a rectangle, we have two ways,
One way is to add all the sides of the rectangle, which is
7+9+7+9= 32 cm.
And the other way is, as the two sides of the rectangle are equal, we wil use formula
Perimeter = 2( Length + Breadth)
= 2× Length + 2 × Breadth
= 2×9 + 2× 7
= 18+ 14
= 32 cm.

Show and Grow

Find the perimeter of the polygon.
Answer:

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides.

Question 1.

The perimeter is ___.
Answer:
The perimeter is 12 m.

Explanation:
To find the perimeter of the polygon, we will add the length of all sides of the polygon.
So the length of the sides is 5 m, 3 m, 3 m, 2 m
The perimeter is 5+3+3+2
= 12 m.

Question 2.

The perimeter is ___.

Answer:
The perimeter is 24 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×6
= 24 ft.

Apply and Grow: Practice

Find the perimeter of the polygon.

Question 3.

Perimeter = ___
Answer:
The perimeter of the polygon is 28 m.

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides of the polygon.
So the length of the sides is 4 m, 9 m, 10 m, 5 m
The perimeter is 4+9+10+5
= 28 m.

Question 4.

Perimeter = ___
Answer:
The perimeter of the figure is 40 in.

Explanation:
To find the perimeter of the figure, we will add the length of all sides of the figure.
So the length of the sides is 12 in, 4 in, 8 in, 7 in,9 in.
The perimeter is 12+4+8+7+9
= 40 in.

Question 5.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 36 cm.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 9 cm,
and the breadth is 10 cm,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(10+9)
= 2(19)
= 38 cm.

Question 6.
Rhombus

Perimeter = ___
Answer:
The perimeter of the Rhombus is 4 in.

Explanation:
As all sides of the Rhombus are equal, so the formula of the Rhombus is 4a
and the side of the Rhombus is 1 in,
so the perimeter is 4a
= 4×1
= 4 in.

Question 7.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 22 cm.

Explanation:
As the opposite sides of the parallelogram are equal,
so the perimeter of the parallelogram is 2(a+b)
the length of the parallelogram is 3 cm,
and the breadth of the parallelogram is 8 cm
So the perimeter of the parallelogram is
= 2(3+8)
= 2(11)
= 22 cm.

Question 8.
Square

Perimeter = ___
Answer:
The perimeter of the square is 16 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×4
= 16 ft.

Question 9.
You build a pentagon out of wire for a social studies project. Each side is 8 centimeters long. What is the perimeter of the pentagon?
Answer:
The perimeter of the pentagon is 40 cm.

Explanation:
The length of the pentagon is 8 cm,
so the perimeter of the pentagon is 5a,
which is 5×8
= 40 cm

Question 10.
Number Sense
The top length of the trapezoid is 4 feet. The bottom length is double the top. The left and right lengths are each 2 feet less than the bottom. Label the side lengths and find the perimeter of the trapezoid.

Answer:
The sides of the trapezoid are 4ft, 6ft, 8ft, 6ft, and the perimeter is 24 ft.

Explanation:
As the top length of the trapezoid is 4 feet and the bottom length is double the top,
which is 4×2= 8 feet. And the left and right lengths are each 2 feet less than the bottom,
which means 8 – 2 = 6 ft each.

So to find the perimeter of the trapezoid is we will add the length of all sides.
The perimeter of the trapezoid is 4+6+8+6= 24 ft.

Question 11.
Writing
Explain how finding the perimeter of a rectangle is different from finding its area.
Answer:
Perimeter= 2(length + breadth)
Area = length × breadth.

Explanation:
To find the perimeter of the rectangle,
we will add the length and breadth and will multiply the result with 2
Perimeter= 2(length + breadth)
and to find the area of the rectangle,
we will multiply the length and breadth of the rectangle.
Area = length × breadth.

Question 12.
Dig Deeper!
A rectangle has a perimeter of 12 feet. What could its side lengths be ?
Answer:
The length of the rectangle is 4 feet
and the breadth of the rectangle is 2 feet

Explanation:
Given the perimeter is 12 feet,
so the let the length be 4 feet
and the breadth be 2 feet
Let’s check on the length and breadth is correct are not
perimeter = 2( length + breadth)
= 2( 4+2)
= 2(6)
= 12 feet.

Think and Grow: Modeling Real Life

The rectangular sign is 34 feet longer than it is wide. What is the perimeter of the sign?

Understand the problem:
Make a plan:
Solve:
The perimeter is ___.
Answer:
The perimeter of the rectangular sign is 124 ft.

Explanation:
Given the rectangular sign is 34 feet longer than it’s wide
and the wide is 14 ft,
so the length is 34ft + 14ft= 48 ft.
the perimeter of the rectangular sign is
P= 2(48ft + 14ft)
= 2(62ft)
= 124 ft.
The perimeter of the rectangular sign is 124 ft.

Show and Grow

Question 13.
A city has a rectangular sidewalk in a park. The sidewalk is 4 feet wide and is 96 feet longer than it is wide. What is the perimeter of the sidewalk?
Answer:
The perimeter of the sidewalk is 208 feet.

Explanation:
As given the rectangular sidewalk’s wide is 4 feet and the length is 96 feet longer than it’s wide,
which means 96+4= 100 feet is the length of the rectangular sidewalk,
so the perimeter of the rectangular sidewalk is 2( length + breadth)
= 2( 4+100)
= 2(104)
= 208 feet.

Question 14.
A team jogs around a rectangular field three times. The field is 80 yards long and 60 yards wide. How many yards does the team jog?
Answer:
The number of yards does the team jog is 3×280= 840 yards.

Explanation:
The length of the rectangular field is 80 yards,
The breadth of the rectangular field is 60 yards,
So, the perimeter of the rectangular field is 2(length + breadth)
= 2(80+60)
= 2(140)
= 280 yards.
As the team jogs around a rectangular field three times,
so the number of yards does the team jog is 3×280= 840 yards.

Question 15.
Each side of the tiles is 8 centimeters long. What is the sum of the perimeters?

You put the tiles together as shown. Is the perimeter of this new shape the same as the sum of the perimeters above? Explain.

Answer:
The sum of the perimeters is 96 cm.
Yes, there will be a change in the perimeter of the new figure. The perimeter is 80 cm.

Explanation:
In the above figure, we can see a hexagon that has six sides.
So the formula for the perimeter of a hexagon is 6a,
the perimeter of the tiles is 6×8
= 48 cm.
So the sum of the tiles 48+48= 96 cm.
Yes, there will be a change in the perimeter of the new figure. As there are ten sides in the new figure, so the perimeter of the new figure is 10×8= 80 cm.

Find Perimeters of Polygons Homework & Practice 15.2

Find the perimeter of the polygon

Question 1.

Perimeter = ___
Answer:
The perimeter of the polygon is 29 in.

Explanation:
The perimeter of the polygon is the sum of the length of its sides,
So the sides of the polygon are 9 in, 4 in, 6 in, 10 in,
and the perimeter of the polygon is 9+4+6+10= 29 in.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 38 cm.

Explanation:
The perimeter of the figure is the sum of the length of its sides,
So the sides of the figure are 6 cm, 5 cm, 7 cm, 8 cm, 7 cm, 5 cm,
and the perimeter of the figure is 6+5+7+8+7+5= 38 cm.

Question 3.
Square

Perimeter = ___
Answer:
The perimeter of the square is 8 ft.

Explanation:
The length of the square is 2 ft,
and the perimeter of the square is 4a
= 4×2
= 8 ft.
so the perimeter of the square is 8 ft.

Question 4.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 8m.

Explanation:
The length of the parallelogram is 1 m,
and the breadth of the parallelogram is 3 m,
so the perimeter of the parallelogram is 2(length + breadth)
= 2(1+3)
= 2(4)
= 8 m.

Question 5.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 40 ft.

Explanation:
The length of the side of the rhombus is 10 ft,
and the perimeter of the rhombus is 4a
= 4×10
= 40 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the rectangle is 22 in.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 4 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+4)
= 2(11)
= 22 in.

Question 7.
Each side of a triangle is 5 centimeters long. What is the perimeter of the triangle?
Answer:
The perimeter of the triangle is 15 cm.

Explanation:
Given the length of the side of the triangle is 5 cm,
and the perimeter of the triangle is 3a
= 3×5
= 15 cm.

Question 8.
You Be The Teacher
Descartes says that a square will always have a greater perimeter than a triangle because it has more sides. Is he correct? Explain.
Answer:
Yes, Descartes is correct.

Explanation:
Yes, Descartes is correct. As the square has four sides and the perimeter of the square is 4a.
Whereas the triangle has 3 sides and the perimeter of the triangle is 3a.
So the square will always have a greater perimeter than a triangle.

Question 9.
Structure
Draw a pentagon and label its sides so that it has the same perimeter as the rectangle.

Answer:
The length of the side of the pentagon is 4m.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
So the perimeter of the rectangle is 2( length + breadth)
= 2(7+3)
= 2(10)
= 20 m.
Here we have the perimeter of the pentagon which is 20 m,
so we should find the sides of the pentagon,
As we know the perimeter of the pentagon is 5a
5a= 20
a= 20/5
= 4 m.
So the length of the side of the pentagon is 4m.

Question 10.
Modeling Real Life
An Olympic swimming pool is 82 feet longer than it is wide. What is the perimeter of the swimming pool?

Answer:
The perimeter of the swimming pool is 492 ft.

Explanation:
Given the Olympic swimming pool is 82 feet longer than it’s wide
and the wide is 82 ft,
so the length of the swimming pool is 82+82= 164 ft
the perimeter of the swimming pool is
p= 2(length+breadth)
= 2(164+82)
= 2(246)
= 492 ft.
The perimeter of the swimming pool is 492 ft.

Question 11.
Modeling Real Life
You put painter’s tape around two rectangular windows. The windows are each 52 inches long and 28 inches wide. How much painter’s tape do you need?
Answer:
The painter’s tape 160 inches.

Explanation:
Given the length of the window is 52 inches
and the width of the window is 28 inches
so the perimeter of the window is
p= 2(length + breadth)
= 2(52+28)
= 2(80)
= 160 inches.
So the painter’s tape 160 inches.

Review & Refresh

Question 12.

Answer:
737.

Explanation:
On adding 590+147 we will get 737.

Question 13.

Answer:
894.

Explanation:
On adding 636+258 we will get 894

Question 14.

Answer:
805.

Explanation:
On adding 476+329 we will get 805

Lesson 15.3 Find Unknown Side Lengths

Explore and Grow

You have a map with the three side lengths shown. The perimeter of the map is 20 feet. Describe how you can find the fourth side length of your map without measuring.

Answer:
The fourth side of the map is 4ft.

Explanation:
We can find the value of the fourth side in two methods
First method:
Given the perimeter of the map is 20 feet,
and the sides of the map is 6 ft, 4 ft, 6ft, X ft.
As we know the perimeter of the rectangle is
6+4+6+X= 20 feet
16+X= 20
X= 20- 16
= 4 ft.
Second method:
As we know that the opposite sides of the rectangle are equal, as we know that the length of the side is 4 ft so the other side will also be 4 ft.

Repeated Reasoning
How is finding the unknown side length of a square different from finding the unknown side length of a rectangle?
Answer:
Refer below for a detailed explanation.

Explanation:
To find the unknown side length of the square
if we know the perimeter of the square then
the perimeter of the square is
p= 4a
we will substitute the value of p, on solving we will get the length of the square.
and to find the unknown side length of the rectangle,
we need to know the area or perimeter of the rectangle
and the other side of the rectangle.
so the formula of the perimeter of the rectangle is
p = 2(length + breadth)
we will substitute the perimeter value and the other side value
then we can find the length of the rectangle.

Think and Grow: Find Unknown Side Lengths
Example
The perimeter of the trapezoid is 26 feet. Find the unknown side length.


Write an equation for the perimeter.
Add the known side lengths.
What number plus 16 equals 26?
The unknown side length is ___.
Answer:
K= 10,
The number 16+10 equals 26.
The unknown side length is 10.

Explanation:
Given the perimeter of the trapezoid is 26 ft,
So the perimeter of the trapezoid is
K+5+6+5= 26
K+16= 26
K= 10.
The number 16+10 equals 26.
The unknown side length is 10.

Example

The perimeter of the square is 32 centimeters. Find the length of each side of the square.


Write an equation for the perimeter 4 times what number equals 32?
So, the length of each side is ___.
Answer:
n= 8.
The length of each side is 8 cm.

Explanation:
The perimeter of the square is 32 cm
So to find the sides of the square
4a= 32
a= 32/4
= 8 cm.
So, the length of each side is 8 cm.

Show and Grow

Find the unknown side length.

Question 1.
Perimeter = 34 inches

y = ___
Answer:
y = 13 in.

Explanation:
Given the perimeter is 34 inches,
and the sides of the figure are 10 in, 7 in, 4 in, y in.
so the perimeter of the figure is
34 in = 10+7+4+y
34 = 21+ y
y = 34 – 21
y = 13.

Question 2.
Perimeter = 20 meters

j = ___
Answer:
The length of the sides of the square is 5 m.

Explanation:
As we can see the above figure is a square and the perimeter of the square 20 meters,
so the sides of the square are
perimeter = 4a
20 = 4 j
j= 5 m.

Apply and Grow: Practice

Find the unknown side length.

Question 3.
Perimeter = 19 feet

y = ___
Answer:
The perimeter of the figure is 8 ft.

Explanation:
The perimeter of the figure is 19 feet,
and the length of the sides of the figure is 8 ft, 3 ft, y ft.
so perimeter = 8 ft + 3 ft + y ft
19= 11 ft + y ft
y= 19 ft – 11 ft
= 8 ft.

Question 4.
Perimeter = 26 centimeters

d = ___
Answer:
d= 4 cm.

Explanation:
The perimeter of the figure is 26 cm,
and the length of the sides of the figure is 10 cm, 5 cm, 7 cm, d cm.
so the perimeter of the figure is
p = 10+5+7+d
26 = 22 + d
d= 26-22
= 4

Question 5.
Perimeter = 30 feet

k = ___
Answer:
k = 11 ft.

Explanation:
Given the perimeter of the figure is 30 feet,
and the length of the sides is 5ft, 12 ft, 2 ft, k ft
So the perimeter of the figure is
p = 5 + 12 + 2 + k
30 ft = 19 ft + k
k = 11 ft.

Question 6.
Perimeter = 32 inches

k = ___
Answer:
k= 4 in.

Explanation:
Given the perimeter of the figure is 32 inches,
and the lengths of all sides is 10 in, 4 in, 5 in, 4 in, 5 in, k in.
So the perimeter of the figure is
p= 10+4+5+4+5+k
32 in =  28 in + k
k= 4 in.

Question 7.
Perimeter = 8 meters

y = ___
Answer:
y = 2 m.

Explanation:
Given the perimeter of the rhombus is 8 feet,
and the length of the side is y m,
So the perimeter of the rhombus is
p = 4a
8 m = 4×y
y = 2 m.

Question 8.
Perimeter = 48 inches

d = ___
Answer:
d = 8 in.

Explanation:
Given the perimeter of the Hexagon is 48 inches,
and the length of the sides is d in
So the perimeter of the hexagon is
p = 6 a
48 in = 6 × d in
d= 8 in.

Question 9.
Number Sense
A rectangle has a perimeter of 30 centimeters. The left side is 7 centimeters long. What is the length of the top side?
Answer:
The length of the top side is 8 cm.

Explanation:
Given the perimeter of the rectangle is 30 cm,
and the length of the left side of the rectangle is 7 cm,
So let the length of the top side be X,
Perimeter of the rectangle is
P = 2 (Length + breadth)
30 = 2 ( 7 cm + X cm)
30 / 2 = 7 cm + X cm
15 = 7 cm + X cm
X = 15 cm – 7 cm
X = 8 cm.
so, the length of the top side is 8 cm.

Question 10.
Writing
A triangle has three equal sides and a perimeter of 21 meters. Explain how to use division to find the side lengths.
Answer:
The length of the side is 7 m.

Explanation:
Given the perimeter of the triangle is 21 m,
and we need to find the side of the lengths,
so the perimeter of the triangle is
p = 3a
21 m = 3×a
a = 21/3
= 7 m
So the length of the side is 7 m.

Question 11.
DIG DEEPER!
Newton draws and labels the square and rectangle below. The perimeter of the combined shape is 36 feet. Find the unknown side length.

Answer:
The unknown side of the length is 14 ft.

Explanation:
As the perimeter of the combined shape is 36 feet,
and the length of the side of the rectangle is 4 ft, and the other side be X ft
and the perimeter of the rectangle is 36 ft,
so perimeter = 2 (length + breadth)
36 ft  = 2( 4 ft + X ft)
36/2 = 4 ft + X ft
18 = 4 ft + X ft
X= 14 ft.
The unknown side of the length is 14 ft.

Think and Grow: Modeling Real Life

The perimeter of the rectangular vegetable garden is 30 meters. What are the lengths of the other three sides?

Understand the problem:
Make a plan:
Solve:
The lengths of the other three sides are ___, ___, and ___.
Answer:
The lengths of other three sides is 6 m, 9 m, 9 m.

Explanation:
The perimeter of the rectangular vegetable garden is 30 m
as it is in a rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular vegetable garden is 6m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
30 m = 2( 6 m+ X m)
30/2 = 6 + X
15 = 6 + X
X= 15 – 6
= 9 m.
So the lengths of other three sides is 6 m, 9 m, 9 m.

Show and Grow

Question 12.
The perimeter of the rectangular zoo enclosure is 34 meters. What are the lengths of the other three sides?

Answer:
The lengths of the other three sides are 12 m, 11 m, 11 m.

Explanation:
The perimeter of the rectangular zoo is 34 m
as it is in rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular zoo is 12 m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
34 m = 2( 12 m+ X m)
34/2 = 12 + X
17 = 6 + X
X= 17 – 6
= 11 m.
So the lengths of the other three sides is 12 m, 11 m, 11 m.

Question 13.
The floor of an apartment is made of two rectangles. The Perimeter is 154 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 7 ft.
The other length of the side of the big rectangle is 30 ft.

Explanation:
Given the perimeter of the apartment is 154 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal and the length of the big rectangle is 30 ft
so the other length is also 30 ft.
as the perimeter of the small rectangle is 38 ft
and the length of the one side of the rectangle is 12 ft
so the other length of the small rectangle is
p = 2(length+breadth)
38 = 2(12 + breadth)
38/2 = 12+ breadth
19= 12 + breadth
breadth= 19 – 12
= 7 ft.
The other length of the side is 7 ft.

Question 14.
DIG DEEPER!
You want to make a flower bed in the shape of a pentagon. Two sides of the flower bed are each 7 inches long, and two sides are each 16 inches long. The perimeter is 57 inches. Sketch the flower bed and label all of the side lengths.
Answer:
The length of the other side is 11 ft.

Explanation:

Given the perimeter of the flower bed shaped pentagon is 57 inches
and the two sides of the flower bed each is 7 inches long
and the other two sides of the flower bed each is 16 inches long
the other side of the flower bed be X
the perimeter of the flower bed is
p = 7+7+16+16+X
57= 46+X
X= 11 in.
The length of the other side is 11 ft.

Find Unknown Side Lengths Homework & Practice 15.3

Find the unknown side length.

Question 1.
Perimeter = 24 feet

d = ___
Answer:
The length of the unknown side is 8 ft.

Explanation:
The perimeter of the triangle is 24 ft,
and the lengths of the sides is 10 ft, 6 ft, d ft
so the perimeter of the triangle is
p = 10+6+d
24 = 10+6+d
24 = 16+d
d = 24 – 16
=  8 ft.
The length of the unknown side is 8 ft.

Question 2.
Perimeter = 46 inches

k = ___
Answer:
The length of the unknown side is 15 in.

Explanation:
The perimeter of the figure is 46 inches,
and the lengths of the sides is 13 in, 5 in, 13 in, k in
so the perimeter of the figure is
p = 13+5+13+k
46 = 31+k
46-31 = k
k = 15
=  15 in.
The length of the unknown side is 15 in.

Question 3.
Perimeter = 21 centimeters

y = ___
Answer:
The length of the unknown side is 7 cm.

Explanation:
The perimeter of the figure is 21 cm,
and the lengths of the sides is 4 cm, 1 cm, 9 cm, y cm
so the perimeter of the figure is
p = 4 cm+ 1 cm+ 9 cm+ y cm
21 = 14+ y
y = 21 – 14
y = 7
=  7 cm.
The length of the unknown side is 7 cm.

Question 4.
Perimeter = 41 meters

y = ___
Answer:
The length of the unknown side is 8 m.

Explanation:
The perimeter of the figure is 41 m,
and the lengths of the sides is 3 m, 12 m, 10 m, 8 m, y m
so the perimeter of the figure is
p = 3+12+10+8+y
41 = 33 + y
y = 41-33
y = 8
=  8 m.
The length of the unknown side is 8 m.

Question 5.
Perimeter = 12 feet

d = ___
Answer:
The length of the sides of the triangle is 4 ft.

Explanation:
The perimeter of the triangle is 12 feet,
and the length of the side of the triangle is d ft,
so the perimeter of the triangle is
p = 3 a
12 = 3 × d
d = 12/3
= 4 ft.
The length of the sides of the triangle is 4 ft.

Question 6.
Perimeter = 50 inches

k = ___
Answer:
The length of the sides of the Hexagon is 4 in.

Explanation:
The perimeter of the Hexagon is 50 inches,
and the length of the side of the Hexagon is k in,
so the perimeter of the hexagon is
p = 5 a
50 = 5 × k
k = 50/5
= 10 in.
The length of the sides of the triangle is 10 in.

Question 7.
DIG DEEPER!
Each polygon has equal side lengths that are whole numbers. Which polygon could have a perimeter of 16 centimeters? Explain.

Answer:
The length of the sides of the octagon is 2 cm.

Explanation:
In the above three polygons, the second figure is an octagon, which has eight sides.
and the perimeter of the octagon is
p = 8 a
16 cm = 8 a
a = 16 /8
= 2 cm.

Question 8.
Number Sense
The area of a square is 25 square inches. What is its perimeter?
Answer:
The perimeter of the square is 20 inches.

Explanation:
The area of the square is 25 square inches, so
area = s^2
25 = s^2
s= 5 inches
so the perimeter of the square is
p = 4s
= 4×5
= 20 inches.

Question 9.
Modeling Real Life
The perimeter of the rectangular side walk is 260 meters. What are the lengths of the other three sides?

Answer:
The length of the other three sides is 10 m,10m,120 m.

Explanation:
The perimeter of the rectangular side walk is 260 meters,
and the length of the one side of the side walk is 120 m,
so the perimeter of the rectangular side walk is
p = 2( length + breadth)
260 = 2 ( 120 + breadth)
260/2 = 120 + breadth
130 = 120 + breadth
breadth = 130 – 120
= 10 m.
The length of the other three sides is 10 m,10m,120 m.

Question 10
Modeling Real Life
Two rectangular tables are pushed together. The perimeter is 40 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 3 ft.
The other length of the side of the big rectangle is 5 ft.

Explanation:
Given the perimeter of the apartment is 40 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal
and the length of the big rectangle is 5 ft
so the other length is also 5 ft.
as the perimeter of the small rectangle is 10 ft
and the length of the one side of the rectangle is 2 ft
so the other length of the small rectangle is
p = 2(length+breadth)
10 = 2(2 + breadth)
10/2 = 2+ breadth
5= 2 + breadth
breadth= 5 – 2
= 3 ft.
The other length of the side is 3 ft.

Review & Refresh

Write the time. Write another way to say the time.

Question 11.

Answer:
06: 48

Explanation:
Another way to say time is 06: 48

Question 12.

Answer:
03: 24

Explanation:
Another way to say time is 03: 24

Question 13.

Answer:

Explanation:
Another way to say the time is 11: 48

Lesson 15.4 Same Perimeter, Different Areas

Use color tiles to create two different rectangles that each have a perimeter of 16 units. Then draw your rectangles and label their dimensions. Do the rectangles have the same area? Explain how you know.

Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

Given the perimeter of the rectangle is 16 units
The length of rectangle 1 is 5 units
and the breadth of rectangle 1 is 3 units
so the area of rectangle 1 is
area = length × breadth
= 5×3
= 15 square units.
The length of rectangle 2 is 6 units
and the breadth of rectangle 2 is 2 units
so the area of rectangle 2 is
area = length × breadth
= 6×2
= 12 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Repeated Reasoning
Draw another rectangle that has the same perimeter but different dimensions. Compare the area of the new rectangle to the rectangles above. What do you notice?
Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

The length of rectangle 1 is 4 units
and the breadth of rectangle 1 is 3 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 units.
so the area of rectangle 1 is
area = length × breadth
= 4×3
= 12 square units.
The length of rectangle 2 is 5 units
and the breadth of rectangle 2 is 2 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 units.
so the area of rectangle 2 is
area = length × breadth
= 5×2
= 10 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Think and Grow : Same Perimeter, Different Areas

Example :
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?


Rectangle ___ has a greater area.
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 24 m²
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 16 m²
The rectangle A has a greater area.

Explanation:
Given the length of the rectangle is 6m
and the breadth of the rectangle is 4m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 6 × 4
= 24 m²

we can see in the above figure another rectangle with a length of 8m,
and the breadth is 2m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8 + 2)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 8×2
= 16 m²
So the rectangle A has a greater area.

Show and Grow

Question 1.
Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:The perimeter of the rectangle A is 14 in
and the area of the rectangle A is 10 in²
The perimeter of rectangle B is 14 in
and the area of the rectangle B is 12 in²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 in
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 2)
= 2(7)
= 14 in.
And the area of the rectangle is
a = length × breadth
= 5×2
= 10 in²

we can see in the above figure another rectangle with a length of 4 in,
and the breadth is 3 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 in.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 in²
So the rectangle B has a greater area.

Apply and Grow: Practice

Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Question 2.
Rectangle A

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 22 cm
and the area of the rectangle A is 11 cm²
The perimeter of rectangle B is 22 cm
and the area of the rectangle B is 30 cm²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 10 cm,
and the breadth of the rectangle is 1 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 1)
= 2(11)
= 22 cm.
And the area of the rectangle is
a = length × breadth
= 11 × 1
= 11 cm²

we can see in the above figure another rectangle with a length of 6 cm,
and the breadth is 5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 5)
= 2(11)
= 22 cm.
And the area of the rectangle is
area = length × breadth
= 6×5
= 30 cm²
So the rectangle B has a greater area.

Question 3.
Rectangle A

Perimeter = ___
Area = ___

Rectangle B

Perimeter = ___
Area = ___
Rectangle ___ has the greater area
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 21 m²
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 24 m²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 3)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 7 × 3
= 21 m²

we can see in the above figure another rectangle with a length of 6 m,
and the breadth is 4 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 6×4
= 24 m²
So the rectangle B has a greater area.

Question 4.
MP Structure
Draw a rectangle that has the same perimeter as the one shown, but with a lesser area. What is the area ?

Answer:
The perimeter of the rectangle A is 26 ft
and the area of the rectangle A is 40 ft²
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 ft,
and the breadth of the rectangle is 8 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 8)
= 2(13)
= 26 ft.
And the area of the rectangle is
a = length × breadth
= 5 × 8
= 40 ft

we can see in the above figure another rectangle with a length of 10 ft,
and the breadth is 3 ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(10 + 3)
= 2(13)
= 26 ft.
And the area of the rectangle is
area = length × breadth
= 10×3
= 30 ft²
So the rectangle A has a greater area.

Think and Grow: Modeling Real Life

A paleontologist has 12 meters of twine to rope off a rectangular section of the ground. How long and wide should she make the roped-off section so it has the greatest possible area?
Draw to show:
She should make the roped-off section ___ meters long and ___ meters wide.
Answer:
She should make the roped-off section 4 meters long and 2 meters wide.

Explanation:
Given that a paleontologist has 12 meters of twine to rope off a rectangular section,
so if we take the length as 4 m and width as 2 m then we can get the greatest possible area,
so the area of the rectangular section is
area = length × breadth
= 4 m ×2 m
= 8 m²

Show and Grow

Question 5.
Newton has 16 feet of wood to make a rectangular sandbox. How long and wide should he make the sandbox so it has the greatest possible area?

Answer:
The greatest possible area of the rectangular sandbox is 15 ft²

Explanation:
As Newton has 16 feet of wood to make a rectangular sandbox,
so let the length be 5 ft and the wide be 3 ft to get the greatest possible area,
so the area of the rectangular sandbox is
area = length × breadth
= 5 ft × 3 ft
= 15 ft²
The greatest possible area of the rectangular sandbox is 15 ft²

Question 6.
DIG DEEPER!
You and Newton are building forts. You each have the same length of rope to make a rectangular perimeter for the forton the ground. Your roped-off section is shown. Newton’s section has a greater area than yours. Draw one way Newton could rope off his fort.

Descartes also builds a fort. He has the same length of rope as you to make a perimeter around his fort. Descartes’s roped-off section has a lesser area than yours. Draw one way Descartes could rope off his fort.

Answer:
Refer the below for detailed explanation.

Explanation:
The length of the rope is 7 ft
and the breadth of the rope is 3 ft
the perimeter is
p = 2(length+breadth)
= 2(7+3)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 7×3
= 21 square feet.

Let the length of the Newton’s rope is 6 ft
and the breadth of the Newton’s rope is 4 ft
the perimeter is
p = 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 6×4
= 24 square feet.
And here Newton’s area is greater.

Let the length of the Descarte’s rope is 8 ft
and the breadth of the Descarte’s rope is 2 ft
the perimeter is
p = 2(length+breadth)
= 2(8+2)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 8×2
= 16 square feet.
And here Descartes area is lesser.

Same Perimeter, Different Areas Homework & Practice 15.4

Question 1.
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter? Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle __ has the greater area.
Answer:
The perimeter of the rectangle A is 7 cm
and the area of the rectangle A is 24 cm²
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 cm,
and the breadth of the rectangle is 5 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 5)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 7 × 5
= 35 cm².

we can see in the above figure rectangle with a length of 6.5 cm,
and the breadth is 5.5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6.5 + 5.5)
= 2(12)
= 24 ft.
And the area of the rectangle is
area = length × breadth
= 6.5×5.5
= 35.75 square feet
So the rectangle B has a greater area.

Question 2.
Patterns
Complete the pattern. Find the area of each rectangle.

Each rectangle has the same perimeter. As the area increases, what do you notice about the shape of the rectangle?
Answer:
As the area increases the shape of the figure was changed, we can see the figure was changed from rectangle to square.

Explanation:
As we know that the perimeter of the above figure is the same,
so the perimeter of the above figures is
p = 2 (length + breadth)
= 2 (1 m+9 m)
= 2(10 m)
= 20 m.
So, the perimeter of the above figures is 20 m.
The area of figure 1 is
area = length × breadth
= 1 m × 9 m
= 9 m².
The area of figure 2 is
= 8m × 2m
= 16 m².
The area of figure 3 is
= 7m × 3m
= 21 m².
Let the length of figure 4 be 6m and the breadth be 4m,
The area of figure 4 is
= 6m × 4m
= 24 m².
Let the length of figure 5 be 5m and the breadth be 5m,
The area of figure 5 is
= 5m × 5m
= 25 m².

Question 3.
Modeling Real Life
You are making a card with a 36-centimeter ribbon border. How long and wide should you make the card so you have the greatest possible area to write?

Answer:
The length and the breadth of the card is 9 cm.

Explanation:
Given the perimeter of the card with a ribbon border is 36 cm
so the length of the card is
p =4a
36 = 4a
a= 36/4
= 9 cm.
The length and the breadth of the card is 9 cm.
The area of the card is
a = length×breadth
= 9×9
= 81 square cm.

Question 4.
DIG DEEPER!
A school has two rectangular playgrounds that each have the same perimeter. The first playground is shown. The second has a lesser area than the first. Draw one way the second playground could look.

The school builds another playground. It has the same perimeter as the first. The third playground has a greater area than the first. Draw one way the third playground could look
Answer:

Explanation:

Review & Refresh

Question 5.
2 × 30 = ___
Answer:
60

Explanation:
On multiplying 2 × 30 we will get 60.

Question 6.
6 × 20 = ___
Answer:
120

Explanation:
On multiplying 6 × 20 we will get 120.

Question 7.
3 × 90 = ___
Answer:
270

Explanation:
On multiplying 3 × 90 we will get 270.

Lesson 15.5 Same Area, Different Perimeters

Explore and Grow

Use color tiles to create two different rectangles that each have an area of 18 square units. Then draw your rectangles and label their dimensions. Do the rectangles have the same perimeter? Explain how you know

Answer:
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Explanation:

In the above figure, we can see two colored rectangles
the length of rectangle 1 is 6 units
and the breadth of rectangle 1 is 3 units
so the perimeter of rectangle 1 is
p = 2(length+breadth)
= 2(6+3)
= 2(9)
= 18 units
and the area of rectangle 1 is
area= length×breadth
= 6×3
= 18 square units
The length of rectangle 2 is 9 units
and the breadth of rectangle 2 is 2 units
so the perimeter of rectangle 2 is
p = 2(length+breadth)
= 2(9+2)
= 2(11)
= 22 units.
and the area of rectangle 2 is
area= length×breadth
= 9×2
= 18 square units.
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Repeated Reasoning
As the perimeter increases and the area stays the same, what do you notice about the shape of the rectangle?

Think and Grow : Same Area, Different Perimeters

Example
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = 2 × 6
= _____
Perimeter = 6 + 2 + 6 + 2
= ______

Area = ___ × ___
= ____
Perimeter = ___ + ___ + ___ + ___
= ____
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft²
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft²
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 ft,
and the breadth of the rectangle is 2 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 2)
= 2(8)
= 16 ft.
And the area of the rectangle is
a = length × breadth
= 6 × 2
= 12 ft²

we can see in the above figure rectangle B and the length be 4 ft,
and the breadth be 3ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 ft.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 ft²
So the rectangle A has a lesser area.

Show and Grow

Question 1.
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft²
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft²
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 cm,
and the breadth of the rectangle is 6 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 6)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 6 × 6
= 36 cm²

we can see in the above figure rectangle B and the length be 9 cm,
and the breadth be 4 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(9 + 4)
= 2(13)
= 26 cm.
And the area of the rectangle is
area = length × breadth
= 9×4
= 36 cm²
So the rectangle A has a lesser perimeter.

Apply and Grow: Practice

Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?
Answer:

Question 2.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 24 in
and the area of the rectangle A is 20 in²
The perimeter of rectangle B is 18 in
and the area of the rectangle B is 20 in²
The rectangle B has a lesser perimeter.

Explanation:
Given the length of the rectangle is 10 in,
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 2)
= 2(12)
= 24 in.
And the area of the rectangle is
a = length × breadth
= 10 × 2
= 20 in²

we can see in the above figure rectangle B and the length be 5 in,
and the breadth be 4 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(5 + 4)
= 2(9)
= 18 in.
And the area of the rectangle is
area = length × breadth
= 5×4
= 20 in²
So the rectangle B has a lesser perimeter.

Question 3.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 12 m
and the area of the rectangle A is 8 m²
The perimeter of rectangle B is 18 m
and the area of the rectangle B is 8 m²
The rectangle A has a lesser perimeter.

Explanation:
Given the length of the rectangle is 4 m,
and the breadth of the rectangle is 2 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4 + 2)
= 2(6)
= 12 m.
And the area of the rectangle is
a = length × breadth
= 4 × 2
= 8 m²

we can see in the above figure rectangle B and the length be 8 m,
and the breadth be 1 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8+ 1)
= 2(9)
= 18 m.
And the area of the rectangle is
area = length × breadth
= 8×1
= 8 m²
So the rectangle A has a lesser perimeter.

Question 4.
DIG DEEPER!
The perimeter of a blue rectangle is 10 feet. The perimeter of a green rectangle is 14 feet. Both rectangles have the same area. Find the area and the dimensions of each rectangle.
Answer:

Explanation:
Given the perimeter of the blue rectangle is 10 ft and
the perimeter of the green rectangle is 14 ft

Think and Grow: Modeling Real Life

You have 40 square patio bricks that are each 1 foot long and 1 foot wide. You want to make a rectangular patio with all of the bricks. How long and wide should you make the patio so it has the least possible perimeter?

Draw to show:
You should make the patio ___ feet long and ___ feet wide.
Answer:
You should make the patio 8 feet long and 5 feet wide.
The least possible perimeter is 26 feet.

Explanation:

As there are 40 square patio bricks and each brick is 1 foot long and 1 foot wide
so to make a rectangular patio we need
the length of the rectangular patio be 8 feet
and the breadth of the rectangular patio be 5 feet
so the perimeter of the rectangular patio is
p= 2(length+breadth)
= 2(8+5)
= 2(13)
= 26 feet.
The least possible perimeter is 26 feet.

Show and Grow

Question 5.
Your friend has 16 square foam tiles that are each 1 foot long and 1 foot wide. He wants to make a rectangular exercise space with all of the tiles. How long and wide should he make the exercise space so it has the least possible perimeter?

Answer:
The least possible perimeter is 16 feet.

Explanation:
As there are 16 square foam tiles and each foam tile is 1 foot long and 1 foot wide
so to make an exercise space we need
the length of the exercise space be 4 feet
and the breadth of the exercise space be 4 feet
so the perimeter of the exercise space is
p= 2(length+breadth)
= 2(4+4)
= 2(8)
= 16 feet.
The least possible perimeter is 16 feet.

Question 6.
DIG DEEPER!
You and your friend each use fencing to make a rectangular playpen for a puppy. Each pen has the same area. Your pen is shown. Your friend’s pen uses less fencing than yours. Draw one way your friend could make her pen.

Your cousin makes a playpen for a puppy. His pen has the same area as your pen. Your cousin’s pen uses more fencing than yours. Draw one way your cousin could make his pen.
Answer:

 

Same Area, Different Perimeters Homework & Practice 15.5

Question 1.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of rectangle A is 4 in
and the area of rectangle A is 20 square inches.
It is not possible to draw a rectangle that has the same area and different perimeter.

Explanation:
Given the length of the rectangle is 4 in,
and the breadth of the rectangle is 4 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4+4)
= 2(8)
= 16 in.
And the area of the rectangle is
a = length × breadth
= 4 × 4
= 16 square inches.

Question 2.
Structure
The dimensions of a rectangle are 4 feet by 10 feet. Which shape has the same area, but a different perimeter?

Answer:
The yellow shape rectangle has the same area and different perimeter.

Explanation:
Given the dimensions of the rectangle are 4 feet by 10 feet
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+10)
= 2(14)
= 28 feet.
The area of the rectangle is
area = length×breadth
= 10×4
= 40 square feet.
Here, we can see the yellow rectangle has a length of 8 feet
and the breadth of the rectangle is 5 feet
so the perimeter of the rectangle is
p = 2(length+beadth)
= 2(8+5)
= 2(13)
= 26 feet.
and the area of the rectangle is
area = length×breadth
= 8×5
= 40 square feet.

Question 3.
MP Reasoning
The two fields have the same area. Players run one lap around each field. At which field do the players run farther?

Answer:
As the perimeter of field A is greater than field B so in field A the players run father.

Explanation:
Let the length of field A is 10m
and the breadth of field A is 2m
so the perimeter of field A is
p = 2(length+breadth)
= 2(10+2)
= 2(12)
= 24 m.
and the area of field A is
area= length×breadth
= 10×2
= 20 square meters.
Let the length of field B is 5m
and the breadth of field B is 4m
so the perimeter of field B is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 m.
and the area of field A is
area= length×breadth
= 5×4
= 20 square meters.
As the perimeter of field A is greater than field B so in field A the players run father.

Question 4.
Modeling Real Life
You have 24 square pieces of T-shirt that are each 1 foot long and 1 foot wide. You want to make a rectangular T-shirt quilt with all of the pieces. How long and wide should you make the quilt so it has the least possible perimeter?

Answer:
The least possible perimeter is 20 feet.

Explanation:
As there are 24 square pieces of T-shirt and each was 1 foot long and 1 foot wide
so to make a rectangular T-shirt quilt with all of the pieces we need
the length of the rectangular T-shirt quilt be 6 feet
and the breadth of the rectangular T-shirt quilt be 4 feet
so the perimeter of the rectangular T-shirt quilt is
p= 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 feet.
The least possible perimeter is 20 feet.

Question 5.
DIG DEEPER!
You and Descartes each have40 cobblestone tiles to arrange in to a rectangular pathway. Your pathway is shown. Descartes’s pathway has a lesser perimeter than yours. Draw one way Descartes could make his pathway.

Newton also makes a rectangular pathway with 40 cobblestone tiles. His pathway has a greater perimeter than yours. Draw one way Newton could make his pathway.
Answer:

Review & Refresh

Identify the number of right angles and pairs of parallel sides.

Question 6.

Right angles: ___
Pairs of Parallel sides: ___
Answer:
Right angles: 1
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 1, and the pairs of parallel sides are 2.

Question 7.

Right angles: ___
Pairs of parallel sides: ____
Answer:
Right angles: 4.
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 4, and the pairs of parallel sides are 2.

Find Perimeter and Area Performance Task

You and your cousin build a tree house.

Question 1.
The floor of the tree house is in the shape of a quadrilateral with parallel sides that are 4 feet long and 10 feet long. The other 2 sides are equal in length. The perimeter is 24 feet. Sketch the floor and label all of the side lengths.

Answer:
The length of the sides is 6 feet.

Explanation:
The perimeter of the floor is 24 feet
so the length of the floor be 6 feet
as the two other sides are also equal
so the other side length also be 6 feet
and let’s check the perimeter
p = 2(length+breadth)
= 2(6+6)
= 2(12)
= 24 feet.

Question 2.
Each rectangular wall of the tree house is 5 feet tall. How many square feet of wood is needed for all of the walls?
Answer:

Explanation:

Question 3.
You cut out a door in the shape of a rectangle with sides that are whole numbers. Its area is 8 square feet. What is the height of the door?
Answer:
The height of the rectangular door is 4 feet.

Explanation:
The area of the rectangular shape door is 8 square feet
as the sides of the rectangular door are whole numbers
so the length rectangular door be 4 feet
and the breadth be 2 feet
then we can get the area 8 square feet
let’s check the area
area = length×breadth
= 4×2
= 8 square feet.
So the height of the rectangular door is 4 feet.

Question 4.
You want to paint the floor and walls on the inside of your tree house. The area of the floor is 28 square feet. Each quart of paint covers 100 square feet.
a. How many quarts of paint do you need to buy?
b. Do you have enough paint to paint the outside walls of the tree house? Explain.

Answer:
a. 2,800 square feet quarts of paint we need to buy.

Explanation:
a. The area of the floor is 28 square feet and each quart of paint covers 100 square feet, so we need to buy
28×100= 2,800 square feet quarts of paint.

b.

Find Perimeter and Area Activity

Perimeter Roll and Conquer

Directions:
1. Players take turns rolling two dice.
2. On your turn, draw a rectangle on the board using the numbers on the dice as the side lengths. Your rectangle cannot cover another rectangle.
3. Write an equation tofind the perimeter of the rectangle.
4. If you cannot fit a rectangle on the board, then you lose your turn. Play 10 rounds, if possible.
5. Add all of your rectangles’ perimeters together. The player with the greatest sum wins!

Answer:

Explanation:

Find Perimeter and Area Chapter Practice

15.1 Understand Perimeter

Find the perimeter of the figure

Question 1.

Perimeter = ___
Answer:
The perimeter of the rectangle is 18 cm.

Explanation:
In the above figure, we can see the rectangle
with a length of 5 cm,
and the breadth of 4 cm
the perimeter of the rectangle is
p = 2 (length + breadth)
= 2 (5+4)
= 2(9)
= 18 cm.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 26 ft.

Explanation:
To find the perimeter of the above figure,
we will add the lengths of all sides of the figure
the sides of the above figure is 2 ft, 8 ft, 3 ft, 2 ft, 2 ft, 2 ft, 1 ft, 1 ft, 2 ft, 3 ft
the perimeter of the above figure is
p = 2+8+3+2+2+2+1+1+2+3
= 26 ft.

Question 3.
Draw a figure that has a perimeter of 10 inches.

Answer:

15.2 Find Perimeter of Polygons

Find the perimeter of the polygon

Question 4.

Perimeter = ___
Answer:
The perimeter of the polygon is 33 cm.

Explanation:
To find the perimeter of the polygon, we will add all the sides of the polygon
so the sides of the polygon are 9 cm, 6 cm, 8 cm, 10 cm
the perimeter of the polygon is
p = 9 cm +6 cm +8 cm +10 cm
= 33 cm.

Question 5.

Perimeter = ___
Answer:
The perimeter of the figure is 27 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
so the sides of the perimeter is 5 ft, 11 ft, 7 ft, 3 ft, 1 ft
the perimeter of the figure is
p = 5 ft+11 ft+ 7 ft+3 ft+1 ft
= 27 ft.

Question 6.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 12 m.

Explanation:
Given the length of the parallelogram is 4 m
and the breadth of the parallelogram is 2 m
the perimeter of the parallelogram is
p = 2 (length + breadth)
= 2( 4 m+ 2 m)
= 2(6 m)
= 12 m

Find the perimeter of the polygon

Question 7.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 36 cm.

Explanation:
Given the length of the side of the rhombus is 9 cm
and the perimeter of the rhombus is
p = 4a
= 4× 9 cm
= 36 cm.

Question 8.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 26 in.

Explanation:
The length of the rectangle is 5 inch
and the breadth of the rectangle is 8  inch
the perimeter of the rectangle is
p = 2( length + breadth)
= 2( 5 in+ 8 in)
= 2(13 in)
= 26 in.
So the perimeter of the rectangle is 26 in.

Question 9.
Square

Perimeter = ___
Answer:
The perimeter of the square is 28 ft.

Explanation:
The length of the square is 7 ft
so the perimeter of the square is
perimeter= 4a
= 4×7
= 28 ft.

Question 10.
Modeling Real Life 
You want to put lace around the tops of the two rectangular lampshades. How many centimeters of lace do you need?

Answer:
We need 1,120 square centimeters.

Explanation:
The length of the rectangular lampshades is 35 cm
The breadth of the rectangular lampshades is 32 cm
and the area of the rectangular lampshades is
area= length×breadth
= 32×35
= 1,120 square cm.
So 1,120 square centimeters of lace you need.

15.3 Find Unknown Side Lengths

Find the unknown side length.

Question 11.
Perimeter = 22 feet

d = ____
Answer:
The length of the other side is 9 ft.

Explanation:
Given the perimeter of the above figure is 22 feet
and the length of the sides of the figure is 6 ft, 7 ft, d ft
so the perimeter of the figure is
p = 6 ft+ 7 ft+ d ft
22  ft = 13 ft + d ft
d = 22 ft – 13 ft
= 9 ft.
So, the length of the other side is 9 ft.

Question 12.
Perimeter = 31 inches

k = ___
Answer:
The length of the other side is 5 in.

Explanation:
Given the perimeter of the above figure is 31 inches
and the length of the sides of the figure is 10 in, 4 in, 12 in and k in.
so the perimeter of the figure is
p = 10 in+ 4 in+ 12 in+k in
31 in = 26 in + k in
k = 31 in – 26 in
= 5 in.
So, the length of the other side is 5 in.

Question 13.
Perimeter = 34 meters

y = ___
Answer:
The length of the other side is 7 m.

Explanation:
Given the perimeter of the above figure is 34 meters
and the length of the sides of the figure is 11 m, 8 m, 2 m, 1 m, 5 m, y m.
so the perimeter of the figure is
p = 11 m+ 8 m+ 2 m+ 1 m+ 5 m+ y m
34 m = 27 m + y m
y = 34 m – 27 m
= 7 m.
So, the length of the other side is 7 m.

Find the unknown side length.

Question 14.
Perimeter = 24 feet

k = ___
Answer:
The length of the sides of the triangle is 8 feet.

Explanation:
The perimeter of the triangle is 24 feet
and the perimeter of the triangle is
p = 3a
24 feet = 3a
a= 24/3
= 8 feet.
So, the length of the sides of the triangle is 8 feet.

Question 15.
Perimeter = 16 meters

y = ___
Answer:
The length of the sides of the square is 4 meters.

Explanation:
The perimeter of the triangle is 16 meters
and the perimeter of the triangle is
p = 4a
16 meters = 4a
a= 16/4
= 4 meters.
So, the length of the sides of the triangle is 4 meters.

Question 16.
Perimeter = 30 inches

d = ___
Answer:
The length of the sides of the pentagon is 6 inches.

Explanation:
The perimeter of the pentagon is 30 inches
and the perimeter of the pentagon is
p = 5a
30 inches = 5a
d= 30/5
= 6 inches.
So, the length of the sides of the pentagon is 6 inches.

Question 17.

Number Sense
A rectangle has a perimeter of 38 centimeters. The left side length is 10 centimeters. What is the length of the top side?
Answer:
The length of the top side is 9 cm.

Explanation:
Given the perimeter of the rectangle is 38 cm and
the left side length is 10 cm
Let the length of the top side be X, so
perimeter of the rectangle is
p = 2( length +breadth)
38 = 2(10+X)
38/2 = 10 + X
19 = 10 + X
X = 19 – 10
= 9 cm.
So the length of the top side is 9 cm.

15.4 Same Perimeter, Different Area

Question 18.
Find the perimeter and area of Rectangle A. Drawa different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ____
Area = ___

Perimeter = ____
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 5m
and the breadth of the rectangle is 2m
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 m
and the area of the rectangle is
area = length×breadth
= 2×5
= 10 square meters.

In the above image, we can see the length of the rectangle is 4 m
and the breadth of the rectangle is 3 m
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 m.
and the area of the rectangle is
area = length×breadth
= 4×3
= 12 square meters.
The rectangle B has greater area.

Question 19.
Patterns
Each Rectangle has the same perimeter. Are the areas increasing or decreasing ? Explain.

Answer:
As we can see in the above images the length of the images was increasing one by one and the breadth is decreasing, so the areas increasing or decreasing will depend upon the breadth of the rectangle. So, if the breadth is also increasing then the area will also be increasing. And if the breadth was decreasing then the area will also be decreasing.

15.5 Same Area, Different Perimeters

Question 20.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 10 in
and the breadth of the rectangle is 5 in
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(10+5)
= 2(15)
= 30 in
and the area of the rectangle is
area = length×breadth
= 10×5
= 50 square inches.

Question 21.
Reasoning
The two dirt-bike parks have the same area. Kids ride dirt bikes around the outside of each park. At which park do the kids ride farther ? Explain.

Answer:
As the length of the park B is longer, so at the park B kids rid farther than the park A.

Find Perimeter and Area Cumulative practice 1 – 15

Question 1.
A mango has a mass that is 369 grams greater than the apple. What is the mass of the mango?
A. 471 grams
B. 369 grams
C. 267 grams
D. 461 grams

Answer:
B.

Explanation:
The mass of the mango is 369 grams greater than apple

Question 2.
Which term describes two of the shapes shown, but all three of the shapes?

A. polygon
B. rectangle
C. square
D. parallelogram
Answer:
B, C, D.

Explanation:
In the above figures, we can see the parallelogram and the square. As the square is also known as a rectangle so we will choose option B also.

Question 3.
A rectangular note card has an area of 35 square inches. The length of one of its sides is 7 inches. What is the perimeter of the note card?
A. 5 inches
B. 24 inches
C. 84 inches
D. 12 inches
Answer:
The breadth of the rectangular note card is 24 inches.

Explanation:
The area of the rectangular note card is 35 square inches and the length of one of its sides is 7 inches
so the breadth of the rectangular note card is
area = length × breadth
35 = 7 × breadth
breadth = 35/7
= 5 inches.
The perimeter of the rectangular note card is
p = 2(length+breadth)
= 2(7+5)
= 2(12)
= 24 inches.
The breadth of the rectangular note card is 24 inches.

Question 4.
How many minutes are equivalent to4 hours?
A. 400 minutes
B. 240 minutes
C. 24 minutes
D. 40 minutes
Answer:
B

Explanation:
The number of minutes is equivalent to 4 hours is
4× 60= 240 minutes.

Question 5.
A balloon artist has 108 balloons. He has 72 white balloons, and an equal number of red, blue, green, and purple balloons. How many purple balloons does he have?
A. 36
B. 180
C. 9
D. 32

Answer:
9 balloons.

Explanation:
As a balloon artist has 108 balloons and he has 72 white balloons
and the remaining balloons are 108 – 72= 36 balloons
and an equal number of red, blue, green, and purple balloons
which means 36 balloons are equally divided by 4 colors of balloons, so
36÷4 = 9 balloons.
So the purple balloons are 9.

Question 6.
Which statements about the figures are true?

Answer:
The shapes have different perimeters.
The shapes have the same area.

Explanation:
The length of the side of the square is 6 in,
and the perimeter of the square is
p = 4a
= 4×6 in
= 24 in
The area of the square is a^2
= 6 in×6 in
36 in^2.

Question 7.
The graph show many students ordered each lunch option.

Part A How many students ordered lunch?
Part B Choose a lesser value for the key. How will the graph change?
Answer:
Part A: 60 students ordered lunch.
Part B: Turkey hot dog has a lesser value.

Explanation:
Part A:
The number of students who ordered lunch is
the grilled chicken was ordered by 21 students
Turkey hot dog was ordered by 9 students
A peanut butter and jelly sandwich was ordered by 12 students
the salad bar was ordered by 18 students
so the number of students who ordered lunch is
21+9+12+18= 60 students.

Part B:
The turkey hot dog was ordered by 9 students which is a lesser value.

Question 8.
Find the sum

Answer:
935

Explanation:
The sum of the above given numbers is 935

Question 9.
What is the perimeter of the figure?

A. 26 units
B. 22 units
C. 20 units
D. 16 unit
Answer:
B

Explanation:
The sides of the figure is 2,4,1,2,1,1,2,2,1,1,2,1,1,1
and the perimeter of the figure is
p = 2+4+1+2+1+1+2+2+1+1+2+1+1+1
= 22 units.

Question 10.
Which bar graph correctly shows the data?

Answer:
Graph B.

Explanation:
Graph B shows the correct graph data.

Question 11.
Which polygons have at least one pair of parallel sides?

Answer:
Red color polygon.

Explanation:
The red color polygon has one pair of parallel sides, as it is a trapezoid.

Question 12.
The perimeter of the polygon is 50 yards. What is the missing side length?
A. 41 yards
B. 10 yards
C. 91 yards
D. 9 yards

Answer:
The missing side length is 9 yards.

Explanation:
Given the perimeter of the polygon is 50 yards
Let the missing side length be X yd
and the lengths of the sides of the polygon is 15 yds, 6 yds, 13 yds, 7 yds, and X yd,
So the perimeter of the polygon is
p = 15 yd+6 yd+13 yd+ 7 yd+ X yd
50 yards = 41 yards + X Yards
X = 9 yards.
So the missing side length is 9 yards.

Question 13.
Which line plot correctly shows the data?

Answer:
A

Explanation:
Option A line plot shows the correct data.

Question 14.
Your friend is asked to draw a quadrilateral with four right angles. She says it can only be a square. Is she correct?
A. Yes, there is no other shape it can be.
B. No, it could also be a rectangle.
C. No, it could also be a hexagon.
D. No, it could also be a trapezoid.
Answer:
Yes, there is no other shape it can be.

Explanation:
Yes, she is correct. There is no other shape than the square with four right angles.

Question 15.
Which numbers round to480 when rounded to the nearest ten?

Answer:
484, 480, 478

Explanation:
The numbers that are rounded to 480 to the nearest ten is 484, 480, 478.

Find Perimeter and Area Cumulative Steam Performance Task 1 – 15

Question 1.
Use the Internet or some other resource to learn more about crested geckos.
a. Write three interesting facts about geckos.
b. Geckos need to drink water every day. Is this amount of water milliliters or liters? Explain.
c. Geckos can live in a terrarium. Is the capacity of this terrarium milliliters liters measured in or?

Answer:
a) The three interesting facts about geckos are:
i) Geckos are a type of lizards and their toes help them to stick to any surface except Teflon.
ii) Gecko’s eyes are 350 times more sensitive than human eyes to light.
iii) Some of the pieces of Geckos have no legs and look more like snakes.

b) The number of water Geckos will have is in milliliters only as Geckos will not often drink water.

c)Yes, geckos can live in a terrarium and the capacity of this terrarium is between 120 liters to 200 liters.

Question 2.
Your class designs a terrarium for a gecko.
a. The base of the terrarium is a hexagon. Each side of the hexagon is 6 inches long. What is the perimeter of the base?
b. The terrarium is 20 inches tall. All of the side walls are made of glass. How many square inches of glass is needed for the terrarium?
c. Another class designs a terrarium with a rectangular base. All of its sides are equal in length. The base has the same perimeter as the base your class designs. What is the perimeter of the base? What is the area?

Answer:
a. 36 inches.
b.

Explanation:

a.
Given the length of the sides of the hexagon is 6 inches, and
the perimeter of the hexagon is
p = 6a
= 6 × 6
= 36 inches.

Question 3.
An online store sells crested geckos. The store owner measures the length of each gecko in the store. The results are shown in the table.


a. Use the table to complete the line plot.

b. How many geckos did the store owner measure?
c. What is the difference in the lengths of the longest gecko and the shortest gecko?
d. How many geckos are shorter than 6\(\frac{1}{4}\) inches?
e.The length of a gecko’s tail is about 3 inches. How would the line plot change if the store owner measured the length of each gecko without its tail?
Answer:
b. The number of geckos the store owner measures is 24.

d. 12

Explanation:

a.

b. The number of geckos the store owner measures is 24.

d. The number of geckos shorter than 6\(\frac{1}{4}\) inches are 12.