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Change of Subject of Formula: The main intention while changing the subject of a formula is to make the variable becoming the subject stand alone and be on the left side of the expression. Want to know the procedure to change the subject of an equation or formula along with the methods used during rearranging formulae? Refer to this productive article and gain extra knowledge about math equations and formulas.

Subject of the Formula Meaning

The variables that states in other variables is known as the Subject of a Formula. Also, finding the solution of the unknown quantity using the hints and values of known variables in the given context is considered to be the subject formula.

Example:
a = lb, where a is the subject of the formula. This equation express in terms of the product of the l and b. If you want to know how to do the Change of Subject of Formula, then go with the below steps and learn efficiently along with solved examples.

How do you solve the Change of Subject Formula?

Changing the subject of the formula is simply done by rearranging the formula using the arithmetic operations and getting the required subject.

• To change the subject formula, first, we have to change its side and change operations.
• If the variable of one side moved to the other side of equal sign then the operation becomes inverse.
• Likewise, all variables need to move to the opposite side and we have to get the required subject of the formula or equation.

For Example:
Change y as a subject of the formula in x + y = 2a + 2b.
Solution:
Given that x + y = 2a + 2b
x is added to y
To solve the y as the subject of the formula, we need to subtract x on both sides ie.,
x + y – x = 2a + 2b – x
y = 2a + 2b – x
Therefore, final answer is y = 2(a + b) – x.

Do Check: Establishing an Equation

Solved Examples on Change of Subject of Formula in Mathematics

Example 1:
The volume of a cuboid is the product of the length and breadth and height of the cuboid. Change ‘h’ as a subject of the formula?

Solution:
Given that the volume of a cuboid is the product of length and breadth and height of the cuboid.
The volume of a cuboid = v
The length of the cuboid = l
The breadth of the cuboid = b
The height of the cuboid = h
v = l x b x h
Now, find the change of subject of formula with h.
To make h subject of the formula, rearrange the equation by performing common operations.
Now, divide with l x b on both sides of the expression
\(\frac { v }{ l x b } \) =\(\frac { l x b x h }{ l x b } \)
\(\frac { v }{ l x b } \) = h
Hence, h = \(\frac { v }{ l x b } \).

Example 2:
Change the subject of the formula below to Q;
P = √\(\frac { 2Qh }{ 100 } \)

Solution:
Given that P = √\(\frac { 2Qh }{ 100 } \)
Step 1: The target variable to change the subject of the formula is Q. Hence, to make Q subject, we have to remove the square root. This can be possible by applying square on both sides;
P² = \(\frac { 2Qh }{ 100 } \)
Step 2: Now, cross multiply both sides;
2Qh x 1 = P² x 100
2Qh = 100P²
Step 3: Finally, divide both sides by 2h;
\(\frac { 2Qh }{ 2h } \) = \(\frac { 100P² }{ 2h } \)
Q = \(\frac { 100P² }{ 2h } \)
Since 2 can divide 100, perform that operation for getting a final result;
Q = \(\frac { 50P² }{ h } \)
The above equation is the required equation in which the subject is ‘Q’.

FAQs on Changing the Subject of a Formula

1. How to solve the change of subject equation or formula?

In order to change the subject of a formula, first, find the new subject variable from the given expression. In the next step, apply the inverse operation to deal with other variables in the equation. At last, rearrange the formula and remain the subject of the formula on the left side of the expression.

2. What is the rule to apply while changing the subject of an equation? 

The important rule followed while rearranging formulae is Operation that is utilized to make the change of subject had to be applied to both sides of the expression.

3. How do you change the subject of the area of a circle formula? 

Subject of a Formula: A formula is a rule or a fact written with mathematical symbols. In mathematics, students find many formulas for various concepts. The main objective of the formula is to solve the problem easily & quickly. We think you all may have an idea about establishing an equation if not, look at our previous article.

The subject of a formula is the part of Changing the subject of a formula concept. Students who are looking to learn about the subject of formula and how to change it can refer to this article without any delay.

It completely helps you to understand what is meant by the subject of the formula and how to make something the subject of a formula. So, jump into the below sections and gain proper knowledge.

What is the Subject of a Formula?

A linear equation which is stated in variables and literals with mathematical operators is known as a formula. Hence, the variable that we need to calculate & discover as per the given context hints is called the Subject of the Formula.

For instance, let’s assume Newton’s law of motion as an equation ie., v²-u² = 2as

Here, v, u, a, and s are the variables such as final velocity, initial velocity, acceleration, and displacement of the particle respectively.

This equation can be rearranged as:

s = \(\frac { v²-u² } { 2a } \), s become the subject of the formula.

OR

a = \(\frac { v²-u² }{ 2s } \), a become the subject of the formula.

How to make the subject of the formula?

The steps that are involved in making one variable as a subject of the formula are as given.

• First, Isolate the variable by adding or subtracting terms near to the variable, eliminating any fractions, dividing by the coefficient of the variable, and removing a root or power of both sides of the equation.
• Rearrange the equation and the subject variable should be on the left side of the equal sign.
• In case, the same subject variable contains in multiple terms, then perform factorization or any other related operation to make an end result as a single variable left as a subject of the formula on the left side of the equal sign.

Subject of a Formula Questions | Examples on Change the Subject of the Formula

Example 1:
If z is the subject of a formula, it expresses the product of x and y. Find x as the subject of the formula?

Solution:
Given formula as per the context is z = xy
Now, we have to find out the x as a subject of the formula.
z = xy
Divide y on both sides ie., (÷)y
\(\frac { z} { y } \) = \(\frac { xy } { y } \)
\(\frac { z } { y } \) = x
Hence, x = \(\frac { z } { y } \).

Example 2:
Change the subject of the equation in terms of p: z = x² + 2y +p

Solution: 
Given equation is z = x² + 2y +p
We have to change the subject of an equation in terms of p:
z = x² + 2y +p
After applying the steps of how to subject the formula, we get the result as
p = z – x² – 2y

Example 3:
Make ‘x’ the subject of the formula, s = x + bt

Solutions:
Given expression is s = x + bt
x is added to the bt.
Firstly, subtract bt on both sides.
s – bt = x + bt – bt
s – bt = x, Hence, the subject x = s – bt.

FAQs on Rearrange Formulas or Equations

1. How to find the subject of the formula in x = 6a + 2b?

The variable that expresses in other variables is known as the subject of a formula. In the given formula, the x variable is the subject of the formula.

2. What happens when a positive variable moves to the other side of an equal sign?

When the positive variable moves to the other side of the equal sign it changes to a negative variable or subtracts.

3. What is changing the subject of the formula? 

Changing the subject of the formula is nothing but rearranging the formula and getting the different subject as per the given hints in the problem.

Factorization is a method of breaking down an entity into a product of another entity. It is simply the process of resolution of an integer or polynomial into factors such that when multiplied together results in the original polynomial or integer.

Method of Factorization is quite useful to reduce any algebraic expression or quadratic equation into simpler form rather than writing product of terms within brackets. In this article, you will get acquainted with the definition of Factorization, Necessary Factorization Formulas, Methods on How to Factorize with Examples, etc. all explained in detail.

What is meant by Factorization in Mathematics?

Factorization is nothing but writing the given expression in terms of the product of its factors and the factors can be either numbers, variables, or even algebraic expressions. To Factor is to break down a number into products of other numbers that when multiplied can result in the original number.

For Example Factorization of 36 is 9*4 where 9, 4 are the factors. Now, that you are aware of finding the factorization of numbers let us move ahead and determine the factorization of a quadratic polynomial.

Factorization in Algebra

Numbers 1, 2, 3, 4, 6, 8, 12, 24 are all factors of 24 as they divide 24 without leaving a remainder. This is a vital topic to be learned for learning other concepts like simplifying expressions, simplifying fractions, and solving equations. It is referred to as Algebra Factorization.

General Factorization Formulas List

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca

Factorization Methods

Algebraic Expressions can be factorized using 4 different methods and we have stated all of them below for your knowledge. They are as follows

• Common factors method
• Regrouping terms method
• Factorization using identities
• Factors of the form (x+a) (x+b)

Factorization using Common Factors Method

In this method, we will simply find the common factors for each term of the given expression. For Example Factorization of 4x+16 can be written as 4(x+4). Here we have taken 4 i.e. common factor for both the terms

Factorization using Regrouping of Terms Method

Regrouping of Terms is a technique in which we will rearrange the given expression on basis of like and unlike terms.

For Example 6xy + 3x + 8y + 4. Rearranging them and expanding to get the factored form 6*x*y +3*x+8*y+4*1
= 3x(2y+1)+4(2y+1)
Now taking out the common factor (2y+1) we have the following terms as factors
= (3x+4)(2y+1)

Factorization using Identities

For Factorization using Identities we can factorize the algebraic expression as follows

Example: Factorize 4x²-16
Using Algebraic Identities we know a² – b² = (a – b)(a + b)
we can write 4x²-16 = (2x)² – (4)²
= (2x+4)(2x-4)

Factorization of Factors in the form of (x+a) (x+b)

If a given expression is in the form of x² + (a + b) x + ab then its factors can be written as (x+a) (x+b).

For Example: x² + 7x + 12
We can rewrite the given expression as x² + (4+3)x + 4.3
Now, after comparing with the general expression form we get
a+b = 7 ….(i)
ab = 12 ….(ii)
a = 4, b =3 both satisfies the given condition thus we can write the factors in the form of (x+4)(x+3)

Factorization Method Examples

Example 1.
Factorize the Quadratic Polynomial x² + 5x + 6?
Solution:
Given Quadratic Polynomial = x² + 5x + 6
We can rewrite the given quadratic polynomial as x² + (2+3)x + 2.3
After comparing with the general expression we have
a+b = 5 ….(i)
ab = 6 ….(ii)
2, 3 satisfies the given quadratic polynomial when substituted so we can write the quadratic polynomial in the form of (x+2)(x+3)

Therefore, Quadratic Polynomial x² + 5x + 6 when factorized results in (x+2)(x+3)

Example 2.
Factorize x² – 16?
Solution:
Given Expression is x² – 16
Factorizing the expression using identities we have a² – b² = (a – b)(a + b)
The above expression can also be written as (x)² – (4)² = (x-4)(x+4)

Example 3.
Factorize  (8 x + 8 x³) + (x⁴ + x⁶)?
Solution:
Given Expression is (8 x + 8 x³) + (x⁴ + x⁶)
Now taking the common factors we get the equation as follows 8x(1+x²)+x⁴(1+x²)
= (8x+x⁴)(1+x²)
These are the required factors.

FAQs on Factorization

1. What is meant by Factorization?

Factorization is a technique of breaking down an entity into products of another entity.

2. What are the different methods of Factorization?

The different methods of Factorization are

• Common factors method
• Regrouping terms method
• Factorization using identities
• Factors of the form (x+a) (x+b)

3. What are Factorisation Formulas?

Formulas of Factorization are listed below

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca

Formulas and equations play an important role in solving basic to advanced mathematical problems. Each and every concept covered in middle and high school maths subject deals with equations & formulas. Having proper knowledge about them is the best way to understand the equations-related concepts. Changing the Subject of a Formula is the best chapter for the students to learn completely Establishing an Equation, subject of the formula, Framing a formula, etc.

Kids who are getting confused about how to make an equation from the given math statement? how to frame a formula? can look at the steps for setting up equations from context problems available in this article. Just dive into this page and gain proper knowledge on how to establish an equation from a word problem.

Establishing an Equation or Framing of a Formula

The relation between variables expressed by equality or inequality in the math statement context is defined as a formula. Whereas, the algebraic expression expressed by equality is called Equation.

Establishing an equation holds a few simple steps. by following those simple points you can easily frame a formula or equation. So, go through the below module and get some idea about the steps to establish an equation.

Steps on How to Establish an Equation?

1. To establish an equation, firstly, you should know that the variables of the context are symbolized by a, x, A, X, etc.
2. Now, you need to utilize or apply the context-related laws or conditions to frame equality (or inequality) between the variables.
3. Finally, we find the formula for the given context or math statement.

Examples on Framing a Formula or Equation

Example 1:
Assume a sum of $ P is spent in a bank at a simple interest rate of r% per annum for a time period of n years. After ending the n years what amount of $ A is achieved.

Solution:
We know that the context of the word problem is kind of arithmetic.
Amount = Principal + Interest.
As we are aware; Interest = \(\frac { Principal × Rate × Time }{ 100 } \)
Hence, A = P + \(\frac { P×r×n }{ 100 } \)
Finally, this is the formula or equation framed from the given context.

Example 2:
If the mathematical statement is Amount (A) is equal to the subtraction of the Interest (I) and Principal (P) then establish an equation.

Solution:
From the given context, the equation or formula that can be framed is A = I – P.

Example 3:
Thrice a number is 90. Find the equation and the value of the number.

Solution:
We know that a number multiplied by three (thrice) equals 90. Here, x can be taken as an unknown number.
The equation is written as:
3x= 90
Now, calculate the value of x ie.,
x = 90 ÷ 3
x = 30
Hence, the value of the number x is 30.

FAQs on Setting Up Equations

1. What is a formula?

A formula is nothing but an equation that expresses a relationship between two or more qualities utilizing literals and symbols.

2. What is the definition of the equation in maths?

A simple statement with two expressions, one on each side of an equal sign in math is known as an equation.

3. What are the basic steps to set up an equation?

• Read what the question is asking.
• Draft the relevant data in simple statements.
• Assign symbols to unknown values that need to be found.
• Discover how the statements are associated with each other mathematically.

4. How do you explain an equation?

5. What are the 4 steps to solve an equation?

Adding, Substracting, multiplication, and division are the four major ways to solve one-step equations.

Estimating numbers is a basic concept that students need to learn at the primary level itself. The result of estimation is less accurate and easy to use especially while doing basic arithmetic operations. With the help of the Worksheet on Estimate, you can better understand the principles of Estimation. Practice the Problems in Estimation Worksheet PDF meticulously and solve problems fastly in your exams. 6th Grade Math Students can enhance their math skills by answering the questions from the Estimation Worksheets PDF.

Also, Read:

• Estimating Product and Quotient
• Estimate to Nearest Thousands
• Estimate to Nearest Hundreds

Estimate Worksheet with Answers

Problem 1:
Estimate the following numbers to nearest Tens,
(i) 8
(ii) 24
(iii) 96
(iv) 729

Problem 2:
Estimate to the nearest Hundreds,
(i) 678
(ii) 3640
(iii)945154

Problem 3:
What is the value of the Estimate to the nearest 1000,
(i) 8912
(ii) 473178

Problem 4: 
Estimate the sum 812 and 752 to the nearest hundred. What is the final sum value?

Problem 5:
What is the estimating difference tens value of 421 and 120

Problem 6:
What is the Estimated Sum value of 52, 94?

Problem 7:
A shopkeeper has 81 packets of chocolates. If each packet has 27 chocolates, then how many chocolates are there in the shop?
(i) 3600 (ii) 2400

Problem 8:
A school track is 13.96 meters wide. It is divided into 7 lanes of equal width. How wide is each lane?

Problem 9:
A museum has 318 marble jars. Each jar has 129 marbles. Find the total number of marbles in jars in the museum?

Problem 10:
Estimate the sum of 59, 69, 48.

From this page, you will learn about Estimating Product and Quotient. Estimation is nothing but taking the values that are approximate to the original number. This Estimating value helps your child to improve mental math. Estimation of products is nothing but rounding the given factors to the required place value.

You can get the approximate value of the product of numbers with an estimation of products. The quotient means the number of times a division is completed fully, and the remainder is the quantity that doesn’t go fully into the divisor. Here, you will learn about how to estimate the quotient of two numbers easily by referring to the solved example questions.

Do Refer:

• Estimate to Nearest Thousands
• Estimate to Nearest Hundreds
• Estimate to Nearest Tens

Estimating Product and Quotient – Definitions

Estimating Product:  It is a process of rounding the given factors to the required place values. In estimating product, first, we need to round off the multiplier and the multiplicand to the nearest tens, hundreds, or thousands and then multiply that rounded numbers.

Estimating Quotient: Estimating Quotient means it’s not doing the division on the actual numbers. First, you need to round off the dividend and divisor to the nearest 10 and continue the division process we will get the estimated quotient value.

How to Estimate Product and Quotient?

The below are the steps to estimating the quotient of two given numbers. The steps are:
Step 1: If the numbers have two or more digits, then round off the numbers to the nearest tens, hundreds, and or thousands.
Step 2: Next, divide those numbers to get the quotient.
Step 3: The obtained quotient is called the estimated quotient of the given numbers.

The following are the steps to find out given numbers of products using Estimation of Products. The steps are as follows:
Step 1: First, take the given number. Next, we have to estimate the multiplier and the multiplicand to the nearest tens, hundreds, or thousands.
Step 2: Finally multiply the rounded numbers and get the obtained number.

Advantages of Estimating Product and Quotient

There are many benefits of estimating product and quotient. A few of advantages are listed below,

• Estimating product and quotient helps to improve mental math.
• Your fluency in mathematical calculations will be improved.
• You can easily understand the concept of rounding off numbers in the number system by learning the concept of estimation.

Problems on Estimating Product and Quotient

Problem 1:
Estimate quotient when 1140 is divided by 40

Solution: 
Given the value is 1140 and 30.
Now, we have to find the estimating quotient value.
First, round off 1140 to the nearest 100 that is 1200.
Now, divide 1200 by 30
So, the value is
1200 ÷ 30 = 40
Therefore, estimating the quotient of given numbers is 40.

Problem 2:
Estimate the products of 43 and 67.

Solution:
As, given in the question the values are 43, 67.
Now, estimate the given numbers to the nearest tens, hundreds, or thousands.
The number 43 is between 40 and 50. But the 43 is near to 40 compared to 50. Therefore, 43 is rounded down to 40.
Next, the number 67 is between 60 and 70. But the 67 is near to 70 compared to 60. Therefore, 67 is rounded up to 70.
So, Multiply 40 and 70.
40 × 70 = 2800.
Hence, the estimated product is 2800.

Problem 3: 
Estimate the following product to the Nearest Ten and Hundred. The product value is 242 × 368

Solution:
As given in the question, the product value is 242 x 368
Now, we have to find the nearest tens value and hundreds value.
(i) The given numbers are 242 and 368.
Now, estimate the given numbers to the nearest hundreds.
So, the number is 242, in between 200 and 300. But the 242 is near to 200 compared to 300. Therefore, 242 is rounded down to 200.
Next, the number 368 is between 300 and 400. But the 368 is near to 400 compared to 300. Therefore, 368 is rounded up to 400.
Multiply 200 and 400.
200 × 400 = 80,000.
Therefore, the estimated product is 80,000.

(ii) The given numbers are 242 and 368.
Now, estimate the given numbers to the nearest tens, hundreds, and thousands.
The number is 242 between 240 and 250. But the 242 is near to 240 compared to 250. Therefore, 242 is rounded down to 240.
Next, the number is 368 between 360 and 370. But the number 368 is near 370 compared to 360. Therefore, 368 is rounded up to 370.
Multiply 240 and 370.
240 × 370 = 88,800.
Thus, the estimated product is 88,800.

Problem 4: 
Find the estimated value of 5160 ÷392.

Solution: 
As given in the question, the value is 5160 ÷392
Now, let us round 5160 and 392 separately and then divide to get the estimated value.
First, take the value 5160,
It is a four-digit number. So, we can round off this to a maximum of the nearest 1000′s.
Now try to round off evenly by 100′s. Let us round off 5160 to 100′s.
First the digit to the hundreds place. Next, determine the digit to its right. If the digit is 5 or greater than 5, then add 1 to it and place zeros in tens place and units place. If the digit is less than 5, then leave it as it is.
So, 6 is greater than 5 then the value is 5200.
Next, the given value is 392. 9 is greater than 5. So, we have to increase 1 by hundreds place digit and zeros in the units place and tens place digit. The value is 400.
Thus, estimating the numbers 5160 and 392 to the nearest hundreds is 5200, and 400.
Now divide 5200 by 400, the quotient value is 13.
Therefore the estimated quotient value is 13.  

Problem 5:
Estimate the product 93 × 46. Write the value.

Solution:
Given numbers are 93 and 46.
Now, estimate the given numbers to the nearest tens, hundreds, and thousands.
The number 93 is between 90 and 100. But the 93 is near to 90 compared to 100. Therefore, 93 is rounded down to 90.
Next, the number 46 is between 40 and 50. But the number 46 is near to 50 compared to 40. Therefore, 46 is rounded up to 50.
Now, Multiply 90 and 50.
90 × 50 = 4500.
Therefore, the estimated product is 4500.

FAQs on Estimating Product and Quotient

1. How can you check if a quotient is reasonable or not using the Estimation?

The following are used for estimation to check if a quotient is reasonable or not:

• • First, Round off the divisor and dividend to the nearest 10 or 100 depending on the number of digits.
  • Next, we have to divide the rounded dividend by divisor to get the estimated quotient.
  • Finally, compare the estimated value and exact answers to check whether the answer is reasonable or not.

2. What is a quotient?

Quotient means the answer to a division problem. In the division parts, you have to divide the dividend to get the quotient.

An integer is a number that doesn’t have decimal or fractional parts. The set of positive and negative numbers including zero are called integers. A number line is a horizontal straight line in which the integers are placed at equal intervals. Both ends of the number line extend indefinitely at both ends. This article is helpful for the 6th-grade math students to understand the math concept i.e number line easily. Check the steps on how to represent integers on the number lines, solved examples.

Do Refer:

• Adding Integers
• Subtracting Integers

What is a Number Line?

A number line is a graphical representation of numbers on a straight horizontal line. The numbers are placed at equal intervals in a number line. Mainly it can be used for comparing and ordering the numbers. We can represent all real numbers on a number line easily. Using the number line, you can also perform arithmetic operations of numbers such as addition, subtraction, multiplication, and division.

Integers – Definition

An integer is a number with no fractional or decimal part, from the set of positive, negative numbers including zero. The integers are represented by the symbol Z. The three types of integers are zero, positive integers, and negative integers. The set of integers are Z = {. . . -7, -6, 5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, . . . . . }.

How to Represent Integers on the Number Line?

Have a look at the simple steps for representing integers on the number line.

• Draw a horizontal line and both ends of the line must extend indefinitely.
• Place vertical lines at equal intervals on that line.
• Label one interval as zero.
• Keep positive integers on the right side of the zero, negative integers on the left side of zero.
• The opposite number like 2, -2 should be at an equal distance from zero.

The above gives the exact explanation for integers and the number line. In the following sections, you will get the example questions.

Questions on Integers and the Number Line

Question 1:
Represent the set of integers {-4, 0, 2, 5} on the number line.

Solution:
Mark the given set of integers on the number line.

Question 2:
Write the opposite integer of each of the following:
(i) -5
(ii) 6
(iii) 1
(iv) 23
(v) 108

Solution:
(i) 5
(ii) -6
(iii) -1
(iv) -23
(v)-108

Question 3:
Write all the integers between the following.
(i) -5 and 5
(ii) 2 and 8
(iii) -7 and 10

Solution:

By observing the above-number graph, we can easily solve this question.
(i) -4, -3, -2, -1, 0, 1, 2, 3, 4
(ii) 3, 4, 5, 6, 7
(iii) -6, -5, -4,-3,-2,-1, 0,1, 2, 3, 4, 5, 6, 7, 8, 9

FAQ’s on Number Line and the Integers

1. Why do we use the number line?

A number line is used to represent integers and compare them. It can also be used to perform simple arithmetic operations.

2. What is a number line example?

A number is nothing but the horizontal straight line that represents integers on it at regular intervals. It has zero in the middle with positive and negative integers on both sides.

3. What are some real-life examples of the number line?

Some of the examples of number lines are ruler, protractor, barometer, pressure gauge, scales, micrometer, and so on.

4. How many numbers can be represented in a number line?

A number line extends indefinitely on both sides. So, we can represent indefinite numbers on a number line.

Worksheet on Concept of Ratio has problems on part-to-part, part-to-whole ratios, reducing ratios, dividing quantities, generating equivalent ratios, and more. Practice the questions in the Ratio Worksheet and solidify your understanding of the topic. We have included several questions in different formats to keep your learning process engaging and interesting. These Math Worksheets on Ratio Concept follow a stepwise format for explaining the questions so that you don’t feel any difficulty in understanding them.

Refer More:

• Worksheet on Basic Problems on Ratio
• Worksheet on Ratio in Simplest Form

Ratio Worksheets with Answers

I. A group of friends went out for dinner. 13 of the diners ordered vegetarian food and 15 ordered non-vegetarian food. What is the ratio of the number of vegetarian meals to the number of nonvegetarian meals?

II. In the group of 60 people, 40 people are educated and the remaining people are not educated. What is the ratio of the number of people who are not educated to those who are educated?

III. 150 employees were working on the computer and 50 employees were playing games on their computers. What is the ratio of the number of employees playing games on the computer to the number of employees working on the computer?

IV. The ratio of coins to notes in the handbag is 2: 5. If there are a total of 12 coins, find the number of notes in the handbag?

V. In a minibus there are 30 seats, there are 18 occupied seats on the bus, remaining are empty. What is the ratio of the number of occupied seats to the number of empty seats?

Vi. In a box, there are oranges and apples. The ratio of oranges and apples is 3:5. If there are 18 oranges, find the number of apples?

Vii. Jay carries a bag of rice which weighs 50 kilograms. If he is going to reduce his bag weight in the ratio 6 : 5, find his new weight of the bag?

Viii. If the angles of a triangle are in the ratio 4:6:10, then find the angles?

IX. Sanjay, Sunil, and Sudheera are three friends. The ratio of average salaries of A and B is 3 : 5and that between A and C is 7: 8. Find the ratio between the average salaries of B and C?

X. Two numbers are respectively 40% and 60% are more than a third number, Find the ratio of the two numbers?

In the Worksheet on Basic Problems on Proportion, you have questions on proportion, continued proportion, finding mean proportional between the numbers, simple proportions, proportions with decimals, etc. Practice the Questions on Proportion Problems Worksheet on a regular basis and enhance your math skills. Answering the Problems in the Proportion Worksheet with Answers students can develop practical skills that are necessary for day-day-life. Try to solve as much as you can and have a clear understanding of the topics within.

Do refer:

• Worksheet on Ratio Problems
• Worksheet on Basic Problems on Ratio
• Worksheet on Ratio in Simplest Form

Solving Proportions Worksheets PDF

I. Find the value of x in each of the following proportions:
(i) x : 6 = 3 : 9
(ii) 30 : x = 6 : 2
(iii) 3 : 9 = x : 6
(iv) 3 : 2 = x : 4
(v) 5 : 2 = 15 : x
(vi) 6 : 8 = 3 : x

II. Find the mean proportional between:
(i) 0.5 and 3.8
(ii) 0.7 and 8.5
(iii)  16 and 25

III. Check whether the following quantities form a proportion or not:
(i) 45:25=35:15
(ii) 3:7=6:14
(iii) 6:3=8:4

IV. Find the unknown value of the following proportion:
i. 3x+2:7=x+4:3

V.  If x : y = 3 : 4 and y : z = 6 : 7, find x : y : z.

VI. If m : n = 2 : 7 and n : s = 3 : 8, find m : s.

VII. Verify if the ratio 2:4::4:8 is proportion?

VIII. If the third proportion of the two numbers is 24. The first number is 6, then find the second number?

IX. One piece of pipe 10 meters long is to be cut into two pieces, with the lengths of the pieces being in a 2 : 3 ratio. What are the lengths of the pieces?

X. The time taken by a vehicle is 2 hours at a speed of 40 miles/hour. What would be the speed taken to cover the same distance at 4 hours?

XI. In a construction company, a supervisor claims that 6 men can complete a task in 36 days. In how many days will 15 men finish the same task?

XII. Suppose x and y are in an inverse proportion such that when x = 100, y = 3. Find the value of y when x = 150 using the inverse proportion formula?

Are you feeling difficulty in calculating the elapsed time? Don’t worry we are here to help you to overcome the difficulties in calculating the time. Before that, you have to know what the elapsed time is. Elapsed time is the calculation of time that passes from starting off a program till its end.

It is very important to know how to calculate the elapsed time to know time. There are different techniques to calculate the elapsed time. One method is the time interval between the start of an event to the end of the event. Another method is the number line on which we break up the time intervals. We suggest the students of 5th grade go through the article and solve the given problems. Learn the concept with suitable examples and score good marks in the exams.

See More: Adding and Subtracting Time

What is Elapsed Time in Math?

Elapsed time is the amount of time duration from the start of the event to the finish of the event. In simple words, the elapsed time is the time that goes from one time to another time.

How to Calculate Elapsed time?

1. For solving the elapsed time first we would find the starting time and ending time.
2. Second, counts the hours and minutes between the starting point to noontime and from noontime to ending time.
3. The third is finding out the elapsed time by adding durations.

Elapsed Time Examples

Example 1.
Find the elapsed time from 7.00 am to 8.00 pm?
Solution:
Given that,
Starting time = 7.00 am
Finishing time = 8.00 pm
The difference between 7.00 am to 12 noon = 5 hours
The difference between 12 noon to 8.00 pm = 8 hours
Duration time = 5 hours + 8 hours = 13 hours.
Duration time is 8 hours

Example 2.
Find the elapsed time from 7 hours 30 minutes to 3 hours 20 minutes?
Solution:
7 hours 30 minutes – 3 hours 20 minutes
7 hours – 3 hours = 4 hours
30 minutes – 20 minutes = 10 minutes
The elapsed time = 4 hours + 10 minutes = 4 hours 10 minutes

Example 3.
What time would it be 3 hours 30 minutes after 8 am?
Solution:
Given that,
Present time = 3 hours 30 minutes
After time = 8 am = 8 hours
Therefore 3 hours 30 minutes + 8 hours
Time would be 11 hours 30 minutes.

Example 4.
The starting time is 11.25 am and the finishing time is 3.40 pm. What is the duration of the time?
Solution:
Given that,
Starting time = 11.25 am
Finishing time = 3.40 pm
The difference between 11.25 am to 12 noon = 35 minutes
The difference between 12 noon to 3.40 pm = 3 hours 40 minutes.
Duration time = 35 minutes + 3 hours 40 minutes
35 minutes + 40 minutes = 1 hour 15 minutes
3 hours + 1 hour 15 minutes = 4 hours 15 minutes
Therefore, the Duration time is 4 hours 15 minutes.

Example 5.
If the movie starts at 4 pm and ends at 8 pm. How long is the movie?
Solution:
Given that,
Starting time of the movie = 4 pm
Finishing time of the movie = 8 pm
Duration of the movie time = 4 pm – 8 pm = 4
Thus, the duration of the movie is 4 hours.

FAQs on Elapsed Time

1. What is elapsed time?

An elapsed time is the amount of time taken to travel or to start from one place to another place.

2. How do you calculate elapsed time?

1. First count in minutes from earlier time to the nearest hour.
2. Count on hours to the hour nearest to the later time.
3. Then count in minutes to reach the later time.

3. How do you subtract elapsed time?

In order to subtract the time, subtract the minutes and then subtract the hours.