## Worksheet on Types of Ratios

I. Find the compound ratio of the following ratios.

(i) 2 : 3 and 5 : 12
(ii) m : n and n : p
(iii) (a−b): (a+b),(a−b)2: (a+b)2
Solution:
(i)  Given that,
2 : 3 and 5 : 12
Compound ratio of 2 : 3 and 5 : 12 is
=2*5:3*12
=10:36
=5:18
(ii) Given that,
m : n and n : p
Compound ratio of m : n and n : p
=m*n:n*p
=mn/np
=m/p
(iii) Given that,
(a−b): (a+b),(a−b)2: (a+b)2
Compound ratio will be
=(a-b)(a−b)2: (a+b)(a+b)2
=(a-b)3:(a+b)3
II. Find the compound ratio of each of the following three ratios.
(i) 1 : 2, 4 : 7 and 7 : 8
(ii) a: b, b: c, and c : d
(iii) 2x : 3y, ab : y2 and y : b
Solution:
(i) Given that,
1: 2, 4: 7, and 7: 8

## Worksheet on Proportion and Continued Proportion | Free Printable Proportion and Continued Proportion Problems with Solutions Worksheet PDF

The worksheet on proportion and continued proportion help students to practice and acquire more knowledge on the concept of proportion where it also assists them in real-life incidents too. The proportion is a mathematical comparison of two numbers, and also we can compare the four quantities. Proportions are represented by the symbol “::” or “=”. If three quantities are in continued proportion then the ratio between the first and second quantity of ratio is equal to the second and the third quantity of the ratio.

For example, if the three quantities a, b, and c are continued proportion, then a: b = b: c i.e., $$\frac{a}{b}$$ = $$\frac{b}{c}$$. This free printable proportion and continued proportion worksheet pdf with answers is fun to practice and express the following ratios whether they are proportion and continued proportion or not.

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## Worksheet on Proportion and Continued Proportion PDF with Answers

1. Check the following numbers are in proportion or not.
(i) 1.5, 5.5, 2.7, 9.9
(ii) 2.5, 4.5, 1$$\frac{1}{4}$$, 2$$\frac{1}{4}$$

Solution:

(i) Given numbers are 1.5, 5.5, 2.7, 9.9.
For example, to say the numbers are in proportion or not. We have to prove a: b = c: d.
Now, we find 1.5: 5.5 = 2.7: 9.9.
1.5: 5.5 = $$\frac{1.5}{5.5}$$ = $$\frac{1.5×10}{5.5×10}$$ = $$\frac{15}{55}$$ = $$\frac{3}{11}$$
2.7: 9.9 = $$\frac{2.7}{9.9}$$ = $$\frac{2.7×10}{9.9×10}$$ = $$\frac{27}{99}$$ = $$\frac{3}{11}$$
Thus, $$\frac{1.5}{5.5}$$ = $$\frac{2.7}{9.9}$$.
Therefore, the numbers 1.5, 5.5, 2.7, 9.9 are in proportion.

(ii) Given numbers are 2.5, 4.5, 1$$\frac{1}{4}$$, 2$$\frac{1}{4}$$.
As you know earlier, the numbers to be in proportion when a: b = c: d.
Now, 2.5: 4.5 = $$\frac{2.5}{4.5}$$ = $$\frac{2.5×10}{4.5×10}$$ = $$\frac{25}{45}$$= $$\frac{5}{9}$$.
1$$\frac{1}{4}$$: 2$$\frac{1}{4}$$ = $$\frac{5}{4}$$: $$\frac{9}{4}$$ = $$\frac{5}{4}$$× 4: $$\frac{9}{4}$$× 4 = 5: 9 = $$\frac{5}{9}$$.
Hence, $$\frac{5}{9}$$ = $$\frac{5}{9}$$.
Therefore, the numbers 2.5, 4.5, 1$$\frac{1}{4}$$, 2$$\frac{1}{4}$$ are in proportion.

2. Find x in the following proportions.
(i) 4: 3 = 8: x
(ii) 2.5: x = 8.1: 1.4
(iii) 1.0 : 2 = x: 4

Solution:

(i) Given 4: 3 = 8: x
⇒ $$\frac{4}{3}$$ = $$\frac{6}{x}$$
⇒ 4x = 8×3
⇒ 4x = 24
⇒ x = $$\frac{24}{4}$$
⇒ x = 6.
Thus, the value of x is 6.

(ii) Given 2.5: x = 8.1: 1.4
⇒ $$\frac{2.5}{x}$$ = $$\frac{8.1}{1.4}$$
⇒ $$\frac{25}{10x}$$ = $$\frac{81}{14}$$
⇒ 25 × 14 = 81 × 10x
⇒ 350 = 810x
⇒ x = $$\frac{810}{350}$$
⇒ x = 2.3
Hence, the x value is 2.3.

(iii) Given 1.0 : 2 = x: 4
⇒ $$\frac{1.0}{2}$$ = $$\frac{x}{4}$$
⇒ $$\frac{10}{20}$$ = $$\frac{x}{4}$$
⇒ 10 × 4 = 20x
⇒ 20x = 40
⇒ x = $$\frac{40}{20}$$
⇒ x = 2.
The value of x is 2.

3. Find the following are in continued proportion or not?
(i) 0.2, 2.4, 4.0
(ii) 4, 6, 9

Solution:

(i) Given numbers are 0.2, 2.4, 4.0.
If we have three numbers to find whether they are continued proportion or not, thus a:b = b:a.
Now, 0.2: 2.4 = 2.4: 4.0
⇒ $$\frac{0.2}{2.4}$$ = $$\frac{2.4}{4.0}$$
⇒ $$\frac{2}{24}$$ = $$\frac{24}{40}$$
⇒ $$\frac{1}{12}$$ ≠ $$\frac{3}{5}$$
Hence, the given numbers are not in continued proportion because 0.2: 2.4 ≠ 2.4: 4.0.

(ii) Given numbers are 4, 6, 9.
As we know, a:b = b:a
Now, 4: 6 = 6:9
⇒ $$\frac{4}{6}$$ = $$\frac{6}{9}$$
⇒ $$\frac{2}{3}$$ = $$\frac{2}{3}$$
Thus, the given numbers 4, 6, and 9 are in continued proportion because 4: 6 = 6: 9.

4. Find m, if the numbers 8, 4, and m are in continued proportion.

Solution:

Given the numbers 8, 4, and m are in continued proportion.
8: 4 = 4: m
⇒ $$\frac{8}{4}$$ = $$\frac{4}{m}$$
⇒ 8m = 16
⇒ m = $$\frac{16}{8}$$
⇒ m = 2.
Hence, the value of m is 2.

5. Find the fourth proportion of 6, 2, and 9.

Solution:

Given proportion numbers are 6, 2, and 9.
Let the fourth proportional be p.
Now, according to given problem 6, 2, 9, and p are proportionality.
Thus,
$$\frac{6}{2}$$ = $$\frac{9}{p}$$
⇒ 6p = 9 × 2
⇒ 6p = 18
⇒ p = $$\frac{18}{6}$$
⇒ p = 3.
Therefore, the fourth proportional number is 3.

6. Find the third proportion to the following numbers.
(i) 4, 6
(ii) 1$$\frac{2}{3}$$, $$\frac{4}{5}$$
(iii) 3.4, 7.2

Solution:

(i) Given proportion numbers 4, 6.
Now, find the third proportion.
Let the third proportion be k.
Thus,
$$\frac{4}{6}$$ = $$\frac{6}{k}$$
⇒ 4k = 6 × 6
⇒ 4k = 36
⇒ k = $$\frac{36}{4}$$
⇒ k = 9.
Therefore, the third proportion number is 9.

(ii) Given proportion numbers are 1$$\frac{2}{3}$$, $$\frac{4}{5}$$
Let the third proportion be k.
Thus,
$$\frac{5}{3}$$ × $$\frac{5}{4}$$ = $$\frac{4}{5}$$ × $$\frac{1}{k}$$
⇒ $$\frac{25}{12}$$ = $$\frac{4}{5k}$$
⇒ 25 × 5k = 4 × 12
⇒ 125k = 48
⇒ k = $$\frac{48}{125}$$
Hence, the third proportion number is $$\frac{48}{125}$$.

(iii) Given proportion numbers are 3.4, 7.2.
Let the third proportion be k.
Now,
$$\frac{3.4}{7.2}$$ = $$\frac{7.2}{k}$$
⇒ $$\frac{34}{72}$$ = $$\frac{72}{10k}$$
⇒ 34 × 10k = 72 × 72
⇒ 340k = 5184
⇒ k = $$\frac{5184}{340}$$
⇒ k = 15.2
Thus, the third proportion value is 15.2.

7. Find q in the following numbers so that the numbers are proportional.
(i) 10, q, 20, 15
(ii) 8, 16, 24, q
(iii) q, 35, 15, 40

Solution:

(i) Given numbers are 10, q, 20, 15
To find the proportion numbers, we use a:b = c:d i.e., $$\frac{a}{b}$$ = $$\frac{c}{d}$$.
Now,
$$\frac{10}{q}$$ = $$\frac{20}{15}$$
⇒ 10 × 15= 20q
⇒ 20q = 150
⇒ q = $$\frac{150}{20}$$
⇒ q = 7.5
Thus, the value of q is 7.5.

(ii) Given numbers are 8, 16, 24, q.
Now,
$$\frac{8}{16}$$ = $$\frac{24}{q}$$
⇒ $$\frac{1}{2}$$ = $$\frac{24}{q}$$
⇒ q = 24 × 2
⇒ q = 48
Hence, the value of q is 48.

(iii) Given numbers are q, 35, 15, 40.
We use the proportion formula, a:b = c:d i.e., $$\frac{a}{b}$$ = $$\frac{c}{d}$$.
Now,
$$\frac{q}{35}$$ = $$\frac{15}{40}$$
⇒ 40q = 15 × 35
⇒ 40q = 525
⇒ q = $$\frac{525}{40}$$
⇒ q = 13.12
Therefore, the value of q for given proportional numbers is 13.12.

## Worksheet on Mean Proportional PDF | Free Printable Mean Proportional Problems with Answers Activity Sheet

Getting confused while solving the mean proportional problems then practice more with this Mean Proportional Worksheet pdf and learn the concept of mean proportional efficiently. Mean proportional is also known as Geometric Mean and it is not similar to the arithmetic mean. The Mean Proportion is calculated between two terms of a ratio by taking the square root of the product of those two quantities in terms of ratio.

Let us understand more about mean proportional by taking help from this free printable mean proportional of two numbers worksheet. For example, if a, b, and c are in continued proportion then b is called the mean proportional of a and c. The mean proportion is expressed as b = √ac. This free printable Worksheet on Mean Proportional with Answers PDF helps you understand the problem-solving techniques and feels fun to practice.

## Mean Proportional Worksheet PDF with Solutions

Example 1:
Find the mean proportional of the following sets of positive integers:
(i) x- y, x³- x²y
(ii) x³y, xy³

Solution:

(i) Given x- y, x³- x²y
Now, to find the mean proportion.
Let p be the mean proportional between x- y, x³- x²y.
So, x- y: p:: p: x³- x²y
Product of extremes = Product of means.
Now,
p² = (x- y)(x³- x²y)
⇒ p² = x²(x- y)(x- y)
⇒ p² = x²(x- y)²
⇒ p = √(x²(x- y)²)
⇒ p = x(x- y)
Thus, the mean proportion of x- y, x³- x²y is x(x- y).

(ii) Given x³y, xy³
Let m be the mean proportion of x³y, xy³.
So, x³y: m:: m: xy³.
Product of extremes = Product of means.
Product of extremes = x³y × xy³
Product of means = m × m = m²
m² = x³y × xy³
⇒ m² = (x³ × x) × (y³ × y)
⇒ m² = x4 × y4
⇒ m = √(x4 × y4)
⇒ m = √(x²y²)²
⇒ m = x²y²
Hence, the mean proportion of x³y, xy³ is x²y².

Example 2:
Find the mean proportional of the following:
(i) 8 and 32
(ii) 0.04 and 0.56
(iii) 4 and 25

Solution:

(i) Given 8 and 32
Let the mean proportion between 8 and 32 is a.
Now, 8: a:: a: 32
We know that, Product of extremes = Product of means.
Here, extremes are 8 and 32 and means are a and a.
So, 8 × 32 = a × a
⇒ a² = 256
⇒ a = √256
⇒ a = 16
Thus, the value of mean proportion ‘a’ is 16.

(ii) Given 0.04 and 0.56
Let the mean proportion be p.
Now, 0.04: p:: p: 0.56
We know that, Product of extremes = Product of means.
The extremes are 0.04 and 0.56 and the means are p and p.
So, 0.04 × 0.56 = p × p
⇒ p² = 0.02
⇒ p = √0.02
⇒ p = 0.14
Therefore, the value of the mean proportion p is 0.14.

(iii) Given 4 and 25
Let the mean proportion between 4 and 25 be x.
Thus, 4: x:: x: 25
We know that, Product of extremes = Product of means.
Here, the extremes are 4 and 25 and the means are x and x.
So, 4 ×25 = x × x
⇒ x² = 100
⇒ x = √100
⇒ x = 10
Hence, the mean proportion of 4 and 25 is 10.

Example 3:
If b is the mean proportion between a and c, show that a4+ a2b2+ b4/b4+ b2c2+ c4 = $$\frac{a²}{c²}$$.

Solution:

Given b is the mean proportion between a and c, then we have b² = ac.
Now, we have to prove that a4+ a2b2+ b4 / b4+ b2c2+ c4 = $$\frac{a²}{c²}$$ i.e., L.H.S = R.H.S.
LHS = a4+ a2b2+ b4/ b4+ b2c2+ c4
Let us substitute b² = ac in LHS.
LHS = a4+ a²(ac)+ (ac)²/(ac)²+ (ac)c²+ c4
⇒ LHS = $$\frac{a²( a²+ ac+ c² )}{c²( a²+ ac+ c² )}$$
⇒ LHS = $$\frac{a²}{c²}$$ = RHS
⇒ LHS = RHS
Therefore, a4+ a2b2+ b4/ b4+ b2c2+ c4= $$\frac{a²}{c²}$$.

Example 4:
Find the mean proportion of the following
(i) 4$$\frac{4}{5}$$, 2$$\frac{1}{2}$$
(ii) a²b, ab²

Solution:

(i) Given 4$$\frac{4}{5}$$, 2$$\frac{1}{2}$$
Now, change the mixed fraction into proper fraction
4$$\frac{4}{5}$$ = $$\frac{24}{5}$$
2$$\frac{1}{2}$$ = $$\frac{5}{2}$$
Let m be the mean proportion of $$\frac{24}{5}$$, $$\frac{5}{2}$$.
Product of extremes = Prouct of means
Here, the extremes are $$\frac{24}{5}$$ and $$\frac{5}{2}$$, the means are m and m.
$$\frac{24}{5}$$ × $$\frac{5}{2}$$ = m × m
⇒ m² = $$\frac{24}{2}$$
⇒ m² = 12
⇒ m = √12
⇒ m = 3.46
Thus, the value of mean proportion is 3.46.

(ii) Given a²b, ab²
Let k be the mean proportion of a²b and ab².
So, a²b: k:: k: ab².
In mean proportion, Product of extremes = Product of means.
Now, a²b × ab² = k²
⇒ k² = a³b³
⇒ k² = (ab)¹ × (ab)²
⇒ k = √((ab)¹)²
⇒ k = ab
Therefore, ab is the mean proportion of a²b, ab².

## Worksheet on Ratio of Two or More Quantities | Sharing a Quantity in a Given Ratio Worksheet with Answers PDF

Worksheet on Ratio of Two or More Quantities gives information of ratio in different types and quantities. The ratio indicates their relative sizes and always expresses the ratio numbers in the lowest terms possible. A ratio compares two or more different quantities that have the same units of measure. In this ratio of two or more quantities word problems worksheet pdf, students can get the questions that are relevant to how to divide a quantity into two parts or more than the quantities of a given ratio with detailed solutions.

Practice with these activity sheet questions to have the hold on the concept of a ratio if we have more than two quantities of the same kind. This free printable activity sheet on the ratio of two or more quantities is a fun-learning workbook to solve all the questions. Also, students can able to create the questions themselves in different types by using this ratio of two or more quantities worksheet.

Do Check:

## Free Printable Ratio of Two or More Quantities Worksheet PDF with Answers

Example 1:
Find p: s, if p: q = 4: 6 and r: s = 3: 12.

Solution:

Given ratios are p: q = 4:6 and r: s = 3: 12
Now, to find the ratio of p: s.
As we know, if there is a ratio of quantities a: b and c: d. To get a: d = $$\frac{a}{b}$$ × $$\frac{c}{d}$$ = $$\frac{a}{d}$$.
Similarly, we apply the same formula
p: s = $$\frac{p}{q}$$ × $$\frac{r}{s}$$
⇒ p: s = $$\frac{4}{6}$$ × $$\frac{3}{12}$$
⇒ p: s = 1: 6
Therefore, the value of the ratio p: s is 1: 6.

Example 2:
If x: y = 5: 4 and y: z = 2: 6, then find (i) x: z and (ii) x: y: z.

Solution:

(i) Given x: y = 5: 4 and y: z = 2: 6
To get x: z, we use the formula, x: z = $$\frac{x}{y}$$ × $$\frac{y}{z}$$
So,
x: z = $$\frac{5}{4}$$ × $$\frac{2}{6}$$
⇒ x: z = $$\frac{5×2}{4×6}$$
⇒ x: z = $$\frac{10}{24}$$
⇒ x: z = $$\frac{5}{12}$$
Thus, the ratio x: z = 5: 12.

(ii) Given ratio x: y = 5: 4 and y:  = 2: 6
Now, to find the ratio of x: y: z.
Here, the values of y are not the same. To get the ratio of x: y: z, we have to multiply the value of y with a number to get both the values equal.
$$\frac{x}{y}$$ = $$\frac{5}{4}$$ × $$\frac{1}{1}$$ = $$\frac{5}{4}$$
$$\frac{y}{z}$$ = $$\frac{2}{6}$$ × $$\frac{2}{2}$$ = $$\frac{4}{12}$$
Thus, the ratio x: y: z = 5: 4: 12.

Example 3:
Divide ₹ 3876 in the ratio $$\frac{4}{5}$$: 1$$\frac{3}{5}$$.

Solution:

Given ratio $$\frac{4}{5}$$: 1$$\frac{3}{5}$$ i.e., $$\frac{4}{5}$$: $$\frac{8}{5}$$.
Therefore, the ratio becomes 4: 8 = 4+8 = 12.
Let the common multiple be x.
12x = 3876
x = $$\frac{3876}{12}$$
⇒ x = 323
Now, 4x = 4×323 = 1292
8x = 8×323 = 2584
Thus, the ratio is divided into ₹1292 and ₹2584.

Example 4:
If a: b = 3: 7, then find the ratios of the following:
(i) (2a+ 5b): (3a+ 6b)
(ii) (a+ 2b): (7a- b)

Solution:

(i) Given ratio a: b = 3: 7
The given equation is (2a+ 5b): (3a+ 6b).
Now, substitute the values of a and b in the above equation.
(2a+ 5b): (3a+ 6b) = ((2×3)+(5×7)): ((3×3)+(6×7)) = (6+35): (9+42) = 41: 51
Thus, the ratio of the equation (2a+ 5b): (3a+ 6b) is 41: 51.

(ii) Given equation is (a+ 2b): (7a- b)
The values of a and b were given are 3 and 7.
Now, substitute the values in the above equation.
(a+ 2b): (7a- b) = (3+(2×7)): ((7×3)-7) = (3+14): (21-7) = 15: 14
Hence, the ratio is 15: 14.

Example 5:
Find the value of m, if (m-1): (m+3) is the reciprocal ratio of 12: 8.

Solution:

The reciprocal ratio of 12: 8 is 8: 12.
Now, 8: 12 = (m-1): (m+3)
⇒ $$\frac{8}{12}$$ = $$\frac{m-1}{m+3}$$
⇒ 8(m+3) = 12(m-1)
⇒ 8m+24 = 12m-12
⇒ 8m- 12m = -12 – 24
⇒ -4m = -36
⇒ m = $$\frac{36}{4}$$
⇒ m = 9.
Therefore, the value of m is 9.

Example 6:
If 4x = 7y = 2z, find the ratio of x: y: z.

Solution:

Given equation is 4x = 7y = 2z.
Now, 4x = 7y
⇒ x = $$\frac{7y}{4}$$ —- (i)
7y = 2z
⇒ z = $$\frac{7y}{2}$$ —- (ii)
Now,
x: y: z = $$\frac{7y}{4}$$: y : $$\frac{7y}{2}$$ (from (i) and (ii))
⇒ x: y: z = $$\frac{7}{4}$$: 1 : $$\frac{7}{2}$$
⇒ x: y: z = 7: 4: 14
Hence, the ratio of x: y: z is 7: 4: 14.

Example 7:
Divide $3570 in the ratio $$\frac{1}{5}$$: 1$$\frac{2}{3}$$: $$\frac{2}{15}$$. Solution: Given ratio is $$\frac{1}{5}$$: 1$$\frac{2}{3}$$: $$\frac{2}{15}$$ i.e., $$\frac{1}{5}$$: $$\frac{5}{3}$$: $$\frac{2}{15}$$. Let the common multiple be k. Therefore, $$\frac{1k}{5}$$: $$\frac{5k}{3}$$: $$\frac{2k}{15}$$ = 3570 ⇒ $$\frac{3k}{15}$$: $$\frac{25k}{15}$$: $$\frac{2k}{15}$$ = 3570 ⇒ $$\frac{30k}{15}$$ = 3570 ⇒ 2k = 3570 ⇒ k = $$\frac{3570}{2}$$ ⇒ k = 1785 Now, to find the three equal ratios $$\frac{1k}{5}$$ = $$\frac{1×1785}{5}$$ = $$\frac{1785}{5}$$ =357 $$\frac{5k}{3}$$ = $$\frac{5×1785}{3}$$ = $$\frac{8925}{3}$$ = 2975 $$\frac{2k}{15}$$ = $$\frac{2×1785}{15}$$ = $$\frac{3570}{15}$$ = 238 The ratio is divided into$357, $2975, and$238.

Example 8:
If a: b = 4: 1, b: c = 12: 7, and c: d = 7: 9, then find the triplicate ratio of a: d.

Solution:

Given ratios are a: b = 4: 1, b: c = 12: 7, and c: d = 7: 9
Now, to find the ratio of a: d.
Here,
a: d = $$\frac{a}{b}$$ × $$\frac{b}{c}$$ × $$\frac{c}{d}$$
⇒ a: d = $$\frac{4}{1}$$ × $$\frac{12}{7}$$ × $$\frac{7}{9}$$
⇒ a: d = $$\frac{4×12×7}{1×7×9}$$
⇒ a: d = $$\frac{4×4×1}{1×1×}$$
⇒ a: d = 16: 3
The ratio of a: d is 16: 3.
Triplicate ratio = 16³ : 3³ = 4096 : 27.
Therefore, the triplicate ratio of a: d = 4096: 27.

Example 9:
What number should be added to the ratio of 13: 33? So, the ratio becomes 4: 9.

Solution:

Given ratio is 13: 33
Now, we have to add the number so that we can get the ratio of 4: 9.
If we add the number 3 to the ratio of 13: 33
We got, 13+3: 33+3 = 16: 36.
So, the ratio becomes 16: 36 = 4: 9.
Thus, the number 4 should be added to the given ratio.

## Perimeter and Area of Irregular Figures

In this article, you will learn how to find the Perimeter and Area of irregular figures. An irregular shape can be of any size and length. Irregular shapes can be seen all around us, for example, a diamond shape, a kite, a leaf, a flower, etc. The Area of irregular shapes means the space occupied by the shape which is measured in square units. The Perimeter of irregular shapes is by adding the length of their sides. Any shape whose sides and angles are not of equal length is termed an irregular shape.

On this page, learn about the definition of the Area and perimeter of irregular figures, how to find the area and perimeter of irregular figures, solve example problems, and so on.

### Definition of Irregular Figures

Irregular Figures is defined as a figure that is not a standard geometric shape. An irregular shape is simply a shape where not every single side is the same length. But some irregular figures are made up of two or more standard geometric shapes. If the shape is irregular then it has angles that are not all the same size. Based on the number of sides or corners we can decide the irregular figure.

### How to find Perimeter and Area of Irregular Figures?

The following are the steps for finding the area and perimeter of irregular figures:
How to find Area of Irregular Shapes or Figures?
Step 1: First, divide the compound shape into a basic regular shape.
Step 2: Next, find each basic shape area separately.
Step 3: Then Add all the areas of basic shapes together.
Step 4: Now, write the final answer in square units.

How to find the Perimeter of Irregular Figures?
To find the perimeter of the irregular figure, we can simply add up each of its outer sides length of a shape. To find the perimeter of any shape like rectangle, square, and so on you have to add all the lengths of four sides. Consider ‘A’ is in this case the length of the rectangle and ‘B’ is the width of the rectangle.

### Solved Examples on Perimeter and Area of Irregular Figures

Example 1: The Irregular Figure is given below. Find the area of that figure.

Solution:
As given in the question, the irregular figure is given.
Now, we can break the given irregular figure. After splitting the figure we have two rectangles.
Next, we will find the area of the two rectangles. The area of the irregular figure is the sum of the areas of two rectangles.
The width of one rectangle is 12 and the length of the rectangle is 4.
Next, the width of the other rectangle is 2, but its length is not given. By using the upper rectangle length we can find the length of the lower rectangle. So the right side of the figure is the length of the upper rectangle plus the length of the lower rectangle.
Since the total length is 10 units, the right side of the upper rectangle is 4 units long. So the length of the lower rectangle will be 6 units.
So the area of the figure is,
The Area of the figure is the Area of the upper rectangle + Area of the lower rectangle
We know that the Area of the rectangle is, length x width (or) breadth.
So, the area of a figure is , lw + lw = 12(4) + 2(6).
Area of the figure is = 48 + 12 = 60sq.units.
Therefore, the total area of the figure is 60 square units.

Example 2: Find the area of the below-given irregular figure.

Solution:
As given in the question, the given figure is an irregular figure.
Now, we can break the given irregular figure. After splitting the figure, we have one triangle and one rectangle.

Next, we will find the area of the irregular figure. The area of the irregular figure is the sum of the areas of two rectangles.
The rectangle has a length of 8 units and a width of 4 units. We need to find the base and height of the triangle.
On both sides of the rectangle 4units, the vertical side of the triangle is 3 units, which is 7- 4 = 3units.
Next, the length of the rectangle is 8units, so the base of the triangle is 3units, which is 8-5= 4units.
Now, we can add the areas then we get the area of the irregular figure.
So, the Area of the figure is the Area of the rectangle + the Area of the triangle.
We know the formulas, the area of the rectangle is, length x width (or) breadth.
The area of the triangle is 1/2bh.
So, the area of a figure is , lw + 1/2bh = 8(4) + 1/2(3)(3).
Area of the figure is = 32 + 4.5 = 36.5sq.units.
Hence, the total area of the given irregular figure is 36.5square units.

Example 3:

## Worksheet on Dividing a Quantity in a Given Ratio | Dividing Quantities in Given Ratio Worksheet Pdf with Answers

Worksheet on Dividing a Quantity in a Given Ratio Concept provides problems on quantities into two parts and three parts of a given ratio. We will discuss the quantities of a given ratio in various aspects to get a clear understanding of the whole concept to the children. As we know, a ratio is a comparison of quantities of the same unit and it can be described as a fraction. Here, the ratio of X and Y is defined as X : Y = $$\frac{X}{Y}$$. The quantity X in the ratio is called antecedent and Y is the consequent.

Practice the questions given in the below Dividing quantity in a given ratio worksheet pdf and answer all complex calculations with ease. Dividing quantities in a given ratio worksheet makes you learn the concept in a fun-learning & engaging manner. This Math Dividing a Quantity in a Given Ratio Word Problems Worksheet gives a step-to-step explanation so that you don’t feel bored and difficult to study & solve the calculations.

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## Free Printable Worksheet on Dividing a Quantity in a Given Ratio

Example 1:
A number is divided into two parts in the ratio of 4: 9. If the larger part is 270, then find the actual number and the smaller part.

Solution:

The given ratio is 4: 9.
Let the numbers be 4k and 9k
Given larger part of the number is 270.
Now, we find the number by solving 9k = 270.
k = $$\frac{270}{9}$$
⇒ k = 30
Another number is 4k, substitute the k value in 4k.
4k = 4 × 30 = 120.
Thus, the smaller part number is 120 and the actual number is 390.

Example 2:
A bag contains 3 dollars, 50 cents, 4 dollars in the ratio of 5: 4: 2. The total amount is $2450. Find the number of each denomination? Solution: Let the number of each denomination be 5x, 4x, and 2x respectively. The amount of 3 dollars = 5x × 300 cents = 1500x cents The amount of 50 cents = 4x × 50 cents = 200x cents The amount of 4 dollars = 2x × 400 cents = 800x cents The total amount given = 2450 × 100 cents =245000 cents. Now, 1500x + 200x + 800x = 245000 ⇒ 2500x = 245000 ⇒ x = $$\frac{245000}{2500}$$ ⇒ x = 98 Now, we substitute the x value in each denomination. The number of 3 dollars i.e., 5x = 5×98 = 490 The number of 50cents i.e., 4x = 4×98 = 392 The number of 4 dollars i.e., 2x = 2×98 = 196 Therefore, the denominations of each number are 490, 392, and 196. Example 3: The sum of the numbers is 250, and the two numbers are in the ratio 4: 6. Find the numbers? Solution: Given two numbers in the ratio is 4: 6. Let the two numbers be 4a and 6a. Now, 4a+ 6a = 250 ⇒ 10a = 250 ⇒ a = $$\frac{250}{10}$$ ⇒ a = 25 Next, we multiply the value of ‘a’ with the two numbers. 4a = 4×25 = 100 6a = 6×25 = 150 Thus, the numbers are 100 and 150. Example 4: Divide 420 into three parts which are in the ratio 3: 6: 5. Solution: The given number is 420 and the ratio is 3: 6: 5. Now, to calculate the sum of the ratios = 3+6+5 = 14 Next, divide the three parts First part = $$\frac{3}{14}$$×420 = 3×30 = 90 Second part = $$\frac{6}{14}$$×420 = 6×30 = 180 Third part = $$\frac{5}{14}$$×420 = 5×30 = 150 Hence, 420 can be divided into 90, 180, and 150 in the ratio of 3: 6: 5. Example 5: Four workers worked for 7 hours, 6 hours, 5 hours, and 6 hours. The total wage of ₹14640 was divided among the four workers according to their number of hours worked. How much did they get for the hours they worked? Solution: Let the hours of the workers worked to be in the ratio of 7: 6: 5: 6. The sum of the ratio = 7+6+5+6 = 24hours Now, get the amount per hour = Total wage / Hours worked = ₹14640/24 = ₹610 per hour The wage of the first worker = 7×₹610 = ₹4270 The wage of the second worker = 6×₹610 = ₹3660 The wage of the third worker = 5×₹610 = ₹3050 The wage of the fourth worker = 6×₹610 = ₹3660 Therefore, each worker got the amount of ₹4270, ₹3660, ₹3050, and ₹3660. Example 6: Divide 4m 25cm in the ratio 4: 6. Solution: Given quantity 4m 25cm Now, first, we have to convert the given quantity into one unit. As we know, 1m = 100cm 4m = 4×100 = 400cm 4m 25cm = 400cm + 25cm = 425cm. Given ratio is 4: 6 The total number of parts = 4x+6x = 10x Now, 10x = 425 ⇒ x = $$\frac{425}{10}$$ = 42.5 So, The quantity of first number 4x = 4×42.5 = 170cm The quantity of second number 6x = 6×42.5 = 255cm Hence, the two quantities of the two numbers are 170cm and 255cm. Example 7: The length and breadth of a rectangle are in the ratio of 8: 5. If the breadth is 85cm, then find the length of a rectangle? Solution: Given the ratio of length and the breadth is 8: 5 and the breadth is 85cm Let the length be 8k and breadth be 5k. Now, breadth 5k = 85cm ⇒ k = $$\frac{85}{5}$$ = 17cm Here, length = 8k ⇒ 8k = 8×17 = 136cm Thus, the length of a rectangle is 136cm. Example 8: A certain sum of money is divided into three parts in the ratio 4: 3: 2. If the first part is ₹448, then find the total amount, the second part, and the third part? Solution: Given three parts ratio is 4: 3: 2. Let the amount of money be 4x, 3x, and 2x. From the given the first part is ₹448. 4x = ₹448 ⇒ x = $$\frac{448}{4}$$ = 112 Thus, x = 112 Now, 3x = 3×112 = 336 and 2x = 2×112 = 224 Therefore, the second amount = ₹336 the third amount = ₹224 The total amount of money = First amount + Second amount + Third amount = ₹448 + ₹336 + ₹224 = ₹1008. Hence, the total amount of money is ₹1008, the second part is ₹336, and the third part is ₹224. ## Word Problems on Proportion | Free Printable Proportion Word Problems with Solutions PDF The word problems on the proportion aid you in a mathematical comparison between two numbers. The proportion gets majorly based on ratio and fractions. This article on Proportion Word Problems Worksheet with Answers PDF gives you a various number of proportion problems. At the end of this page, you can get clear knowledge of the concept of proportion. It says when two ratios are equivalent they are in proportion. Proportion encourages solving many real-life problems. The ratio and proportion are key foundations to grasp the various concepts in mathematics. Proportions are denoted by the symbol “::”, “=”. This free printable Worksheet on Word Problems on Proportion provides various types of questions with answers to make you understand clearly how to solve the proportions problems in exams. By solving proportion examples with answers pdf, you can also improve your problem-solving skills & math skills. Do Refer: ## Proportion Word Problems with Answers Example 1: If the numbers are 4, 15, 7, and 20. What number should be added to make the numbers proportional? Solution: Given numbers are 4, 15, 7, and 20. Let the needed number be k. Now, we write the numbers according to the problem 4+k, 15+k, 7+k, and 20+k are proportional numbers. Here, $$\frac{4+k}{15+k}$$ = $$\frac{7+k}{20+k}$$ ⇒ (4+k) × (20+k) = (7+k) × (15+k) ⇒ 80+ 4k +20k +k² = 105+ 7k+ 15k+ k² ⇒ 24k+80 = 22k+42 ⇒ 24k- 22k = 105-80 ⇒ 2k = 25 ⇒ k = $$\frac{25}{2}$$ ⇒ k = 12.5 Thus, the required number is 12.5. Example 2: Priya enlarged the size of a photo to a height of 20 inches. What is the new width if it was originally 4 inches tall and 2 inches in width? Solution: The height of a photo is 20 inches. Now, to find the new width. The ratio to calculate is 4: 2 = 20: w ⇒ $$\frac{4}{2}$$ = $$\frac{20}{w}$$ ⇒ 4w = 20×2 ⇒ 4w = 40 ⇒ w = $$\frac{40}{4}$$ ⇒ w = 10. Thus, the new width of the photo is 10inches. Example 3: If P is directly proportional to n when P is 4 and n is 6. Find the value of P when n is 8. Solution: Given P is proportional to n ( P ∝ n ). Now, convert to an equation multiplied by k the constant of variation. ⇒ P = nk To find k use the given condition P = 4 when n = 6 P = nk ⇒ k = $$\frac{n}{P}$$ = $$\frac{6}{4}$$ Now, the equation is P = $$\frac{6}{4}$$ × n = $$\frac{6n}{4}$$ When n = 8, then P= $$\frac{6n}{4}$$ = $$\frac{6×8}{4}$$ = $$\frac{48}{4}$$ = 12. Hence, the value of P is 12 when n is 8. Example 4: Find the third proportion of 14 and 22? Solution: The proportions given are 14 and 22. Let the third proportion be x. According to the problem, 14 and 22 are in continued proportion. Now, $$\frac{14}{22}$$ = $$\frac{22}{x}$$ ⇒ 14× x = 22 × 22 ⇒ 14x = 484 ⇒ x = 484/14 = 34.57 Therefore, the third proportion is 34.57. Example 5: Ram, Prudhvi, and Mahesh have$15, $22, and$25 respectively with them. Father asks them to give him an equal amount so that the money held by them now is in continued proportion. Find the amount taken from each of them?
Solution:
Let the amount taken from each of them be $m. Now, 15-m, 22-m, and 25-m are in continued proportion. Thus, $$\frac{15-m}{22-m}$$ = $$\frac{22-m}{25-m}$$ ⇒ (15-m)(25-m) = (22-m)(22-m) ⇒ 375 – 15m -25m +m² = 484 – 44m +m² ⇒ 375 – 40m = 484 -44m ⇒ 375 – 484 = -44m +40m ⇒ -109 = -4m ⇒ m = 109/4 ⇒ m = 27.25 Therefore, the required amount is$27.25.

Example 6:
Keerti ran 150meters in 25seconds. How long did she take to run 3meters?
Solution:
Given that
keerti ran 150metres in 25seconds.
Now, to find the time she takes to run 3mts.
Let k be the time required.
$$\frac{25}{150}$$ = $$\frac{k}{3}$$
⇒ k = $$\frac{25}{150}$$ × 3
⇒ k = $$\frac{75}{150}$$
⇒ k = 0.5
Thus, keerti took 0.5seconds to complete 3meters.

Example 7:
Check whether the following numbers form a proportion or not.
(i) 4.5, 3.5, 6.6, and 8.8
(ii) 2$$\frac{2}{4}$$, 1$$\frac{3}{2}$$, 2.2, and 5.5
Solution:
(i) Given 4.5: 3.5 = $$\frac{4.5}{3.5}$$ = $$\frac{45}{35}$$ = $$\frac{9}{7}$$
6.6: 8.8 = $$\frac{6.6}{8.8}$$ = $$\frac{66}{88}$$ = $$\frac{6}{8}$$
Therefore, $$\frac{4.5}{3.5}$$ ≠ $$\frac{6.6}{8.8}$$
Thus, 4.5, 3.5, 6.6, and 8.8 are not in proportion.

(ii) Given 2$$\frac{2}{4}$$, 1$$\frac{3}{2}$$, 2.2, and 5.5
2$$\frac{2}{4}$$: 1$$\frac{3}{2}$$
= $$\frac{10}{4}$$: $$\frac{5}{2}$$
= $$\frac{10}{4}$$ × 4: $$\frac{5}{2}$$ × 4
= 10: 10 = 1: 1
2.2: 5.5 = $$\frac{2.2}{5.5}$$ = $$\frac{22}{55}$$ = $$\frac{2}{5}$$ = 2: 5
Thus, the given numbers 2$$\frac{2}{4}$$, 1$$\frac{3}{2}$$, 2.2, and 5.5 are not in proportion.

Example 8:
Find the fourth proportional of numbers 3, 6, and 12?
Solution:
Given three proportional numbers are 5, 6, and 12.
Let the fourth proportional be x.
Now, 3, 6,12, and x be the proportionality numbers.
Thus,
$$\frac{3}{6}$$ = $$\frac{12}{x}$$
⇒ 3x = 12×6
⇒ 3x = 72
⇒ x = $$\frac{72}{3}$$
⇒ x = 24.
Hence, the fourth proportional number is 24.

Example 9:
If the mean proportion is 16 and the third proportional is 64 then find the two numbers?
Solution:
Given mean proportion is 16 and the third proportion is 64.
Let the required numbers be a and b.
According to the given problem,
√ab = 16
⇒ ab = 16²
⇒ ab = 256
Now, $$\frac{b²}{a}$$ = 64
⇒ b² = 64a
⇒ a = $$\frac{b²}{64}$$
Substitute, a = $$\frac{b²}{64}$$ in ab = 256
⇒ $$\frac{b²}{64}$$ × b = 256
⇒ $$\frac{b³}{64}$$ = 256
⇒ b³ = 256 × 64
⇒ b³ = 28 × 26
⇒ b³ = 212 × 2²
⇒ b = 24 ×2²
⇒ b = 26
⇒ b = 64
So, from the equation a = $$\frac{b²}{64}$$, we get
a = $$\frac{64²}{64}$$
⇒ a = $$\frac{4096}{64}$$
⇒ a = 64
Therefore, the required two numbers are 64 and 64.

## Word Problems on Ratio | Free Printable Ratio Word Problems with Answers

Are you searching for the word problems on ratio, this page helps you to get answers to your questions. Here, students or teachers can get many types of ratios word problems which helps to understand the concept thoroughly. The word ‘Ratio’ is a term used to compare two or more numbers or quantities.

By using ratio it is very simple to solve the problems and gives the result in the simplest form. We use ‘:‘ to denote the ratio while comparing the quantities and read as ” is to “. Let us practice some examples from this ratio word problems worksheet with answers pdf and get an idea of how to solve the problems on ratio.

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## Ratio Word Problems with Solutions

Example 1:
Find the quantity if it is divided in the ratio 6: 4, the smaller part is 48.
Solution:
Let the quantity be ‘m’.
Then the ratio of two parts is written as $$\frac{6m}{6+4}$$ and $$\frac{4m}{6+4}$$.
Here, the smaller part is 48, we get
$$\frac{4m}{6+4}$$ = 48
⇒ $$\frac{4m}{10}$$ = 48
⇒ 4m = 48×10
⇒ 4m = 480
⇒ m = $$\frac{480}{4}$$
⇒ m = 120
Therefore, the quantity is 120.

Example 2:
A herd of 50 cows and buffaloes has 14 cows and some buffaloes. What is the ratio of buffaloes to cows?
Solution:
Given a total of 50 cows and buffaloes.
There are 14 cows given.
To get the count of buffaloes = 50 – 14 = 36.
The ratio of buffaloes to cows is 36: 14 = 18: 7.
Hence, the simplest form of ratio is 18: 7.

Example 3:
Mr. Ram’s class has 40 students, of which 18 are girls. Find the ratio of girls to boys?
Solution:
Total students in a class = 40.
No. of girls in a class = 18 (given)
No. of boys in a class = Total students – No. of girls = 40 – 18 = 22.
There are 22 boys in a class.
The ratio of girls to boys is 18: 22.
Here, there is a common factor 2. So, we can write the ratio in the simplest form by dividing it with the factor 2.
The simplest form of ratio is 9: 11.

Example 4:
If the ratio of p: q = 4: 3, then find (2p- 3q) : (5p+q)?
Solution:
Given p: q = 5: 3, then p = 5m and q = 3m (m ≠ 0 is a common multiplier).
Now, (2p- 3q) : (5p+q) = $$\frac{2p-3q}{5p+q}$$
= $$\frac{(2×5m)-(3×3m)}{(5×4m)+3m}$$
= $$\frac{10m-9m}{20m+3m}$$
= $$\frac{1m}{23m}$$
= $$\frac{1}{23}$$
= 1: 23.
Therefore, the ratio (2p- 3q) : (5p+q) = 1: 23.

Example 5:
Two numbers are in the ratio 3: 4. If 3 is added to the first number and 8 is added to the second number, they are in the ratio of 3: 5. Find the numbers?
Solution:
Given the ratio of two numbers is 3: 4.
Let the numbers be 3x and 4x.
According to the problem given,
$$\frac{3x+3}{4x+8}$$ = $$\frac{3}{5}$$
⇒ 5(3x+3) = 3(4x+8)
⇒ 15x+15 = 12x+24
⇒ 15x-12x = 24-15
⇒ 3x = 9
⇒ x = $$\frac{9}{3}$$
⇒ x = 3.
Hence, the original numbers are 3x = 3×3 = 9 and 4x = 4×3 = 12.
Therefore, the numbers are 9 and 12.

Example 6:
Find the ratio of a: c from the quantities a: b = 4: 5, b: c = 2: 6?
Solution:
Given the ratios a: b = 4: 5, b: c = 2: 6.
a: b = 4: 5 ⇒ $$\frac{a}{b}$$ = $$\frac{4}{5}$$ ——(i)
b: c = 2: 6 ⇒ $$\frac{b}{c}$$ = $$\frac{2}{6}$$ ——(ii)
Now, multiply the equations (i) and (ii), we get
$$\frac{a}{b}$$ × $$\frac{b}{c}$$ = $$\frac{4}{5}$$ × $$\frac{2}{6}$$
⇒ $$\frac{a}{c}$$ = $$\frac{8}{30}$$
The ratio a: c = 8: 30 = 4: 15.
Thus, the ratio a: c = 4: 15.

Example 7:
If we have the ratio of tomatoes to apples is 2: 4. If there are 18 tomatoes. How many apples are there?
Solution:
The ratio of tomatoes to apples is 2: 4 means that for every 2 tomatoes, you have 4 apples.
Here, you have 18 tomatoes, or we can say 9 times as much.
So, you need to multiply the apples by 9.
⇒ The apples we have is 4 = 4 × 9 = 36.
Thus, there are 36 apples.

Example 8:
If the equation (2x+5y): (6x-4y) = 8: 5. Find the ratio of x: y?
Solution:
Given, (2x+5y): (6x-4y) = 8: 5
Now,
$$\frac{2x+5y}{6x-4y}$$ = $$\frac{8}{5}$$
⇒ 5(2x+5y) = 8(6x-4y)
⇒ 10x+25y = 48x-32y
⇒ 10x – 48x = -32y – 25y
⇒ -38x = -57y
⇒ 38x = 57y
⇒ $$\frac{x}{y}$$ = $$\frac{57}{38}$$
⇒ x: y = 57: 38.
Hence, that ratio of x: y is 57: 38.

Example 9:
Manoj leaves $2461600 behind. Manoj’s wish was the money is to be divided between his son and daughter in the ratio of 3: 2. Find the money received by his son and daughter? Solution: Manoj has money of$ 2461600 and is to be shared with his son and daughter in the ratio of 3: 2.
We know if a quantity x is divided in the ratio of a: b, then the two parts are look alike. i.e., $$\frac{ax}{a+b}$$ and $$\frac{bx}{a+b}$$.
Now, the money received by his son = $$\frac{3}{3+2}$$ × $2461600 = $$\frac{3}{5}$$ ×$ 2461600
= 3 × $492320 =$ 1476960
Next, the money received by his daughter = $$\frac{2}{3+2}$$ × $2461600 = $$\frac{2}{5}$$ ×$ 2461600
= 2 × $492320 =$ 984640
Thus, the money received by Manoj’s son is $1476960 and by his daughter is$ 984640.