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## Free Worksheets on Solving Proportions

1. Check if each of the following is in proportion.

(a) 12, 42, 24, 8

(b) 8, 6, 12, 90

(c) 10, 16, 400, 250

(d) 60, 32, 90, 50

(e) 36, 16, 108, 48

(f) 12, 16, 48, 60

## Solution:

(a) Given numbers are 12, 42, 24, 8.

From the given data, extreme terms are 12 and 8, mean terms are 42 and 24.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 12 × 8 = 96

Product of mean terms = 42 × 24 = 1008.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

96 is not equal to 1008.

Therefore, the given numbers 12, 42, 24, 8 are not in proportion.

(b) Given numbers are 8, 6, 12, 90.

From the given data, extreme terms are 8 and 90, mean terms are 6 and 12.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 8 × 90 = 720

Product of mean terms = 6 × 12 = 72.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

720 is not equal to 72.

Therefore, the given numbers 8, 6, 12, 90 are not in proportion.

(c) Given numbers are 10, 16, 400, 250.

From the given data, extreme terms are 10 and 250, mean terms are 16 and 400.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 10 × 250 = 2500

Product of mean terms = 16 × 400 = 6400.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

2500 is not equal to 6400.

Therefore, the given numbers 10, 16, 400, 250 are not in proportion.

(d) Given numbers are 60, 32, 90, 50.

From the given data, extreme terms are 60 and 50, mean terms are 32 and 90.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 60 × 50 = 3000

Product of mean terms = 32 × 90 = 2880.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

3000 is not equal to 2880.

Therefore, the given numbers 60, 32, 90, 50 are not in proportion.

(e) Given numbers are 36, 16, 108, 48.

From the given data, extreme terms are 36 and 48, mean terms are 16 and 108.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 36 × 48 = 1728

Product of mean terms = 16 × 108 = 1728.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

1728 is not equal to 1728.

Therefore, the given numbers 36, 16, 108, 48 are in proportion.

(f) Given numbers are 12, 16, 48, 60.

From the given data, extreme terms are 12 and 60, mean terms are 16 and 48.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 12 × 60 = 720

Product of mean terms = 16 × 48 = 768.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

720 is not equal to 768.

Therefore, the given numbers 12, 16, 48, 60 are not in proportion.

2. Find the value of x in the following?

(a) 12 : 18 :: 2x : 30

(b) x : 7 :: 12 : 21

(c) 2 : 2x :: 16 : 28

(d) 3x : 9 :: 1.2 : 1.5

(e) 2/3 ∶ 2/4 : : 2/9 : 2x

(f) 8 : x :: 4 : 10

(g) 4 : 14 :: 2x : 84

(h) 3 : 2 :: 3 : x

(i) 2x : 3 :: 12.6 : 9

## Solution:

(a) Given ratios are 12 : 18 :: 2x : 30.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 12 and 30, mean terms are 18 and 2x.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 12 × 30 = 360

Product of mean terms = 18 × 2x = 36x.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

36x = 360

x = \(\frac { 360 }{ 36 } \)

x = 10

Therefore, the value of x is 10.

(b) Given ratios are x : 7 :: 12 : 21.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are x and 21, mean terms are 7 and 12.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 21 × x = 21x

Product of mean terms = 7 × 12 = 84.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

21x = 84

x = \(\frac { 84 }{ 21 } \)

x = 4

Therefore, the value of x is 4.

(c) Given ratios are 2 : 2x :: 16 : 28.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 2 and 28, mean terms are 2x and 16.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 2 × 28 = 56

Product of mean terms = 2x × 16 = 32x.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

32x = 56

x = \(\frac { 56 }{ 32 } \)

x = 1.75

Therefore, the value of x is 1.75.

(d) Given ratios are 3x : 9 :: 1.2 : 1.5.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 3x and 1.5, mean terms are 9 and 1.2.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 3x × 1.5 = 4.5x

Product of mean terms = 9 × 1.2 = 10.8.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

4.5x = 10.8

x = \(\frac { 10.8 }{ 4.5 } \)

x = 2.4

Therefore, the value of x is 2.4.

(e) Given ratios are 2/3 ∶ 2/4 : : 2/9 : 2x.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 2/3 and 2x, mean terms are 2/4 and 2/9.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 2/3 × 2x = 4/3x

Product of mean terms = 2/4 × 2/9 = 1/9.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

4/3x = 1/9

x = \(\frac { 1/9 }{ 4/3 } \)

x = \(\frac { 1 }{ 12 } \)

Therefore, the value of x is \(\frac { 1 }{ 12 } \).

(f) Given ratios are 8 : x :: 4 : 10.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 8 and 10, mean terms are x and 4.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 8 × 10 = 80

Product of mean terms = x × 4 = 4x.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

4x = 80

x = \(\frac { 80 }{ 4 } \)

x = 20

Therefore, the value of x is 20.

(g) Given ratios are 4 : 14 :: 2x : 84.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 4 and 84, mean terms are 14 and 2x.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 4 × 84 = 336

Product of mean terms = 14 × 2x = 28x.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

28x = 336

x = \(\frac { 336 }{ 28 } \)

x = 12

Therefore, the value of x is 12.

(h) Given ratios are 3 : 2 :: 3 : x.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 3 and x, mean terms are 2 and 3.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 3 × x = 3x

Product of mean terms = 2 × 3 = 6.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

3x = 6

x = \(\frac { 6 }{ 3 } \)

x = 2

Therefore, the value of x is 2.

(i) Given ratios are 2x : 3 :: 12.6 : 9.

We need to find the value of x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 2x and 9, mean terms are 3 and 12.6.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 2x × 9 = 18x

Product of mean terms = 3 × 12.6 = 37.8.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

18x = 37.8

x = \(\frac { 37.8 }{ 18 } \)

x = 2.1

Therefore, the value of x is 2.1.

3. Find the fourth proportional to

(a) 36, 108, 12

(b) 60, 64, 84

(c) 8, 6, 12

(d) 4, 12, 2

(e) 64, 48, 16

## Solution:

(a) Given numbers are 36, 108, 12.

Let the fourth proportional to 36, 108, 12 be x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 36 and x, mean terms are 108 and 12.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 36 × x = 36x

Product of mean terms = 108 × 12 = 1296.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

36x = 1296

x = \(\frac { 1296 }{ 36 } \)

x = 36

Therefore, the fourth proportional is 36.

(b) Given numbers are 60, 64, 84.

Let the fourth proportional to 60, 64, 84 be x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 60 and x, mean terms are 64 and 84.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 60 × x = 60x

Product of mean terms = 64 × 84 = 5376.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

60x = 5376

x = \(\frac { 5376 }{ 60 } \)

x = 89.6

Therefore, the fourth proportional is 89.6.

(c) Given numbers are 8, 6, 12.

Let the fourth proportional to 8, 6, 12 be x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 8 and x, mean terms are 6 and 12.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 8 × x = 8x

Product of mean terms = 6 × 12 = 72.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

8x = 72

x = \(\frac { 72 }{ 8 } \)

x = 9

Therefore, the fourth proportional is 9.

(d) Given numbers are 4, 12, 2.

Let the fourth proportional to 4, 12, 2 be x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 4 and x, mean terms are 12 and 2.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 4 × x = 4x

Product of mean terms = 12 × 2 = 24.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

4x = 24

x = \(\frac { 24 }{ 4 } \)

x = 6

Therefore, the fourth proportional is 6.

(e) Given numbers are 64, 48, 16.

Let the fourth proportional to 64, 48, 16 be x.

The four numbers are in proportion.

Therefore, The product of means = product of extremes

From the given data, extreme terms are 64 and x, mean terms are 48 and 16.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 64 × x = 64x

Product of mean terms = 48 × 16 = 768.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

64x = 768

x = \(\frac { 768 }{ 64 } \)

x = 12

Therefore, the fourth proportional is 12.

4. Find the third proportional to

(a) 4, 8

(b) 6, 18

(c) 72, 84

(d) 32, 72

(e) 28, 56

(f) 96, 192

## Solution:

(a) Given numbers are 4, and 8.

Let the third proportional to 4, and 8 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 8, first and the last terms are 4 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = 8² = 8 × 8 = 64

Product of first and last term = 4 × x = 4x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

64 = 4x

x = \(\frac { 64 }{ 4 } \)

x = 16

Therefore, the third proportional is 16.

(b) Given numbers are 6, and 18.

Let the third proportional to 6, and 18 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 18, first and the last terms are 6 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = 18² = 18 × 18 = 324

Product of first and last term = 6 × x = 6x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

324 = 6x

x = \(\frac { 324 }{ 6 } \)

x = 54

Therefore, the third proportional is 54.

(c) Given numbers are 72, and 84.

Let the third proportional to 72, and 84 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 48, first and the last terms are 72 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = 84² = 84 × 84 = 7056

Product of first and last term = 72 × x = 72x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

7056 = 72x

x = \(\frac { 7056 }{ 72 } \)

x = 98

Therefore, the third proportional is 98.

(d) Given numbers are 32, and 72.

Let the third proportional to 32, and 72 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 72, first and the last terms are 32 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = 72² = 72 × 72 = 5184

Product of first and last term = 32 × x = 32x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

5184 = 32x

x = \(\frac { 5184 }{ 32 } \)

x = 162

Therefore, the third proportional is 162.

(e) Given numbers are 28, and 56.

Let the third proportional to 28, and 56 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 56, first and the last terms are 28 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = 56² = 56 × 56 = 3136

Product of first and last term = 28 × x = 28x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

3136 = 28x

x = \(\frac { 3136 }{ 28 } \)

x = 112

Therefore, the third proportional is 112.

(f) Given numbers are 96, and 192.

Let the third proportional to 96, and 192 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is 192, first and the last terms are 96 and x.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = (192)² = 192 × 192 = 36864

Product of first and last term = 96 × x = 96x.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

36864 = 96x

x = \(\frac { 36864 }{ 96 } \)

x = 384

Therefore, the third proportional is 384.

5. Find the mean proportional between

(a) 72 and 8

(b) 2.4 and 0.6

(c) 50 and 18

(d) 48 and 108

(e) 2/9 and 8

## Solution:

(a) Given numbers are 72 and 8.

Let the mean proportional to 72 and 8 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is x, first and the last terms are 72 and 8.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = x²

Product of first and last term = 72 × 8 = 576.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

576 = x²

x = √576

x = 24

Therefore, the mean proportional is 24.

(b) Given numbers are 2.4 and 0.6.

Let the mean proportional to 2.4 and 0.6.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is x, first and the last terms are 2.4 and 0.6.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = x²

Product of first and last term = 2.4 × 0.6 = 1.44.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

1.44 = x²

x = √1.44

x = 1.2

Therefore, the mean proportional is 1.2.

(c) Given numbers are 50 and 18.

Let the mean proportional to 50 and 18 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is x, first and the last terms are 50 and 18.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = x²

Product of first and last term = 50 × 18 = 900.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

900 = x²

x = √900

x = 30

Therefore, the mean proportional is 30.

(d) Given numbers are 48 and 108.

Let the mean proportional to 48 and 108 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is x, first and the last terms are 48 and 108.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = x²

Product of first and last term = 48 × 108 = 5184.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

5184 = x²

x = √5184

x = 72

Therefore, the mean proportional is 72.

(e) Given numbers are 2/9 and 8.

Let the mean proportional to 2/9 and 8 be x.

The three numbers are in proportion.

Therefore, the square of middle term = multiplication of first and last term.

b² = ac

From the given data, the middle term is x, first and the last terms are 2/9 and 8.

Find the square of the middle term and the product of the first and last term.

Square of the middle term = x²

Product of first and last term = 2/9 × 8 = 16/9.

Compare the Square of the middle term and Product of the first and last term.

Square of the middle term = Product of the first and last term

16/9 = x²

x = √(16/9)

x = 4/3

Therefore, the mean proportional is 4/3.

6. The first, second, and fourth terms of the proportion are 18, 42, 154. Find the third term.

## Solution:

Given that the first, second, and fourth terms of the proportion are 18, 42, 154.

Let the third term be x.

From the given data, extreme terms are 18 and 154, mean terms are 42 and x.

Find the Product of extreme terms and mean terms.

Product of extreme terms = 18 × 154 = 2772

Product of mean terms = 42 × x = 42x.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

42x = 2772

x = \(\frac { 2772 }{ 42 } \)

x = 66.

Therefore, the third term is 66.

7. The second, third, and fourth terns of the proportion are 54, 70, 126. Find the first term.

## Solution:

Given that the second, third, and fourth terns of the proportion are 54, 70, 126.

Let the first term be x.

From the given data, extreme terms are x and 126, mean terms are 54 and 70.

Find the Product of extreme terms and mean terms.

Product of extreme terms = x × 126 = 126x

Product of mean terms = 54 × 70 = 3780.

Compare the Product of extreme terms and the Product of mean terms.

The product of means = product of extremes

126x = 3780

x = \(\frac { 3780 }{ 126 } \)

x = 30.

Therefore, the first term is 30.

8. In the school library, the ratio of maths books to science studies books is the same as the ratio of science studies books to English books. If there are 900 books in maths and 600 books in science studies, find the number of books in English?

## Solution:

Given that In the school library, the ratio of maths books to science studies books is the same as the ratio of science studies books to English books.

There are 900 books in maths and 600 books in science studies.

The ratio of maths and science studies books = 900 : 600 = 3 : 2

Let the English books be x.

The ratio of science studies books to English books = 600 : x.

Given the ratio of maths books to science studies books is the same as the ratio of science studies books to English books.

Therefore, 3 : 2 = 600 : x

3x = 1200

x = \(\frac { 1200 }{ 3 } \)

x = 400.

Therefore, the number of books in English is 400.

9. If 8 books cost $56, what will the cost of 40 books be?

## Solution:

Given that 8 books cost $56.

Find one book’s cost from the above information.

1 book cost = $56/8 = $7.

Now, find the cost of 40 books.

To find the cost of 40 books, multiply 40 books by $7.

The cost of 40 books = 40 × $7 = $280.

Therefore, the cost of 40 books is $280.

10. The ratio of length and breadth of a playground is 6 : 4. Find the length if breadth is 72 m.

## Solution:

Given that the ratio of length and breadth of a playground is 6 : 4.

The breadth is 72 m

length : breadth = 6 : 4

length : 72m = 6 : 4

The length = \(\frac { 6 }{ 4 } \) × 72m

The length = 108m

Therefore, the length is 108m.