Ratio & Proportion — Exercise 9(B)

In each of the following, check whether or not the given ratios form a proportion :

(i) 8 : 16 and 12 : 15

(ii) 16 : 28 and 24 : 42

(iii) 98 : 112 and 14 : 16

Answer

(i) 8 : 16 and 12 : 15

Simplifying first ratio :

$\Rightarrow \dfrac{8}{16} = \dfrac{1}{2}$

Simplifying second ratio :

$\Rightarrow \dfrac{12}{15} = \dfrac{4}{5}$

∴ Ratio 8 : 16 ≠ 12 : 15.

Hence, 8 : 16 and 12 : 15 do not form a proportion.

(ii) 16 : 28 and 24 : 42

Simplifying first ratio :

$\Rightarrow \dfrac{16}{28} = \dfrac{4}{7}$

Simplifying second ratio :

$\Rightarrow \dfrac{24}{42} = \dfrac{4}{7}$

∴ Ratio 16 : 28 = 24 : 42.

Hence, 16 : 28 and 24 : 42 form a proportion.

(iii) 98 : 112 and 14 : 16

Simplifying first ratio :

$\Rightarrow \dfrac{98}{112} = \dfrac{7}{8}$

Simplifying second ratio :

$\Rightarrow \dfrac{14}{16} = \dfrac{7}{8}$

∴ Ratio 98 : 112 = 14 : 16.

Hence, 98 : 112 and 14 : 16 form a proportion.

Find the value of x so that the given four numbers are in proportion :

x, 6, 10 and 15

Answer

Four numbers a, b, c, d are in proportion if a × d = b × c.

x, 6, 10 and 15

⇒ x × 15 = 6 × 10

⇒ 15x = 60

⇒ x = $\dfrac{60}{15}$ = 4

Hence, x = 4.

Find the value of x so that the given four numbers are in proportion :

2, x, 10 and 25

Answer

Four numbers a, b, c, d are in proportion if a × d = b × c.

2, x, 10 and 25

⇒ 2 × 25 = x × 10

⇒ 50 = 10x

⇒ x = $\dfrac{50}{10}$ = 5

Hence, x = 5.

Find the value of x so that the given four numbers are in proportion :

9, 12, x and 8

Answer

Four numbers a, b, c, d are in proportion if a × d = b × c.

9, 12, x and 8

⇒ 9 × 8 = 12 × x

⇒ 72 = 12x

⇒ x = $\dfrac{72}{12}$ = 6

Hence, x = 6.

Find the value of x so that the given four numbers are in proportion :

6, 4, 18 and x

Answer

Four numbers a, b, c, d are in proportion if a × d = b × c.

6, 4, 18 and x

⇒ 6 × x = 4 × 18

⇒ 6x = 72

⇒ x = $\dfrac{72}{6}$ = 12

Hence, x = 12.

Find the value of x in the following proportion :

(x + 5) : 24 = 3 : 4

Answer

In a proportion, product of extremes = product of means.

(x + 5) : 24 = 3 : 4

⇒ 4(x + 5) = 24 × 3

⇒ 4(x + 5) = 72

⇒ x + 5 = $\dfrac{72}{4}$

⇒ x + 5 = 18

⇒ x = 18 − 5 = 13

Hence, x = 13.

Find the value of x in the following proportion :

14 : (x − 1) = 7 : 9

Answer

In a proportion, product of extremes = product of means.

14 : (x − 1) = 7 : 9

⇒ 14 × 9 = 7(x − 1)

⇒ 126 = 7(x − 1)

⇒ x − 1 = $\dfrac{126}{7}$

⇒ x − 1 = 18

⇒ x = 18 + 1 = 19

Hence, x = 19.

Find the value of x in the following proportion :

2 : 3 = (x − 2) : 18

Answer

In a proportion, product of extremes = product of means.

2 : 3 = (x − 2) : 18

⇒ 2 × 18 = 3(x − 2)

⇒ 36 = 3(x − 2)

⇒ x − 2 = $\dfrac{36}{3}$

⇒ x − 2 = 12

⇒ x = 12 + 2 = 14

Hence, x = 14.

Find the value of x in the following proportion :

3 : 4 = 30 : (x + 10)

Answer

In a proportion, product of extremes = product of means.

3 : 4 = 30 : (x + 10)

⇒ 3(x + 10) = 4 × 30

⇒ 3(x + 10) = 120

⇒ x + 10 = $\dfrac{120}{3}$

⇒ x + 10 = 40

⇒ x = 40 − 10 = 30

Hence, x = 30.

Find the value of x in the following proportion :

1 : 2 = (x − 1) : 26

Answer

In a proportion, product of extremes = product of means.

1 : 2 = (x − 1) : 26

⇒ 1 × 26 = 2(x − 1)

⇒ 26 = 2(x − 1)

⇒ x − 1 = $\dfrac{26}{2}$

⇒ x − 1 = 13

⇒ x = 13 + 1 = 14

Hence, x = 14.

The first, second and the fourth terms of a proportion are 6, 18 and 75 respectively. Find its third term.

Answer

Let the third term be x.

∴ 6, 18, x and 75 are in proportion, i.e. 6 : 18 : : x : 75

In a proportion, product of extremes = product of means.

⇒ 6 × 75 = 18 × x

⇒ 450 = 18x

⇒ x = $\dfrac{450}{18}$ = 25

Hence, the third term is 25.

The ratio of copper and zinc in an alloy by weight is 9 : 8. If the weight of zinc in the alloy is 9.6 kg, find the weight of copper in the alloy.

Answer

Given,

Ratio (Copper : Zinc) = 9 : 8

Weight of zinc = 9.6 kg

Since 8 parts represent the zinc, 8 parts = 9.6 kg.

1 part = $\dfrac{9.6}{8}$ = 1.2 kg.

Weight of copper = 9 parts = 9 × 1.2 = 10.8 kg.

Hence, the weight of copper in the alloy is 10.8 kg.

The ratio of the number of girls to the number of boys in a school is 2 : 5. If the number of boys is 225, find :

(i) the number of girls in the school.

(ii) the number of total students in the school.

Answer

Given,

Ratio (Girls : Boys) = 2 : 5

Number of boys = 225

Since 5 parts represent the boys, 5 parts = 225.

1 part = $\dfrac{225}{5}$ = 45.

(i) Number of girls = 2 parts = 2 × 45 = 90.

Hence, the number of girls in the school is 90.

(ii) Total students = Girls + Boys = 90 + 225 = 315.

Hence, the total number of students in the school is 315.

If $(3x − 4) : (x + 2)$ = 2 : 1, find the value of $x$.

Answer

Given,

$(3x - 4) : (x + 2)$ = 2 : 1

⇒ $\dfrac{3x - 4}{x + 2} = \dfrac{2}{1}$

By cross multiplication :

$\Rightarrow 3x − 4 = 2(x + 2) \\[1em] \Rightarrow 3x − 4 = 2x + 4 \\[1em] \Rightarrow 3x − 2x = 4 + 4 \\[1em] \Rightarrow x = 8$

Hence, the value of $\bm{x = 8}$.

If (x − 7) : 2 : : (x − 3) : 3, find the value of x − 5.

Answer

Given,

(x − 7) : 2 : : (x − 3) : 3

In a proportion, product of extremes = product of means.

⇒ 3(x − 7) = 2(x − 3)

⇒ 3x − 21 = 2x − 6

⇒ 3x − 2x = 21 − 6

⇒ x = 15

Now, x − 5 = 15 − 5 = 10

Hence, the value of x − 5 is 10.

In a school three out of every 5 students play cricket. If there are 2,400 students in the school, find how many of them play cricket?

Answer

Given,

Out of every 5 students, 3 play cricket.

∴ Fraction of students who play cricket = $\dfrac{3}{5}$

Number of students who play cricket = $\dfrac{3}{5} \times 2400$

= 3 × 480

= 1440.

Hence, 1440 students play cricket.