Ratio & Proportion — Exercise 8(B)

Verify that:

12 : 18 = 8 : 12

Answer

12 : 18 = 8 : 12

We express each of the given ratios in simplest form.

Consider 12 : 18.

$\begin{array}{l} 12\overline{\smash{\big)}18 \smash{\big(}}1 \\ \phantom{111}{12} \\ \phantom{111}\overline{6\smash{\big)}12 \smash{\big(}2} \\ \phantom{11111}\underline{12\phantom{1}} \\ \phantom{111111}0\ \end{array}$

H.C.F. of 12 and 18 = 6.

$\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3} = 2 : 3$

Again, consider 8 : 12.

$\begin{array}{l} 8\overline{\smash{\big)}12 \smash{\big(}}1 \\ \phantom{111}{8} \\ \phantom{111}\overline{4\smash{\big)}8 \smash{\big(}2} \\ \phantom{11111}\underline{8\phantom{1}} \\ \phantom{11111}0\ \end{array}$

H.C.F. of 8 and 12 = 4.

$\dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3} = 2 : 3$

Thus, both the ratios reduce to 2 : 3, so they are equal.

Hence, 12 : 18 = 8 : 12 is verified.

Verify that:

81 kg : 45 kg = 18 men : 10 men

Answer

81 kg : 45 kg = 18 men : 10 men

We express each of the given ratios in simplest form.

Consider 81 : 45.

$\begin{array}{l} 45\overline{\smash{\big)}81 \smash{\big(}}1 \\ \phantom{111}{45} \\ \phantom{111}\overline{36\smash{\big)}45 \smash{\big(}1} \\ \phantom{111111}{36} \\ \phantom{111111}\overline{9\smash{\big)}36 \smash{\big(}4} \\ \phantom{11111111}\underline{36\phantom{1}} \\ \phantom{111111111}0\ \end{array}$

H.C.F. of 81 and 45 = 9.

$\dfrac{81 \div 9}{45 \div 9} = \dfrac{9}{5} = 9 : 5$

Again, consider 18 : 10.

$\begin{array}{l} 10\overline{\smash{\big)}18 \smash{\big(}}1 \\ \phantom{111}{10} \\ \phantom{111}\overline{8\smash{\big)}10 \smash{\big(}1} \\ \phantom{111111}{8} \\ \phantom{111111}\overline{2\smash{\big)}8 \smash{\big(}4} \\ \phantom{11111111}\underline{8\phantom{1}} \\ \phantom{11111111}0\ \end{array}$

H.C.F. of 18 and 10 = 2.

$\dfrac{18 \div 2}{10 \div 2} = \dfrac{9}{5} = 9 : 5$

Thus, both the ratios reduce to 9 : 5, so they are equal.

Hence, 81 kg : 45 kg = 18 men : 10 men is verified.

Verify that:

$3\dfrac{1}{3} : 2\dfrac{1}{2} = 12 : 9$

Answer

$3\dfrac{1}{3} : 2\dfrac{1}{2} = 12 : 9$

We express each of the given ratios in simplest form.

Convert the mixed numbers into improper fractions: $3\dfrac{1}{3} = \dfrac{10}{3}$ and $2\dfrac{1}{2} = \dfrac{5}{2}$.

Consider $\dfrac{10}{3} : \dfrac{5}{2}$.

L.C.M. of 3 and 2 is 6. Multiplying each term by 6:

$\dfrac{10}{3} : \dfrac{5}{2} = \left(\dfrac{10}{3} \times 6\right) : \left(\dfrac{5}{2} \times 6\right) = 20 : 15$

$\begin{array}{l} 15\overline{\smash{\big)}20 \smash{\big(}}1 \\ \phantom{111}{15} \\ \phantom{111}\overline{5\smash{\big)}15 \smash{\big(}3} \\ \phantom{11111}\underline{15\phantom{1}} \\ \phantom{111111}0\ \end{array}$

H.C.F. of 20 and 15 = 5.

$\dfrac{20 \div 5}{15 \div 5} = \dfrac{4}{3} = 4 : 3$

Again, consider 12 : 9.

$\begin{array}{l} 9\overline{\smash{\big)}12 \smash{\big(}}1 \\ \phantom{111}{9} \\ \phantom{111}\overline{3\smash{\big)}9 \smash{\big(}3} \\ \phantom{11111}\underline{9\phantom{1}} \\ \phantom{11111}0\ \end{array}$

H.C.F. of 12 and 9 = 3.

$\dfrac{12 \div 3}{9 \div 3} = \dfrac{4}{3} = 4 : 3$

Thus, both the ratios reduce to 4 : 3, so they are equal.

Hence, $3\dfrac{1}{3} : 2\dfrac{1}{2} = 12 : 9$ is verified.

Verify that:

6 : 45 ≠ 4.8 : 3.6

Answer

6 : 45 ≠ 4.8 : 3.6

We express each of the given ratios in simplest form.

Consider 6 : 45.

$\begin{array}{l} 6\overline{\smash{\big)}45 \smash{\big(}}7 \\ \phantom{11}{42} \\ \phantom{11}\overline{3\smash{\big)}6 \smash{\big(}2} \\ \phantom{1111}\underline{6\phantom{1}} \\ \phantom{1111}0\ \end{array}$

H.C.F. of 6 and 45 = 3.

$\dfrac{6 \div 3}{45 \div 3} = \dfrac{2}{15} = 2 : 15$

Again, consider 4.8 : 3.6.

Multiplying each term by 10, we get 48 : 36.

$\begin{array}{l} 36\overline{\smash{\big)}48 \smash{\big(}}1 \\ \phantom{111}{36} \\ \phantom{111}\overline{12\smash{\big)}36 \smash{\big(}3} \\ \phantom{111111}\underline{36\phantom{1}} \\ \phantom{1111111}0\ \end{array}$

H.C.F. of 48 and 36 = 12.

$\dfrac{48 \div 12}{36 \div 12} = \dfrac{4}{3} = 4 : 3$

Thus, 2 : 15 ≠ 4 : 3, so the two ratios are not equal.

Hence, 6 : 45 ≠ 4.8 : 3.6 is verified.

Which of the following numbers are in proportion?

(i) 30, 42, 5, 7

(ii) 4, 11, 22, 33

(iii) 9, 13, 10, 14

(iv) 16, 18, 24, 27

Answer

Four numbers a, b, c, d are in proportion if and only if a × d = b × c (product of extremes = product of means).

(i) 30, 42, 5, 7

Product of extremes = 30 × 7 = 210

Product of means = 42 × 5 = 210

Since 210 = 210, the numbers are in proportion.

Hence, 30, 42, 5, 7 are in proportion.

(ii) 4, 11, 22, 33

Product of extremes = 4 × 33 = 132

Product of means = 11 × 22 = 242

Since 132 ≠ 242, the numbers are not in proportion.

Hence, 4, 11, 22, 33 are not in proportion.

(iii) 9, 13, 10, 14

Product of extremes = 9 × 14 = 126

Product of means = 13 × 10 = 130

Since 126 ≠ 130, the numbers are not in proportion.

Hence, 9, 13, 10, 14 are not in proportion.

(iv) 16, 18, 24, 27

Product of extremes = 16 × 27 = 432

Product of means = 18 × 24 = 432

Since 432 = 432, the numbers are in proportion.

Hence, 16, 18, 24, 27 are in proportion.

Find the value of x in the following proportions:

36 : 81 :: x : 63

Answer

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

36 : 81 :: x : 63

Product of extremes = product of means

$\Rightarrow 36 \times 63 = 81 \times x\\[1em] \Rightarrow x = \dfrac{36 \times 63}{81} \\[1em] = \dfrac{2268}{81} \\[1em] = 28$

Hence, x = 28.

Find the value of x in the following proportions:

27 : x :: 63 : 84

Answer

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

27 : x :: 63 : 84

Product of extremes = product of means

$\Rightarrow 27 \times 84 = x \times 63\\[1em] \Rightarrow x = \dfrac{27 \times 84}{63} \\[1em] = \dfrac{2268}{63} \\[1em] = 36$

Hence, x = 36.

Find the value of x in the following proportions:

x : 92 :: 87 : 116

Answer

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

x : 92 :: 87 : 116

Product of extremes = product of means

$\Rightarrow x \times 116 = 92 \times 87\\[1em] \Rightarrow x = \dfrac{92 \times 87}{116} \\[1em] = \dfrac{8004}{116} \\[1em] = 69$

Hence, x = 69.

Find the value of x in the following proportions:

45 : x :: 25 : 35

Answer

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

45 : x :: 25 : 35

Product of extremes = product of means

$\Rightarrow 45 \times 35 = x \times 25\\[1em] \Rightarrow x = \dfrac{45 \times 35}{25} \\[1em] = \dfrac{1575}{25} \\[1em] = 63$

Hence, x = 63.

In a proportion, the 1st, 2nd and 4th terms are 32, 112 and 217 respectively. Find the 3rd term.

Answer

Given:

1st term = 32, 2nd term = 112, 4th term = 217

Let the 3rd term be x. Then, 32 : 112 :: x : 217.

Product of extremes = product of means

$\Rightarrow 32 \times 217 = 112 \times x\\[1em] \Rightarrow x = \dfrac{32 \times 217}{112} \\[1em] = \dfrac{6944}{112} \\[1em] = 62$

Hence, the 3rd term of the proportion is 62.

In a proportion, the 1st, 3rd and 4th terms are 51, 81 and 108 respectively. Find the 2nd term.

Answer

Given:

1st term = 51, 3rd term = 81, 4th term = 108

Let the 2nd term be x. Then, 51 : x :: 81 : 108.

Product of extremes = product of means

$\Rightarrow 51 \times 108 = x \times 81\\[1em] \Rightarrow x = \dfrac{51 \times 108}{81} \\[1em] = \dfrac{5508}{81} \\[1em] = 68$

Hence, the 2nd term of the proportion is 68.

The incomes of Ruchi and Rajan are in the ratio 4 : 7. If Ruchi earns ₹16,800 per month, how much does Rajan earn per month?

Answer

Given:

Ratio (Ruchi : Rajan) = 4 : 7

Ruchi's income = ₹16,800

Let Rajan's income be ₹x. Then, 4 : 7 :: 16800 : x.

Product of extremes = product of means

$\Rightarrow 4 \times x = 7 \times 16{,}800\\[1em] \Rightarrow x = \dfrac{7 \times 16{,}800}{4} \\[1em] = \dfrac{1{,}17{,}600}{4} \\[1em] = 29{,}400$

Hence, Rajan earns ₹29,400 per month.

An electric pole casts a shadow of length 20 metres at a time when a tree 6 metres high casts a shadow of length 8 metres. Find the height of the pole.

Answer

Given:

Shadow of the pole = 20 m

Height of the tree = 6 m and shadow of the tree = 8 m

At the same time, the ratio of height to shadow is the same for all objects.

Let the height of the pole be x metres. Then,

Height of pole : Shadow of pole = Height of tree : Shadow of tree

$x : 20 = 6 : 8$

Product of extremes = product of means

$\Rightarrow x \times 8 = 20 \times 6\\[1em] \Rightarrow x = \dfrac{20 \times 6}{8} \\[1em] = \dfrac{120}{8} \\[1em] = 15$

Hence, the height of the pole is 15 metres.

The ratio of copper and zinc in an alloy is 9 : 5. If the weight of zinc in the alloy is 9.5 g, what is the weight of copper in it?

Answer

Given:

Ratio (Copper : Zinc) = 9 : 5

Weight of zinc = 9.5 g

Let the weight of copper be x g. Then, 9 : 5 :: x : 9.5.

Product of extremes = product of means

$\Rightarrow 9 \times 9.5 = 5 \times x\\[1em] \Rightarrow x = \dfrac{9 \times 9.5}{5} \\[1em] = \dfrac{85.5}{5} \\[1em] = 17.1$

Hence, the weight of copper in the alloy is 17.1 g.

The ratio of length and breadth of a rectangular plot is 9 : 5. If its breadth is 60 m, find its length.

[Hint: 9 : 5 :: x : 60. Find x in metres.]

Answer

Given:

Ratio (Length : Breadth) = 9 : 5

Breadth = 60 m

Let the length be x metres. Then, 9 : 5 :: x : 60.

Product of extremes = product of means

$\Rightarrow 9 \times 60 = 5 \times x\\[1em] \Rightarrow x = \dfrac{9 \times 60}{5} \\[1em] = \dfrac{540}{5} \\[1em] = 108$

Hence, the length of the rectangular plot is 108 m.