Class 8: Ratio and Proportion – Exercise 9B – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Question 1: Which of the following statements are true?

i) $27:36=4.5:6$ ii) $\frac{3}{4} : \frac{15}{16}= \frac{2}{3} : \frac{5}{6}$

iii) $Rs. 14 : \ Rs 21=2 \ pens :3 \ pens$ iv) $6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24$

Answer:

i) $27:36=4.5:6$

$4.5:6 =$ $\frac{4.5\times 6}{6\times 6}$ $=$ $\frac{27}{36}$

Hence True.

ii) $\frac{3}{4}$ $:$ $\frac{15}{16}$ $=$ $\frac{2}{3}$ $:$ $\frac{5}{6}$

$\frac{3}{4}$ $:$ $\frac{15}{16}$ $=$ $\frac{3}{4}$ $\times$ $\frac{16}{15}$ $=$ $\frac{4}{5}$

$\frac{2}{3}$ $:$ $\frac{5}{6}$ $=$ $\frac{2}{3}$ $\times$ $\frac{6}{5}$ $=$ $\frac{4}{5}$

Hence True

iii) $Rs. 14 : \ Rs 21=2 \ pens :3 \ pens$

$\frac{Rs. \ 14}{Rs. \ 21}$ $=$ $\frac{2}{3}$

$\frac{2 \ pens}{3 \ pens}$ $=$ $\frac{2}{3}$

Hence True

iv) $6.5 \ km :2.6 \ km=Rs. \ 60 :Rs \ 24$

$\frac{6.5 \ km}{2.6 \ km}$ $=$ $\frac{5}{2}$

$\frac{Rs. \ 60}{Rs. \ 24}$ $=$ $\frac{5}{2}$

Hence True.

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Q.2. Check whether the following numbers are in proportion or not:

i) $8, 12, 18, 24$ ii) $6.4, 3.6, 4.8, 2. 7$ iii) $11$ $\frac{1}{3}$ $, 9$ $\frac{1}{3}$ $, 8$ $\frac{1}{2}$ $, 7$

iv) $0.36, 1.8, 6.4, 32$ v) $\frac{3}{4}$ $,$ $\frac{5}{6}$ $,$ $\frac{7}{8}$ $,$ $\frac{9}{10}$

Answer:

i) $8, 12, 18, 24$

We have:

$8 : 12 = 2 : 3$

$18 : 24 = 3: 4$

Therefore $8 : 12 \neq 18 : 24$

Hence $8, 12, 18, 24$ are not in proportion

ii) $6.4, 3.6, 4.8, 2. 7$

We have

$6.4:3.6=$ $\frac{64}{36}$ $=$ $\frac{32}{18}$ $=$ $\frac{16}{9}$

$4.8 :2.7=$ $\frac{48}{27}$ $=$ $\frac{16}{9}$

Therefore $6.4 :3.6=4.8 :27$

Hence $6.4, 3.6, 4.8, 2.7$ are in proportion.

iii) $11$ $\frac{1}{3}$ $, 9$ $\frac{1}{3}$ $, 8$ $\frac{1}{2}$ $, 7$

We have

$11$ $\frac{1}{3}$ $: 9$ $\frac{1}{3}$ $=$ $\frac{34}{3}$ $:$ $\frac{28}{3}$ $=$ $\frac{34}{28}$ $=$ $\frac{17}{14}$

$8$ $\frac{1}{2}$ $: 7=$ $\frac{17}{2}$ $:7=$ $\frac{17}{14}$

Therefore $11$ $\frac{1}{3}$ $:9$ $\frac{1}{3}$ $= 8$ $\frac{1}{2}$ $: 7$

Hence $11$ $\frac{1}{3}$ $, 9$ $\frac{1}{3}$ $, 8$ $\frac{1}{2}$ $, 7$ are in proportional

iv) $0.36, 1.8, 6.4, 32$

We have:

$0.36 :1.8=$ $\frac{36}{180}$ $=$ $\frac{1}{5}$

$6.4 :32=$ $\frac{64}{320}$ $=$ $\frac{1}{5}$

Therefore $0.36 : 1.8 = 6.4 : 32$

Hence $0.36, 1.8, 6.4, 32$ are in proportion

v) $\frac{3}{4}$ $,$ $\frac{5}{6}$ $,$ $\frac{7}{8}$ $,$ $\frac{9}{10}$

We have:

$\frac{3}{4} :\frac{5}{6}$ $=$ $\frac{3}{4}\times \frac{6}{5}$ $=$ $\frac{9}{10}$

$\frac{7}{8}:\frac{9}{10}$ $=$ $\frac{7}{8}\times \frac{10}{9}$ $=$ $\frac{70}{72}$

Therefore $\frac{3}{4}$ $:$ $\frac{5}{6}$ $\neq$ $\frac{7}{8}$ $:$ $\frac{9}{10}$

Hence $\frac{3}{4}$ $,$ $\frac{5}{6}$ $,$ $\frac{7}{8}$ $,$ $\frac{9}{10}$ are not in proportion

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Q.3. Find the value of x in each of the following:

i) $8 :x :: 6:27$ ii) $5.6 : 3.5 : :x : 1.25$ iii) $1$ $\frac{4}{5}$ $: 2$ $\frac{4}{5}$ $:: x :3$ $\frac{1}{2}$ iv) $\frac{2}{3}$ $:$ $\frac{4}{7}$ $::1$ $\frac{5}{6}$ $:x$

Answer:

i) $8 :x :: 6:27$

We have $\frac{8}{x}$ $=$ $\frac{6}{27}$ $\ or\ x=$ $\frac{ 8 \times 27}{6}$ $=36$

ii) $5.6 : 3.5 : :x : 1.25$

We have $\frac{56}{35}$ $=$ $\frac{x}{1.25}$ $\ or\ x=$ $\frac{56\times 1.25}{35}$ $= 2$

iii) $1 \frac{4}{5}$ $: 2$ $\frac{4}{5}$ $:: x :3$ $\frac{1}{2}$

We have $\frac{9}{5}$ $:$ $\frac{14}{5}$ $x :$ $\frac{7}{2}$

$\frac{9}{14}$ $=$ $\frac{2x}{7}$ $\ or\ x=$ $\frac{9\times 7}{14\times 2}$ $=$ $\frac{9}{4}$

iv) $\frac{2}{3}$ $:$ $\frac{4}{7}$ $::1$ $\frac{5}{6}$ $:x$

We have $\frac{2}{3}$ $\times$ $\frac{7}{4}=$ $\frac{11}{6x}$ $\ or\ x=$ $\frac{11}{7}$

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Q.4. Find the fourth proportional to:

i) $2.8, 14 \ and\ 3.5$ ii) $3$ $\frac{1}{3}$ $, 1$ $\frac{2}{3}$ $, 2$ $\frac{1}{2}$ iii) $1$ $\frac{5}{7}$ $, 2$ $\frac{3}{14}$ $, 3$ $\frac{3}{5}$ iv) $1$ $\frac{1}{5}$ $, 1$ $\frac{3}{5}$ $, 2.1$

Answer:

i) $2.8, 14 \ and\ 3.5$

Let the fourth proportional term be $x$

We have $\frac{2.8}{14}$ $=$ $\frac{3.5}{x}$ $\ or\ x=$ $\frac{35}{2}$

ii) $3$ $\frac{1}{3}$ $, 1$ $\frac{2}{3}$ $, 2$ $\frac{1}{2}$

Let the fourth proportional term be $x$

$3$ $\frac{1}{3}$ $:1$ $\frac{2}{3}$ $= 2$ $\frac{1}{2}$ $:x$

$\frac{10}{3}$ $:$ $\frac{5}{3}$ $=$ $\frac{5}{2}$ $:x$

$\frac{10}{3}$ $\times$ $\frac{3}{5}$ $=$ $\frac{5}{2x}$ $\ or\ x=$ $\frac{5}{4}$

iii) $1$ $\frac{5}{7}$ $, 2$ $\frac{3}{14}$ $, 3$ $\frac{3}{5}$

Let the fourth proportional term be $x$

$1$ $\frac{5}{7}$ $: 2$ $\frac{3}{14}$ $= 3$ $\frac{3}{5}$ $:x$

$\frac{12}{7}$ $:$ $\frac{31}{14}$ $=$ $\frac{18}{5}$ $:x \ or\ x=$ $\frac{93}{20}$

iv) $1$ $\frac{1}{5}$ $, 1$ $\frac{3}{5}$ $, 2.1$

Let the fourth proportional term be $x$

$1$ $\frac{1}{5}$ $:1$ $\frac{3}{5}$ $:: 2.1 :x$

$\frac{6}{5}$ $:$ $\frac{8}{5}$ $:: 2.1 :\ or\ x=2.8$

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Q.5. Find the third proportional to:

i) $12, 16$ ii) $4.5, 6$ iii) $5$ $\frac{1}{2}$ $, 16$ $\frac{1}{2}$ iv) $3$ $\frac{1}{2}$ $,8$ $\frac{3}{4}$

Answer:

i) $12, 16$

Let the third proportional to $12 \ and\ 16 \ be \ x$

Then, $12 :16 :: 16 :x \ or\ x=$ $\frac{64}{3}$

ii) $4.5, 6$

Let the third proportional to $4.5 \ and\ 6 \ be\ x$

Then $4.5 :6 ::6 :x \ or\ x=8$

iii) $5$ $\frac{1}{2}$ $, 16$ $\frac{1}{2}$

Let the third proportional to $5$ $\frac{1}{2}$ $, 16$ $\frac{1}{2}$ $\ be\ x$

Then $5$ $\frac{1}{2}$ $: 16$ $\frac{1}{2}$ $:: 16$ $\frac{1}{2}$ $:x \ or\ x=$ $\frac{99}{2}$

iv) $3$ $\frac{1}{2}$ $,8$ $\frac{3}{4}$

Let the third proportional to $3$ $\frac{1}{2}$ $, 8$ $\frac{3}{4}$ $\ be\ x$

Then $3$ $\frac{1}{2}$ $: 8$ $\frac{3}{4}$ $:: 8$ $\frac{3}{4}$ $:x \ or\ x= 21$ $\frac{7}{8}$

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Q.6. Find the mean proportional between:

i) $8 \ and\ 18$ ii) $0.3 \ and\ 2. 7$ iii) $66$ $\frac{2}{3}$ $\ and\ 6$ iv) $1.25 \ and\ 0.45$ v) $\frac{1}{7}$ $\ and \$ $\frac{4}{63}$

Answer:

i) $8 \ and\ 18$

Mean proportional between $8 \ and\ 18 = \sqrt{8\times 18} = \sqrt{144} =12$

ii) $0.3 \ and\ 2. 7$

Mean proportional between $0.3 \ and\ 2.7 =$ $\frac{\sqrt{3\times 27}}{10}$ $\frac{9}{10}$ $=0.9$

iii) $66$ $\frac{2}{3}$ $\ and\ 6$

Mean proportional between $66$ $\frac{2}{3}$ $\ and\ 6 =$ $\sqrt{\frac{200}{3}\times 6}$ $\sqrt{400} = 20$

iv) $1.25 \ and\ 0.45$

Mean proportional between $1.25 \ and \ 0.45 = \sqrt{1.25 \times 0.45}=0.75$

v) $\frac{1}{7}$ $\ and \$ $\frac{4}{63}$

Mean proportional between $\frac{1}{7}$ $\ and\$ $\frac{4}{63}$ $=$ $\frac{\sqrt{1\times 4}}{7\times 63}$ $=$ $\frac{2}{21}$

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Q.7. If $28$ is the third proportional to $7 \ and\ x$ , find the value of $x$

Answer:

$7:x ::x :28 \ or\ x^2=7 \times 28 \ or\ x=14$

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Q.8. If $18, x, 50$ are in continued proportion, find the value of $x$

Answer:

$x= \sqrt{18\times 50}=30$

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Q.9. A rod was cut into two pieces in the ratio $7: 5$ . If the length of the smaller piece was $45.5 \ cm$ , then find the length of the longer piece.

Answer:

We have $7 :5 ::x :45.5 \ or\ x=63.7$

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Q.10. The areas of two rectangular fields are in the ratio $5: 9$ . Find the area of the smaller field if that of the larger field is $2331$ sq. meters.

Answer:

We have $5 :9 ::x :2331 \ or\ x=1295$

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Q.11. What number must be subtracted from each of the numbers $41, 55, 36, 48$ so that the differences are proportional?

Answer:

Let the number to be subtracted $= x$

Therefore

$\frac{41-x}{55-x}$ $=$ $\frac{36-x}{48-x}$

$41 \times 48-48 x-41 x+ x^2=36\times 55-55x-36x+ x^2$

solving for $x \ we \ get \ x=2$

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Q.12. An alloy is to contain copper and zinc in the ratio $9 : 4$ . Find the quantity of zinc to be melted with $2$ $\frac{2}{5}$ kg of copper, to get the desired alloy.