myhelper: RD Sharma solution class 7 chapter 9 Ratio and Proportion Objective Type Questions

RD Sharma solution class 7 chapter 9 Ratio and Proportion Objective Type Questions

Objective Type Questions

Page-9.14

Question 1:

Mark the correct alternative in the following question:

If a : b = 3 : 4, then 4a : 3b =

(a) 4 : 3 (b) 3 : 4 (c) 1 : 1 (d) None of these

Answer 1:

$As, a : b = 3 : 4 \Rightarrow \frac{a}{b} = \frac{3}{4} So, 4 a : 3 b = \frac{4 a}{3 b} = \frac{4}{3} \times \frac{a}{b} = \frac{4}{3} \times \frac{3}{4} = \frac{12}{12} = \frac{1}{1} = 1 : 1$

Hence, the correct alternative is option (c).

Question 2:

Mark the correct alternative in the following question:

$\frac{1}{12} : \frac{1}{60} =$

(a) 4 : 1 (b) 1 : 4 (c) 5 : 1 (d) 1 : 5

Answer 2:

Since,

$\frac{1}{12} : \frac{1}{60} = \frac{1}{12} \div \frac{1}{60} = \frac{1}{12} \times \frac{60}{1} = \frac{60}{12} = \frac{5}{1} = 5 : 1$

Hence, the correct alternative is option (c).

Question 3:

Mark the correct alternative in the following question:

The simplest form of 24 : 36 is

(a) 9 : 4 (b) 4 : 9 (c) 3 : 2 (d) 2 : 3

Answer 3:

$As, 24 : 36 = \frac{24}{36} = \frac{2}{3} = 2 : 3 So, the simplest form of 24 : 36 = 2 : 3 .$

Hence, the correct alternative is option (d)

Question 4:

Mark the correct alternative in the following question:

If a : b = 4 : 5 and b : c = 2 : 3, then a : c =

(a) 4 : 3 (b) 8 : 15 (c) 8 : 9 (d) 5 : 3

Answer 4:

$As, a : b = 4 : 5 \Rightarrow \frac{a}{b} = \frac{4}{5} Also, b : c = 2 : 3 \Rightarrow \frac{b}{c} = \frac{2}{3} So, a : c = \frac{a}{c} = \frac{a b}{b c} = \frac{a}{b} \times \frac{b}{c} = \frac{4}{5} \times \frac{2}{3} = \frac{8}{15} = 8 : 15$

Hence, the correct alternative is option (b).

Question 5:

Mark the correct alternative in the following question:

$If p : q = 2 : 5, then \frac{25 p + 14 q}{5 p + 7 q} = (a) 8 : 5 (b) 5 : 8 (c) 8 : 3 (d) 3 : 8$

Answer 5:

$As, p : q = 2 : 5 \Rightarrow \frac{p}{q} = \frac{2}{5} Let p = 2 x and q = 5 x Now, \frac{25 p + 14 q}{5 p + 7 q} = \frac{25 \times 2 x + 14 \times 5 x}{5 \times 2 x + 7 \times 5 x} = \frac{50 x + 70 x}{10 x + 35 x} = \frac{120 x}{45 x} = \frac{8}{3} = 8 : 3$

Hence, the correct alternative is option (c).

Question 6:

Mark the correct alternative in the following question:

A ratio equivalent to 2 : 5 is

(a) 6 : 15 (b) 4 : 5 (c) 5 : 2 (d) 5 : 4

Answer 6:

$Since, 2 : 5 = \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} = 6 : 15$

So, the ratio equivalent to 2 : 5 is 6 : 15.

Hence, the correct alternative is option (a).

Question 7:

Mark the correct alternative in the following question:

lf 2a = 3b = 4c, then a : b : c =

(a) 2 : 3 : 4 (b) 3 : 4 : 6 (c) 4 : 3 : 2 (d) 6 : 4 : 3

Answer 7:

$As, 2 a = 3 b = 4 c \Rightarrow 2 a = 3 b and 3 b = 4 c \Rightarrow \frac{a}{b} = \frac{3}{2} and \frac{b}{c} = \frac{4}{3} \Rightarrow \frac{a}{b} = \frac{6}{4} and \frac{b}{c} = \frac{4}{3} \Rightarrow a : b = 6 : 4 and b : c = 4 : 3 ∴ a : b : c = 6 : 4 : 3$

Hence, the correct alternative is option (d).

Question 8:

Mark the correct alternative in the following question:

If 2x = 3y and 4y = 5z, then x : z =

(a) 4 : 3 (b) 8 : 15 (c) 3 : 4 (d) 15 : 8

Answer 8:

$As, 2 x = 3 y \Rightarrow \frac{x}{y} = \frac{3}{2} And, 4 y = 5 z \Rightarrow \frac{y}{z} = \frac{5}{4} Now, x : z = \frac{x}{z} = \frac{x y}{y z} = \frac{x}{y} \times \frac{y}{z} = \frac{3}{2} \times \frac{5}{4} = \frac{15}{8} = 15 : 8$

Hence, the correct alternative is option (d).

Page-9.15

Question 9:

Mark the correct alternative in the following question:

$If \frac{a}{2} = \frac{b}{3} = \frac{c}{4}, then a : b : c = (a) 2 : 3 : 4 (b) 4 : 3 : 2 (c) 3 : 2 : 4 (d) None of these$

Answer 9:

$As, \frac{a}{2} = \frac{b}{3} = \frac{c}{4} \Rightarrow \frac{a}{2} = \frac{b}{3} and \frac{b}{3} = \frac{c}{4} \Rightarrow 3 a = 2 b and 4 b = 3 c (By cross multiplication) \Rightarrow \frac{a}{b} = \frac{2}{3} and \frac{b}{c} = \frac{3}{4} \Rightarrow a : b = 2 : 3 and b : c = 3 : 4 ∴ a : b : c = 2 : 3 : 4$

Hence, the correct alternative is option (a).

Question 10:

Mark the correct alternative in the following question:

$If \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = 3 : 4 : 5, then a : b : c = (a) 5 : 4 : 3 (b) 20 : 15 : 12 (c) 9 : 12 : 15 (d) 12 : 15 : 20$

Answer 10:

$As, \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = 3 : 4 : 5 \Rightarrow \frac{1}{a} : \frac{1}{b} = 3 : 4 and \frac{1}{b} : \frac{1}{c} = 4 : 5 \Rightarrow \frac{1}{a} \div \frac{1}{b} = \frac{3}{4} and \frac{1}{b} \div \frac{1}{c} = \frac{4}{5} \Rightarrow \frac{1}{a} \times \frac{b}{1} = \frac{3}{4} and \frac{1}{b} \times \frac{c}{1} = \frac{4}{5} \Rightarrow \frac{b}{a} = \frac{3}{4} and \frac{c}{b} = \frac{4}{5} \Rightarrow \frac{a}{b} = \frac{4}{3} and \frac{b}{c} = \frac{5}{4} (Reciprocal of both sides) \Rightarrow \frac{a}{b} = \frac{4 \times 5}{3 \times 5} and \frac{b}{c} = \frac{5 \times 3}{4 \times 3} \Rightarrow \frac{a}{b} = \frac{20}{15} and \frac{b}{c} = \frac{15}{12} \Rightarrow a : b = 20 : 15 and b : c = 15 : 12 ∴ a : b : c = 20 : 15 : 12$

Hence, the correct alternative is option (b).

Question 11:

Mark the correct alternative in the following question:

If a : b = 5 : 7 and b : c = 6 : 11, then a : b : c =

(a) 35 : 49 : 66 (b) 30 : 42 : 77 (c) 30 : 42 :55 (d) None of these

Answer 11:

$As, a : b = 5 : 7 and b : c = 6 : 11 \Rightarrow \frac{a}{b} = \frac{5}{7} and \frac{b}{c} = \frac{6}{11} \Rightarrow \frac{a}{b} = \frac{5 \times 6}{7 \times 6} and \frac{b}{c} = \frac{6 \times 7}{11 \times 7} \Rightarrow \frac{a}{b} = \frac{30}{42} and \frac{b}{c} = \frac{42}{77} \Rightarrow a : b = 30 : 42 and b : c = 42 : 77 ∴ a : b : c = 30 : 42 : 77$

Hence, the correct alternative is option (b).

Question 12:

Mark the correct alternative in the following question:

$If x : y = 1 : 1, then \frac{3 x + 4 y}{5 x + 6 y} = (a) \frac{7}{11} (b) \frac{17}{11} (c) \frac{17}{23} (d) \frac{4}{5}$

Answer 12:

$As, x : y = 1 : 1 \Rightarrow \frac{x}{y} = \frac{1}{1} \Rightarrow x = y Now, \frac{3 x + 4 y}{5 x + 6 y} = \frac{3 x + 4 x}{5 x + 6 x} (As, x = y) = \frac{7 x}{11 x} = \frac{7}{11}$

Hence, the correct alternative is option (a).

Question 13:

Mark the correct alternative in the following question:

$If a : b = 2 : 5, then \frac{3 a + 2 b}{4 a + b} = (a) \frac{16}{13} (b) \frac{13}{16} (c) \frac{25}{22} (d) \frac{20}{21}$

Answer 13:

$As, a : b = 2 : 5 \Rightarrow \frac{a}{b} = \frac{2}{5} Let a = 2 x and b = 5 x . Then, \frac{3 a + 2 b}{4 a + b} = \frac{3 \times 2 x + 2 \times 5 x}{4 \times 2 x + 5 x} = \frac{6 x + 10 x}{8 x + 5 x} = \frac{16 x}{13 x} = \frac{16}{13}$

Hence, the correct alternative is option (a).

Question 14:

Mark the correct alternative in the following question:

The mean proportional of a and b is 10 and the value of a is four times the value of b. The value of a + b (a > 0, b > 0) is

(a) 20 (b) 25 (c) 101 (d) 29

Answer 14:

$Since, the mean proportional of two positive numbers a and b is the positive number x such that \frac{a}{x} = \frac{x}{b} . \Rightarrow \frac{a}{10} = \frac{10}{b} \Rightarrow a b = 100 But a = 4 b \Rightarrow 4 b \times b = 100 \Rightarrow b^{2} = \frac{100}{4} \Rightarrow b^{2} = 25 \Rightarrow b = \sqrt{25} \Rightarrow b = 5 \Rightarrow a = 4 \times 5 = 20 ∴ a + b = 20 + 5 = 25$

Hence, the correct alternative is option (b).

Question 15:

Mark the correct alternative in the following question:

If 8 : x : : 16 : 35, then x =

(a) 35 (b) 70 (c) $\frac{35}{2}$ (d) 24

Answer 15:

$As, 8 : x : : 16 : 35 \Rightarrow \frac{8}{x} = \frac{16}{35} \Rightarrow 16 x = 8 \times 35 (By cross multiplication) \Rightarrow x = \frac{8 \times 35}{16} (Transposing 16 to RHS) ∴ x = \frac{35}{2}$

Hence, the correct alternative is option (c).

Question 16:

Mark the correct alternative in the following question:

The mean proportional of 6 and 24 is

(a) 15 (b) 12 (c) 8 (d) 144

Answer 16:

$Let x be the mean proportional of 6 and 24 . Then, \frac{6}{x} = \frac{x}{24} \Rightarrow x^{2} = 6 \times 24 (By cross multiplication) \Rightarrow x^{2} = 144 \Rightarrow x = \sqrt{144} ∴ x = 12$

So, the mean proportional of 6 and 24 is 12.

Hence, the correct alternative is option (b).

Question 17:

Mark the correct alternative in the following question:

The boys and girls in a school are in the ratio 9 : 5. If the number of girls is 320, then the total strength of the school is

(a) 840 (b) 896 (c) 920 (d) 576

Answer 17:

$Let the number of boys in the school be x . Since, the ratio of boys and girls in the school = 9 : 5 \Rightarrow \frac{Number of boys}{Number of girls} = \frac{9}{5} \Rightarrow \frac{x}{320} = \frac{9}{5} \Rightarrow 5 x = 320 \times 9 \Rightarrow x = \frac{320 \times 9}{5} \Rightarrow x = 64 \times 9 \Rightarrow x = 576 ∴ The total strength of the school = 576 + 320 = 896$

Hence, the correct alternative is option (b).

Question 18:

Mark the correct alternative in the following question:

If the first three terms of a proportion are 3, 5 and 21, respectively, then its fourth term is

(a) 21 (b) 35 (c) 15 (d) None of these

Answer 18:

$Let the fourth term be x . As, 3 : 5 : : 21 : x \Rightarrow \frac{3}{5} = \frac{21}{x} \Rightarrow 3 x = 21 \times 5 \Rightarrow x = \frac{21 \times 5}{3} \Rightarrow x = 7 \times 5 ∴ x = 35$

So, the fourth term is 35.

Hence, the correct alternative is option (b).

Question 19:

Mark the correct alternative in the following question:

What must be added to each term of the ratio 9 : 16 to make the ratio 2 : 3?

(a) 5 (b) 3 (c) 4 (d) 6

Answer 19:

$Let the number that must be added to each term of the ratio 9 : 16 be x . Then, (9 + x) : (16 + x) = 2 : 3 \Rightarrow \frac{(9 + x)}{(16 + x)} = \frac{2}{3} \Rightarrow 3 (9 + x) = 2 (16 + x) \Rightarrow 27 + 3 x = 32 + 2 x \Rightarrow 3 x - 2 x = 32 - 27 ∴ x = 5$

So, 5 must be added to each term of the ratio 9 : 16 to make the ratio 2 : 3.

Hence, the correct alternative is option (a).

Question 20:

Mark the correct alternative in the following question:

What least number is to be subtracted from each term of the ratio 15 : 19 to make the ratio 3 : 4?

(a) 3 (b) 5 (c) 6 (d) 9

Answer 20:

$Let the least number that is to be subtracted from each term of the ratio 15 : 19 be x . Then, (15 - x) : (19 - x) = 3 : 4 \Rightarrow \frac{(15 - x)}{(19 - x)} = \frac{3}{4} \Rightarrow 4 (15 - x) = 3 (19 - x) \Rightarrow 60 - 4 x = 57 - 3 x \Rightarrow 3 x - 4 x = 57 - 60 \Rightarrow - x = - 3 ∴ x = 3$

So, 3 is the least number to be subtracted from each term of the ratio 15 : 19 to make the ratio 3 : 4.

Hence, the correct alternative is option (a).

Question 21:

Mark the correct alternative in the following question:

If ₹840 is divided between P and Q in the ratio 3 : 4, then P's share is

(a) ₹340 (b) ₹480 (c) ₹360 (d) ₹400

Answer 21:

$Let P' s share be ₹ x . Then, Q' s share = ₹ (840 - x) As, P' s share : Q' s share = 3 : 4 \Rightarrow \frac{P' s share}{Q' s share} = \frac{3}{4} \Rightarrow \frac{x}{(840 - x)} = \frac{3}{4} \Rightarrow 4 x = 3 (840 - x) \Rightarrow 4 x = 3 \times 840 - 3 x \Rightarrow 4 x + 3 x = 3 \times 840 \Rightarrow 7 x = 3 \times 840 \Rightarrow x = \frac{3 \times 840}{7} \Rightarrow x = 3 \times 120 ∴ x = 360$

So, P's share is ₹360.

Hence, the correct alternative is option (c).

Question 22:

Mark the correct alternative in the following question:

The ages of Ravish and Shikha are in the ratio 3 : 8. Six years hence, their ages will be in the ratio 4 : 9. The present age of Ravish is

(a) 18 years (b) 15 years (c) 12 years (d) 21 years

Answer 22:

$Let the present age of Ravish and Shikha be 3 x and 8 x, respectively . After six years, Age of Ravish = (3 x + 6) years and Age of Shikha = (8 x + 6) years Since, (3 x + 6) : (8 x + 6) = 4 : 9 \Rightarrow \frac{(3 x + 6)}{(8 x + 6)} = \frac{4}{9} \Rightarrow 9 (3 x + 6) = 4 (8 x + 6) \Rightarrow 27 x + 54 = 32 x + 24 \Rightarrow 27 x - 32 x = 24 - 54 \Rightarrow - 5 x = - 30 \Rightarrow x = \frac{- 30}{- 5} \Rightarrow x = 6 ∴ 3 x = 3 \times 6 = 18$

So, the present age of Ravish is 18 years.

Hence, the correct alternative is option (a).

Question 23:

Mark the correct alternative in the following question:

The present ages of Renu and Ravi are in the ratio 5 : 6. The sum of their present ages is 44 in years. The difference of their ages (in years) is

(a) 4 (b) 5 (c) 8 (d) 2

Answer 23:

$Let the present ages of Renu and Ravi be 5 x and 6 x . As, the sum of their present ages = 44 years \Rightarrow 5 x + 6 x = 44 \Rightarrow 11 x = 44 \Rightarrow x = \frac{44}{11} ∴ x = 4 Now, the present age of Renu = 5 \times 4 = 20 years and the present ages of Ravi = 6 \times 4 = 24 years So, the difference of their ages = 24 - 20 = 4 years$

Hence, the correct alternative is option (a).

Page-9.16

Question 24:

Mark the correct alternative in the following question:

The third proportional of 3 and 27 is

(a) 243 (b) 256 (c) 289 (d) 225

Answer 24:

$Let the third proportional of 3 and 27 be x . Then, 3 : 27 : : 27 : x \Rightarrow 3 : 27 = 27 : x \Rightarrow \frac{3}{27} = \frac{27}{x} \Rightarrow 3 x = 27 \times 27 \Rightarrow x = \frac{27 \times 27}{3} \Rightarrow x = 27 \times 9 ∴ x = 243$

So, the third proportional of 3 and 27 is 243.

Hence, the correct alternative is option (a).