Ratio & Proportion Questions and Answers

Answer: A

The value of y is 1. If x varies inversely as the square of y, then \(x y^2\) is constant. From x = 1 and y = 3 the constant is 1 x 9 = 9. For x = 9 we need \[9 y^2 = 9\] so \(y^2 = 1\), and since y is positive, y = 1. The options 9, 2 and 3 give \(xy^2\) values of 729, 36 and 81 at x = 9, which do not equal 9. Check by substitution: with y = 1 the product x times y squared is 9 x 1 = 9, the same as 1 x 3 squared at the start, so the constant holds. The other values of y give 9 x 81 = 729, 9 x 4 = 36 and 9 x 9 = 81, none of which is the constant 9.

Answer: A

The third proportional is 0.0027. If c is the third proportional to a and b, then a : b = b : c, which gives c as the square of b divided by a.

\[ c = \frac{b^2}{a} = \frac{0.09^2}{3} \]

\[ c = \frac{0.0081}{3} = 0.0027 \]

Check: 3 : 0.09 and 0.09 : 0.0027 both equal 100 : 3, since 0.09 divided by 3 is 0.03 and 0.0027 divided by 0.09 is 0.03. The options 0.0018, 0.009 and 0.003 give a different ratio for the second pair, so they are not the third proportional.

Answer: C

The ratio is 8 : 3. The third proportional to 12 and 24 is the number c with 12 : 24 = 24 : c, and the mean proportional between 9 and 36 is the square root of their product.

\[ c = \frac{24 \times 24}{12} \]

\[ c = 48 \]

\[ \text{mean proportional} = \sqrt{9 \times 36} \]

\[ \sqrt{324} = 18 \]

\[ 48 : 18 = 8 : 3 \]

The options 3 : 5, 7 : 5 and 9 : 4 come from mixing up the two proportionals, for example using the mean of 12 and 24 or the third proportional of 9 and 36, and none of them simplifies from 48 : 18.

Answer: B

The ratio of girls to boys is 51 : 49. Girls = 500 - 245 = 255, and the ratio is 255 : 245, which divides by 5 to give 51 : 49.

49 : 51 reverses the order, and 50 : 49 and 51 : 40 are not 255 : 245.

Check: 255 + 245 = 500, and 255 : 245 reduces by the common factor 5 to 51 : 49, which cannot be reduced further because 51 = 3 × 17 and 49 = 7 × 7 share no factor. The question asks for girls to boys, in that order, so 49 : 51 is the wrong way round.

Answer: B

A : D is 2 : 5. Multiply the chain of ratios so that B and C cancel:

5 : 2 is the inverse, and 2 : 3 and 5 : 7 do not follow from the three given ratios.

An equivalent method is to make B and C equal in both ratios: A : B = 2 : 3, B : C = 3 : 4 means A : B : C = 2 : 3 : 4, and C : D = 4 : 5 means A : B : C : D = 2 : 3 : 4 : 5. So A : D = 2 : 5, as the first method showed. The ratio 5 : 2 would be D : A.

Answer: A

There are 80 coins of 25 paise. Let the numbers of coins be 3x, 5x and 2x. Their values in rupees are 3x for the 1 rupee coins, 2.5x for the 50 paise coins and 0.5x for the 25 paise coins, so \[ 3x + 2.5x + 0.5x = 6x = 240 \Rightarrow x = 40 \] The 25 paise coins number 2x = 80.

76, 64 and 52 do not give a total of 240 rupees with this ratio.

Answer: C

The third proportional to a and b is b squared divided by a. For 8 and 12 it is 144/8 = 18, because 8 : 12 = 12 : 18. 21, 24 and 15 do not satisfy 8 : 12 = 12 : x. The third proportional of two numbers a and b is the number c such that a, b, c are in continued proportion, which gives b squared equals a times c. Substituting 12 squared = 144 = 8c gives c = 18.

Answer: A

The ratio r is 5 in the first ratio and 7 in the second, so make r equal by multiplying: p : q : r = 2 : 3 : 5 becomes 14 : 21 : 35 on multiplying by 7, and r : s = 7 : 2 becomes 35 : 10 on multiplying by 5. Then p : q : r : s = 14 : 21 : 35 : 10, option (A). Option (B) has 9 for s, which would not keep r : s = 7 : 2, option (C) has 13 for p, which does not keep p : q = 2 : 3, and option (D) does not follow from either ratio.

Answer: B

There are 27 employees in the ratio 5 : 4, so 15 are seniors and 12 are juniors. A junior gets Rs. 2800, and the shares are in the ratio 7 : 2, so a senior gets 7/2 x 2800 = Rs. 9800. The total share is 15 x 9800 + 12 x 2800 = 147000 + 33600 = Rs. 180600. This is the 30 percent that is divided, since 70 percent is retained, so the total profit is 180600 / 0.3 = Rs. 602000. The answer is Rs. 602000, option (B). The retained 70 percent of the profit is not shared, so only the 30 percent that is divided can be used to work out the shares, as done above.

Answer: A

The value of b is 270. From a : (b + c) = 5 : 7, a is 5/12 of the total, so a = 5/12 x 1944 = 810. From c : (a + b) = 4 : 5, c is 4/9 of the total, so c = 4/9 x 1944 = 864.

\[ b = 1944 - 810 - 864 = 270 \]

The options 320, 350 and 405 do not make the three numbers add up to 1944 with these values of a and c, so the first option is the answer. As a check, a : (b + c) = 810 : 1134 = 5 : 7, and c : (a + b) = 864 : 1080 = 4 : 5, as in the question.