Ratio & Proportion MCQs for UPSC Prelims

Let the price of a brown pair be b, so a black pair costs 3b. Let the original number of brown pairs be x. Original bill = 5 black + x brown = 5(3b) + x(b) = 15b + xb. After interchanging the counts, there are x black pairs and 5 brown pairs, so the wrong bill = x(3b) + 5(b) = 3xb + 5b. A 100% increase means the wrong bill is twice the original: 3xb + 5b = 2(15b + xb). Dividing by b: 3x + 5 = 30 + 2x, so x = 25.

Out of Rs. 2,000 profit with P getting Rs. 400 more than Q: P's share = 1,200 and Q's share = 800, ratio 3 : 2. Profit shares are proportional to capital x time, so (P x 8) : (Q x 10) = 3 : 2, giving 16P = 30Q, i.e., P = 15Q/8. Also P = Q + 14,000, so 15Q/8 - Q = 14,000, i.e., 7Q/8 = 14,000, giving Q = 16,000 and P = 16,000 + 14,000 = Rs. 30,000.

Distance per hour equals speed. Bus speed = 160 / 4 = 40 km/h. Train speed = 320 / 5 = 64 km/h. In one hour the bus covers 40 km and the train covers 64 km, so the ratio is 40 : 64. Dividing both by 8 gives 5 : 8.

Earnings are shared in proportion to work done, i.e. to the rates of work. Let Q's rate = 1 unit. P works thrice as fast as Q, so P's rate = 3. P and Q together work four times as fast as R: (P + Q) = 4R, so R = (3 + 1)/4 = 1. Thus the rates are P : Q : R = 3 : 1 : 1, which is also the ratio of work done (and hence earnings). Hence (a).

A runs the full 500 metres to win. B started 45 m ahead, so B needed only 500 - 45 = 455 m to reach the finish line, but B is still 35 m behind the finish when A wins, meaning B has covered 455 - 35 = 420 m in the same time. Since time is equal, the speed ratio equals the distance ratio: A : B = 500 : 420 = 25 : 21. Hence option (a).

The full capacity equals 18 adults or 30 children, so the space taken by 18 adults equals that of 30 children. Therefore 1 adult occupies the space of 30/18 = 5/3 children. 12 adults occupy 12 x 5/3 = 20 child-spaces. Remaining capacity = 30 - 20 = 10 children. Answer is 10.

Let incomes be 4x and 3x, and expenses be 3y and 2y. Savings = income minus expense, and each saves 6000: 4x - 3y = 6000 and 3x - 2y = 6000. Multiply the first by 2 and the second by 3: 8x - 6y = 12000 and 9x - 6y = 18000. Subtracting gives x = 6000. So Peter's income = 4x = 24000 and Paul's income = 3x = 18000.

There are 120 applicants: 70 male and therefore 50 female. 80 applicants hold a driver's license. To find the minimum number of males with a license, assign licenses to as many females as possible first: all 50 females can hold a license, leaving 80 - 50 = 30 licenses that must go to males. So the minimum number of males with a license is 30. To find the maximum, give licenses to as many males as possible: since there are 70 males and only 80 licenses, all 70 males can hold one. So the maximum is 70. The ratio of minimum to maximum is 30 : 70 = 3 : 7. The official UPSC 2013 key marks (c) 3 to 7, matching the solution.