Ratio, Proportion & Mixtures: previous-year questions

Answer: (D) 72, 96, 108

Multiplying by 12, the ratio $\frac{1}{2} : \frac{2}{3} : \frac{3}{4}$ becomes 6 : 8 : 9. The difference between greatest and smallest is 9k − 6k = 3k = 36, so k = 12 and the numbers are 72, 96, 108. The other sets are not in the ratio 6 : 8 : 9.

Answer: (A) 15

Let the numbers be 5x and 3x; then (5x + 3) : (3x + 3) = 14 : 9 gives 45x + 27 = 42x + 42, so x = 5. The numbers are 25 and 15, so the smaller is 15. 25 is the larger number, and 18 and 28 are those numbers after adding 3.

Answer: (C) ₹ 1,600; ₹ 2,400

Let incomes be 2x, 3x and expenses 5y, 9y; then 2x − 5y = 600 and 3x − 9y = 600. Solving gives y = 200 and x = 800, so the incomes are ₹ 1,600 and ₹ 2,400 (expenses ₹ 1,000 and ₹ 1,800). The other pairs are in ratio 2 : 3 but do not leave ₹ 600 savings each.

Answer: (A) 112

Take the coins as 3x, 8x and 10x; their values are 3x, 4x and x rupees, totalling 8x = 112, so x = 14. The number of 50-paise coins is 8 × 14 = 112. 84 or 96 coins of 50 paise would not make the total value ₹ 112.

No option is correct as printed — this question is disputed. WBPSC awarded the mark to all candidates: no correct option — with sum 92 and ratio 2 : 3 : 4 the 1st number is 20 4/9, not any listed value (a sum of 90 would give 20).

The two ratios combine to 1st : 2nd : 3rd = 2 : 3 : 4, i.e. 9 parts, so the 1st number would be $\frac{2}{9} \times 92 = 20\frac{4}{9}$. That is not a whole number and matches none of the options; with a sum of 90 the answer would have been 20. The Commission therefore awarded the mark to all candidates.

Answer: (D) 70 kg

By alligation, the quantities at 7% and 17% are in the ratio (17 − 10) : (10 − 7) = 7 : 3. So 70 kg was sold at 7% profit and 30 kg at 17%. Option C (30 kg) is the part sold at 17%.

Answer: (A) 11 months

Profit is shared in the ratio of investment × time, so (11 × 8) : (12 × t) = 2 : 3. This gives 24t = 264, so t = 11 months. 8 months would make the profit ratio 11 : 12, the same as the investment ratio.

Answer: (C) ₹ 930

A = $\frac{2}{9}$(B + C) means A is $\frac{2}{11}$ of the total, ₹ 280; B = $\frac{3}{11}$(A + C) means B is $\frac{3}{14}$ of the total, ₹ 330. So C = 1540 − 280 − 330 = ₹ 930. ₹ 280 and ₹ 330 are A's and B's shares.

Answer: (A) ₹ 1,200

If 3A = 4B = 5C, the shares are in the ratio $\frac{1}{3} : \frac{1}{4} : \frac{1}{5}$ = 20 : 15 : 12, i.e. 47 parts. Each part is 2820 ÷ 47 = ₹ 60, so A gets 20 × 60 = ₹ 1,200 (B ₹ 900, C ₹ 720). ₹ 705 is a quarter of the total.

Answer: (C) 20

The mean proportional of a and b is $\sqrt{ab}$, so here it is $\sqrt{16 \times 25} = \sqrt{400} = 20$. 400 is the product itself, not its square root.

Answer: (D) ₹ 4,000

Multiplying by 30, the capitals are 105 : 40 : 36. Over the year A's capital-months are 105 × 4 + 157·5 × 8 = 1680, B's 40 × 12 = 480 and C's 36 × 12 = 432, total 2592. B's share = $\frac{480}{2592} \times 21600$ = ₹ 4,000.

Answer: (B) ₹ 340

Total parts = 4 × 9 + 5 × 8 + 6 × 4 = 36 + 40 + 24 = 100, so each part is ₹ 42·50. A woman gets 8 parts = ₹ 340. ₹ 1,700 is the share of all five women together.

Answer: (D) 41 : 15

Taking one unit of each, milk = $\frac{5}{7} + \frac{3}{4} = \frac{41}{28}$ and water = $\frac{2}{7} + \frac{1}{4} = \frac{15}{28}$. So the new ratio is 41 : 15. 8 : 3 wrongly adds the ratio terms directly.

Answer: (C) ₹ 34·50

The ratio has 2 + 5 + 4 = 11 parts, so one part = 126·50 ÷ 11 = ₹ 11·50. B exceeds A by 5 − 2 = 3 parts = ₹ 34·50. The other options are off by a rupee or two.

Answer: (A) 60 cc

The mixture has 60 cc water and 180 cc glycerine. For water : glycerine = 2 : 3 the water must be 120 cc, so 120 − 60 = 60 cc must be added. 80 cc would make the ratio 7 : 9.

Answer: (B) 16 litres

By alligation, first : second = (17 − 13) : (25 − 17) = 4 : 8 = 1 : 2. So with 8 litres of the first, 16 litres of the second are needed. 4 litres reverses the ratio.

Answer: (A) ₹ 625

After the deductions the total is 1570 − 25 − 45 = ₹ 1,500, shared 2 : 3 as ₹ 600 and ₹ 900. Adding back the deductions, A received 600 + 25 = ₹ 625 (and B ₹ 945). ₹ 945 is B's amount.

Answer: (B) 4 : 6 : 3

Capital × time: A = 2 × 12 = 24, B = 3 × 12 = 36, C = 3 × 6 = 18. So the profit ratio is 24 : 36 : 18 = 4 : 6 : 3. 2 : 3 : 3 ignores that C was in for only 6 months.

Answer: (D) ₹ 5,000

Each person spends 4500 ÷ (15 × 6) = ₹ 50 a month. For 25 persons over 4 months the total is 25 × 4 × 50 = ₹ 5,000. The smaller amounts treat the expenditure as falling with more people.

Answer: (D) ₹ 400

B should get $\frac{5}{8}$ of the amount but gets $\frac{3}{5}$; the difference is $\frac{25 - 24}{40} = \frac{1}{40}$ of the amount. So $\frac{1}{40}$ of the amount = ₹ 10 and the amount is ₹ 400. Check: B gets ₹ 240 instead of ₹ 250.