CMAT questions on Arithmetic - Mixture, Alligation & Replacement

Asked: what fraction of the mixture to draw off and replace with water so that milk becomes half.

Milk is 10/16 = 5/8 of the mixture.

If a fraction f is drawn off and replaced with water, only the milk goes down; it becomes \( \frac{5}{8}(1 - f) \).

Set this to 1/2: \( \frac{5}{8}(1-f) = \frac{1}{2} \), so \( 1 - f = \frac{4}{5} \).

f = 1/5.

Check: take 16 L (6 water, 10 milk). Remove 1/5 = 3.2 L, which carries away 2 L of milk. 8 L of milk is left in 16 L. That is half. ✓

Answer: 1/5

Asked: the original quantities of milk and water.

Let milk = m L. Then water = 160 − m L.

Amount removed = 55% of 160 = 88 L.

0.7m + 0.3(160 − m) = 88

0.4m + 48 = 88, so m = 100.

Milk = 100 L and water = 60 L.

Alligation check: 70% and 30% average to 55%. The ratio is (55 − 30) : (70 − 55) = 25 : 15 = 5 : 3, which gives 100 : 60. ✓

Answer: Milk = 100 L; Water = 60 L

Statement I

Mixing ₹15 and ₹20 pulses in the ratio 7 : 3 gives a price of \( \frac{7 \times 15 + 3 \times 20}{10} = \frac{105 + 60}{10} = 16.5 \) per kg.

(By alligation: (20 − 16.5) : (16.5 − 15) = 3.5 : 1.5 = 7 : 3.) True.

Statement II

Alligation of 8% and 16% to get 14% gives (16 − 14) : (14 − 8) = 2 : 6 = 1 : 3. So the rice at 8% : rice at 16% = 1 : 3.

Rice sold at 16% = \( \frac{3}{4} \times 1000 = 750 \) kg.

Check: 250 × 8% + 750 × 16% = 20 + 120 = 140, which is 14% of 1000. True.

Answer: Both Statement I and Statement II are true (option 1)

Asked: the average marks of the boys.

Boys : Girls = 4 : 6. Out of 50 students, boys = 20 and girls = 30.

Total marks of the batch = 50 × 62 = 3100.

Total marks of the girls = 30 × 60 = 1800.

Total marks of the boys = 3100 − 1800 = 1300.

Average for boys = 1300 / 20 = 65.

Alligation check: the girls are 2 marks below 62, so the boys must be x marks above it, with 30 × 2 = 20 × x. That gives x = 3, and 62 + 3 = 65.

Answer: 65