A housewife has 1000 ml of mixture that contains milk and water in the ratio of 3:1. She adds 250 ml of mixture of milk and water in the ratio of 3:2 in it. Then she uses 250 ml of the combined mixture to make curd. How much of pure milk is left in the mixture remained ?

Correct — option (4), 720 ml.

Step 1 — split the first mixture. In 3 : 1, milk is 3/(3 + 1) = 3/4 of the volume.
Milk = 1000 × 3/4 = 750 ml
Water = 1000 × 1/4 = 250 ml

Step 2 — split the added mixture. In 3 : 2, milk is 3/(3 + 2) = 3/5.
Milk = 250 × 3/5 = 150 ml
Water = 250 × 2/5 = 100 ml

Step 3 — add milk to milk and water to water.
Milk = 750 + 150 = 900 ml
Water = 250 + 100 = 350 ml
Total = 900 + 350 = 1250 ml

Step 4 — take out 250 ml of the combined mixture.
Fraction removed = 250 ÷ 1250 = 1/5
Milk removed = 900 × 1/5 = 180 ml
Milk left = 900 − 180 = 720 ml

Check: water left = 350 × 4/5 = 280 ml, and 720 + 280 = 1000 ml, which is the volume that remains after 250 ml is used.

The idea to remember: a portion taken from a uniform mixture carries every component in the mixture's own ratio, so the 250 ml used for curd holds 180 ml of milk, not 250 ml.

Why the others are wrong

• (1)
  1000 ml — 1000 ml is the whole volume left after the curd is made, milk and water together: 1250 − 250 = 1000 ml.

  The stem asks for pure milk only. Of that 1000 ml, 280 ml is water, so the milk is 720 ml.

• (2)
  912 ml — 912 ml does not come out of any stage of the working. The milk figures along the way are 750 ml, then 900 ml after adding, then 720 ml after removing.

  It also fails a quick check: 912 ml is more than the 900 ml of milk present before any mixture was taken out, so it cannot be the milk left afterwards.

• (3)
  750 ml — 750 ml is the milk in the original 1000 ml of 3 : 1 mixture, before anything was added or removed.

  The added mixture brings in 150 ml of milk, raising it to 900 ml. The 250 ml used for curd then removes 180 ml of that milk, leaving 720 ml.

In a uniform mixture, each component is a fixed fraction of the whole. For milk and water in the ratio a : b, milk is a/(a + b) of the volume and water is b/(a + b).

Two mixtures are combined by adding each component separately. Adding the ratio terms does not work: 3 + 3 : 1 + 2 gives 6 : 3, but 1000 ml at 3 : 1 plus 250 ml at 3 : 2 holds 900 ml milk to 350 ml water, or 18 : 7.

When part of a well-mixed mixture is removed, every component falls by the same fraction as the total volume, so the ratio of what remains does not change.

The same rule handles repeated removal and replacement: if x units are taken out of V units and replaced with water n times, the milk left is the original milk × (1 − x/V)ⁿ.

Key facts

• In a ratio a : b, the first part is a/(a + b) of the whole; 3 : 1 makes milk 3/4, and 3 : 2 makes milk 3/5.
• When two mixtures are combined, milk is added to milk and water to water; the ratios themselves cannot be added.
• Removing part of a uniform mixture reduces every component by the same fraction, so the remaining ratio is unchanged.
• In this item the combined mixture holds 900 ml milk and 350 ml water, a ratio of 18 : 7.
• Removing x units from V units and replacing with water, n times, leaves original milk × (1 − x/V)ⁿ.

Common traps

• Turning 3 : 2 into milk = 3/2 or 3/4 of the volume. The denominator is the sum of the ratio terms, 3 + 2 = 5, so milk is 3/5 of 250 ml = 150 ml.
• Subtracting the whole 250 ml removed from the milk (900 − 250 = 650 ml). The 250 ml taken out is mixture, and 180 ml of it is milk.
• Reporting the volume left, 1000 ml, when the stem asks for pure milk only.

UnlockIAS will link similar questions from RAS Pre 2018 and 2016 here once those papers are published on this site.

• practice — not a real PYQ

  A vessel holds 60 litres of milk and water in the ratio 7 : 3. 10 litres of the mixture are taken out and replaced with 10 litres of water. How much milk is now in the vessel?

  • (a)32 litres
  • (b)35 litres
  • (c)42 litres
  • (d)25 litres
  Answer(2) — Milk = 42 litres; removing 10 of 60 litres takes 1/6 of it, leaving 42 × 5/6 = 35 litres. Added water does not change the milk.

  Option (1) subtracts all 10 litres from the milk; option (3) is the milk before removal; option (4) is the water now (15 + 10 = 25 litres).

• practice — not a real PYQ

  500 ml of a mixture has milk and water in the ratio 4 : 1, and 600 ml of another mixture has them in the ratio 1 : 1. If the two are mixed, the ratio of milk to water in the new mixture is

  • (a)7 : 4
  • (b)5 : 2
  • (c)4 : 1
  • (d)1 : 1
  Answer(1) — Milk = 400 + 300 = 700 ml and water = 100 + 300 = 400 ml, so the ratio is 7 : 4.

  Option (2) adds the ratio terms (4 + 1 : 1 + 1), which ignores the two volumes; options (3) and (4) are the ratios of the separate mixtures.