A cistern contains 100 liters of pure syrup. 10 liters of syrup are drawn out and replaced by water. Then, 10 liters of the mixture are drawn out and replaced by water again. What is the ratio of syrup to water in the final mixture?

Correct — C. Each draw takes out 10% of what the cistern holds, and water brings it back to 100 litres.

After draw 1: syrup = 100 − 10 = 90 L
After draw 2: syrup = 90 − 10% of 90 = 90 − 9 = 81 L

Water = 100 − 81 = 19 L
Syrup : water = 81 : 19 → option (c)

Why the others are wrong

• (a)
  18:1 — 18 : 1 leaves only about 5.3 litres of water in 100. The first replacement alone pours in 10 litres, and the second adds 10 more while removing just 1.
• (b)
  89:14 — 89 : 14 puts the syrup at 89⁄103 of the cistern, about 86.4 litres. The syrup is already down to 90 litres after the first draw and loses 9 more in the second.
• (d)
  18:17 — 18 : 17 makes the syrup only about 51.4 litres. Just 20 litres of liquid leave the cistern across both draws, so at least 80 litres of syrup must remain.

Repeated replacement: when x litres of a V-litre mixture are drawn out and replaced by water, the syrup is multiplied by (1 − x⁄V) each time. After n rounds, syrup = V × (1 − x⁄V)ⁿ.

Here V = 100, x = 10 and n = 2: syrup = 100 × (0.9)² = 81 litres. Water fills the rest of the fixed 100 litres, so it is 19.

Key facts

• Syrup left after n replacements = V × (1 − x⁄V)ⁿ.
• The volume returns to V after each top-up, so water = V − syrup left.
• (0.9)² = 0.81, so two 10% replacements leave 81% of the syrup.

Common traps

• Treating the second draw as pure syrup: 80 litres of syrup and 20 of water, 4 : 1, not an option.
• Forgetting the second top-up, which leaves 81 litres of syrup with only 9 of water.

No directly related past PYQ was found.