From a full cask, 8 litres are drawn and replaced with water. This is repeated 4 times in total. The final wine:water ratio is 16:65. How many litres of wine did the cask originally hold?

Problem restatementRepeatedly remove 8 L of wine and replace with water, 4 times total. Final ratio wine:water = 16:65. Find the initial capacity.

Concept/ApproachFor repeated replacement, the fraction of wine remaining after n operations is (1 − d/V)ⁿ, where d is the amount removed each time and V is the initial volume. The final fraction equals wine/total.

Step-by-step calculationLet V = initial volumeWine fraction after 4 operations = (1 − 8/V)⁴ = 16/(16 + 65) = 16/81(1 − 8/V) = (16/81)^1/4 = 2/38/V = 1/3 ⇒ V = 24 litres

Verification/Alternative(2/3)⁴ = 16/81, consistent with the given final ratio.

Common pitfallsUsing the single-replacement balance instead of the repeated replacement formula, or miscounting the number of operations.

Final Answer24 litres