Discussion
A container contains 30 litres of milk. From this, 3 litres are removed and replaced by water. If this is done three times, what is the ratio of milk to water?
Answer: D
Correct answer: 729 : 271
Each operation leaves the fraction \( 1 - \frac{3}{30} = \frac{9}{10} \) of the milk, so after three operations the milk is \[ 30 \times \left(\frac{9}{10}\right)^{3} = 30 \times \frac{729}{1000} = 21.87 \text{ litres}. \] The total is still 30 litres, so the water is \( 30 - 21.87 = 8.13 \) litres, and the ratio is \[ 21.87 : 8.13 = 729 : 271. \] The ratio can be written down directly without computing the volumes at all: the milk is \( (9/10)^{3} = 729/1000 \) of the whole, so the water is \( 271/1000 \), and the ratio is 729 : 271. That shortcut is the useful one, since the ratio depends only on the fraction replaced and the number of repetitions, never on the actual volume. Subtracting 3 litres three times would give 21 : 9, or 7 : 3, and is wrong because each later withdrawal removes a mixture rather than pure milk.
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