Discussion
A vessel contains 40 litres of milk. 4 litres are removed and replaced by water, and this is done twice. How much milk remains?
Answer: B
Correct answer: 32.4 litres
Each operation leaves a fixed fraction of the milk behind, so the quantity remaining after \( n \) operations is \[ Q\left(1 - \frac{x}{Q}\right)^{n}, \] where \( Q \) is the original quantity and \( x \) the amount removed each time. Here \[ 40\left(1 - \frac{4}{40}\right)^{2} = 40 \times \left(\frac{9}{10}\right)^{2} = 40 \times \frac{81}{100} = 32.4 \text{ litres}. \] The reason a formula is needed rather than simple subtraction is that the second withdrawal removes a mixture, not pure milk: 4 litres of the diluted contents contains only 3.6 litres of milk. Subtracting 4 twice gives 32, which is the distractor - and the commonest error. Following it through by hand confirms the result. After the first replacement 36 litres of milk remain; the second removes a tenth of that, 3.6 litres, leaving 32.4.
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