Discussion
8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of water is 16 : 65. How much wine did the cask hold originally?
Answer: B
The operation is performed four times in all - once, then three more.
Wine : water \( = 16 : 65 \) means wine is \( \tfrac{16}{81} \) of the cask:
\[ \left(1 - \frac{8}{T}\right)^{4} = \frac{16}{81} \]
\[ \left(1 - \frac{8}{T}\right)^{2} = \frac{4}{9} \quad\Longrightarrow\quad 1 - \frac{8}{T} = \frac{2}{3} \]
\[ \frac{8}{T} = \frac{1}{3} \quad\Longrightarrow\quad T = 24 \text{ litres} \]
The ratio 16 : 65 has to be converted to the fraction \( \tfrac{16}{81} \) of the whole before the formula applies.
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