Discussion
Two vessels A and B contain milk and water mixed in the ratio 5 : 2 and 8 : 5 respectively. Find the ratio in which the contents of vessels A and B should be mixed to get a new mixture containing milk and water in the ratio 9 : 4.
Answer: D
The vessels should be mixed in the ratio 7 : 2.
Work with the fraction of milk in each. Vessel A has \(\frac{5}{7}\), vessel B \(\frac{8}{13}\), and the mixture must have \(\frac{9}{13}\). By the rule of alligation:
\[ \begin{aligned} A : B &= \left(\tfrac{9}{13} - \tfrac{8}{13}\right) : \left(\tfrac{5}{7} - \tfrac{9}{13}\right) \\ &= \tfrac{1}{13} : \tfrac{2}{91} \\ &= 7 : 2 \end{aligned} \]
Check: 7 litres of A give 5 of milk, 2 litres of B give \(\frac{16}{13}\), so milk is \(\frac{81}{13}\) of 9 litres, which is \(\frac{9}{13}\).
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