A milk vendor has $2$ cans of milk. The first contains $25 %$ water and the rest milk. The second contains $50 %$ water. How much milk should he mix from each of the containers so as to get $12 litres$ of milk such that the ratio of water to milk is $3 : 5$ ?

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Solution

Step 1: Determine the mean value.
It is given that the ratio of water to milk in the first can is $25 % : 75 %$ i.e. $1 : 3$ and in the second can is $50 % : 50 %$ i.e. $1 : 1$ .
Let the cost of $1 litre$ milk be $Re . 1$ .
The amount of milk contained in $1 litre$ mixture from the first can is $\frac{3}{4} litres$ .
Therefore, the cost price of $1 litre$ mixture from the first can will be $\frac{3}{4} \times Re . 1 = Rs . \frac{3}{4}$ .
Similarly, the amount of milk contained in $1 litre$ mixture from the second can is $\frac{1}{2} litres$ .
Therefore, the cost pricr of $1 litre$ mixture from the second can will be $\frac{1}{2} \times Re . 1 = Rs . \frac{1}{2}$ .
The required ratio of water to milk in $12 litres$ of mixture from both the cans is $3 : 5$ .
Hence, the amount of milk contained in $1 litre$ of the required mixture will be $\frac{5}{8} litres$ .
Therefore, the mean price will be $\frac{5}{8} \times Re . 1 = Rs . \frac{5}{8}$ .
Step 2: Apply the rule of alligation to find the required ratio.
According to the rule of alligation, when different quantities of different ingredients are mixed together to produce a mixture of a mean value, the ratio of their quantities is inversely proportional to the difference in their cost from the mean value.
$\begin{array}{rcl} ∴ \frac{x}{y} & = & \frac{\frac{3}{4} - \frac{5}{8}}{\frac{5}{8} - \frac{1}{2}} \\ = & \frac{\frac{1}{8}}{\frac{1}{8}} \\ = & \frac{1}{1} \end{array}$
Therefore, the required ratio of the quantities of the mixtures from the two cans is $1 : 1$ .
Hence, the quantity of mixture taken from each can should be $\frac{1}{2} \times 12 litres = 6 litres$ .
Therefore, to get $12 litres$ of milk such that the ratio of water to milk is $3 : 5$ , the milk vendor should mix $6 litres$ of milk from each can.

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