Question

# 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 $12\mathrm{litres}$ of milk such that the ratio of water to milk is $3:5$?

Open in App
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 $1\mathrm{litre}$ milk be $\mathrm{Re}.1$.The amount of milk contained in $1\mathrm{litre}$ mixture from the first can is $\frac{3}{4}\mathrm{litres}$.Therefore, the cost price of $1\mathrm{litre}$ mixture from the first can will be $\frac{3}{4}×\mathrm{Re}.1=\mathrm{Rs}.\frac{3}{4}$.Similarly, the amount of milk contained in $1\mathrm{litre}$ mixture from the second can is $\frac{1}{2}\mathrm{litres}$.Therefore, the cost pricr of $1\mathrm{litre}$ mixture from the second can will be $\frac{1}{2}×\mathrm{Re}.1=\mathrm{Rs}.\frac{1}{2}$.The required ratio of water to milk in $12\mathrm{litres}$ of mixture from both the cans is $3:5$.Hence, the amount of milk contained in $1\mathrm{litre}$ of the required mixture will be $\frac{5}{8}\mathrm{litres}$.Therefore, the mean price will be $\frac{5}{8}×\mathrm{Re}.1=\mathrm{Rs}.\frac{5}{8}$.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.$\begin{array}{rcl}\therefore \frac{x}{y}& =& \frac{\frac{3}{4}-\frac{5}{8}}{\frac{5}{8}-\frac{1}{2}}\\ & =& \frac{1}{8}}{1}{8}}\\ & =& \frac{1}{1}\end{array}$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 $\frac{1}{2}×12\mathrm{litres}=6\mathrm{litres}$.Therefore, to get $12\mathrm{litres}$ of milk such that the ratio of water to milk is $3:5$, the milk vendor should mix $6\mathrm{litres}$ of milk from each can.

Suggest Corrections
0
Explore more