Question

# Three containers have their volumes in the ratio 3:4:5. They are full of milk and water. The mixtures contain milk and water in the ratios of 4:1, 3:1 and 5:2, respectively. The ratio of milk and water in the fourth container which contains all these mixtures is [a] 4:1[b] 151:48[c] 157:53[d] 5:2

Hint: Assume that the volumes of the containers are 3x,4x and 5x. Find the content of milk and water in each of these solutions. Hence find the content of milk and water in the final solution. Hence find the ratio of the content of milk to the content of water.

Complete step-by-step solution -

Let the volume of the first container be 3x, the volume of the second container be 4x, and that of the third container be 5x.
Amount of milk in the first container $=\dfrac{4}{1+4}3x=\dfrac{12x}{5}$
Amount of milk in the second container $=\dfrac{3}{3+1}4x=\dfrac{12x}{4}=3x$
Amount of milk in the third container $=\dfrac{5}{5+2}5x=\dfrac{25x}{7}$
Amount of water in the first container $=\dfrac{1}{4+1}3x=\dfrac{3x}{5}$
Amount of water in the second container $=\dfrac{1}{3+1}4x=x$
Amount of water in the third container $=\dfrac{2}{2+5}5x=\dfrac{10x}{7}$
Hence the total amount of milk in the final container $=\dfrac{12x}{5}+3x+\dfrac{25x}{7}=\dfrac{84x+105x+125x}{35}=\dfrac{314x}{35}$
The total amount of water in the final container $=\dfrac{3x}{5}+x+\dfrac{10x}{7}=\dfrac{21x+35x+50x}{35}=\dfrac{106x}{35}$
Hence the ratio of the amount of milk to the amount of water in the fourth container $=\dfrac{314x}{35}:\dfrac{106x}{35}=314:106=157:53$
Hence option [c] is correct.

Note: Alternatively, you can find the ratio of milk to the total volume and hence find the ratio of milk to water.
Let the total volume of the final solution be (3+4+5)=12.
So we have the amount from the first solution = 3
Hence milk from the first solution $=\dfrac{12}{5}$
Similarly milk from the second solution $=3$ and the milk from the third solution $=\dfrac{25}{7}$
Hence total milk $=\dfrac{12}{5}+3+\dfrac{25}{7}=\dfrac{314}{35}$
Hence the ratio of milk to the total volume $=\dfrac{314}{35}:12=\dfrac{157}{35}:6=157:210$
Hence the ratio of milk to water $=157:\left( 210-157 \right)=157:53$