Question

Two vessels A and B contain milk and water in the ratio of 5 : 2 and 2 : 3 respectively. When these mixtures are mixed to form a new mixture containing half milk and half water, they must be taken in the ratio-
A. 2 : 5
B. 7 : 3
C. 1 : 2
D. 4 : 5
E. 7 : 15
F. None of these

Answer 1

Hint:In this question, we will first find the ratio of milk and water in each mixture, and then form an equation for mixing them. It is given that there is half water and half milk finally, so that means milk and water are in equal amounts.Equating the equations we will get the required ratio of milk and water.

Complete step-by-step answer:
Ratio of milk to water in A is 5 : 2.
Fraction of milk = mA
It will be the ratio of the milk to the total amount of liquid in A.
$\dfrac5{5+2}=\dfrac57$
Fraction of water = wA
It will be the ratio of the water to the total amount of liquid in A.
$\dfrac2{5+2}=\dfrac27$

Ratio of milk to water in B is 2 : 3.
Fraction of milk = mB
It will be the ratio of the milk to the total amount of liquid in B.
$\dfrac2{2+3}=\dfrac25$
Fraction of water = wB
It will be the ratio of the water to the total amount of liquid in B.
$\dfrac3{2+3}=\dfrac35$

Now, let us assume that x litres of vessel A is mixed with y litres of vessel B.
The amount of milk in the final mixture = milk in A + milk in B = $\dfrac{5\mathrm x}7+\dfrac{2\mathrm y}5$
The amount of water in the final mixture = water in A + water in B = $\dfrac{2\mathrm x}7+\dfrac{3\mathrm y}5$

It is given that the final mixture has half milk and half water, so the amount of water and milk will be equal. We can write that-
\dfrac{5x}7+\dfrac{2y}5=\dfrac{2x}7+\dfrac{3y}5\\\\\\\
$\dfrac{3x}7=\dfrac y5\\\\\dfrac xy=\dfrac7{15}$
Hence, they must be taken in the ratio 7 : 15. The correct option is E. 7 : 15

Note: To solve the problem in an easier way, remember to convert the ratio in fraction. Also, bring the final answer in ratio form as asked in the question. Initially, it may seem wrong that two variables are used and only one equation is formed. But here we have to find the ratio of the two variables and not the individual values.