Question

In what ratio must water be mixed with milk costing Rs. 12 per litre to obtain a mixture worth Rs. 8 per litre?
a. 1:2
b. 2:1
c. 2:3
d. 3:2

Answer 1

Hint: In order to solve this question, we should know the rule of alligation. So, we will get the formula for finding the ratio as, $\dfrac{\text{Quantity of water}}{\text{Quantity of milk}}\text{=}\dfrac{\text{Cost price of milk-mean value}}{\text{Mean value-cost price of water}}$. By using this concept we can solve this question.
Complete step-by-step answer:
In this question, we have been asked to find the ratio of water to the milk that must be mixed with milk of Rs. 12 per litre to obtain a mixture worth Rs. 8 per litre. To solve this question, we should know the rules of alligation, which states, “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 differences in their cost from the mean value”.
Now, we have been given the cost price of milk as Rs. 12 and the mean value of the mixture as Rs. 8. And we will consider the cost price of water as Rs. 0. So, to find the ratio of the quantities of water and milk, we will apply the rule of allegation, which can be mathematically represented as,
$\dfrac{\text{Quantity of water}}{\text{Quantity of milk}}\text{=}\dfrac{\text{Cost price of milk-mean value}}{\text{Mean value-cost price of water}}$
Now, we will put the value of the cost price of milk, the mean value and cost price of water in the above formula. So, we will get,
\[\dfrac{\text{Quantity of water}}{\text{Quantity of milk}}\text{=}\dfrac{12-8}{8-0}=\dfrac{4}{8}=\dfrac{1}{2}\]
Quantity of water : Quantity of milk = 1:2
Hence, we can say that the ratio of water to be mixed with milk is 1:2. Therefore, option (a) is the correct answer.

Note: We can directly solve this question by applying the formula, $\dfrac{\text{Quantity of ingredient A}}{\text{Quantity of ingredient B}}\text{=}\dfrac{\text{Cost price of B-mean value}}{\text{Mean value-cost price of A}}$ or we can simply represent the given equation as shown below, which implies the formula.