Question

A milk vendor has two 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 liters of milk such that the ratio of water to milk is $3:5$?

Answer 1

Hint: The concept of percentages and ratios will be used in this question. To convert percentages into decimals or fractions, we can simply divide them by 100 as-
${\text{x}}\% = \dfrac{{\text{x}}}{{100}}$
In order to convert ratios into fractions, we simply keep the first term at numerator and the second at th denominator as-
${\text{x}}:{\text{y}} = \dfrac{{\text{x}}}{{\text{y}}}$

Complete step-by-step solution -
Let us assume the two containers to be A and B. We need to find the amount of milk to be mixed from each container. So, let assume that x liters of mixture A is added to y liters of mixture B.
We have been given that for container A-
Percentage of water is 25%, so the fraction of water will be-
 $\dfrac{{25}}{{100}} = \dfrac{1}{4}$
The amount of water in x litres of mixture will be-
${w_{\text{A}}} = \dfrac{1}{4}{\text{x}}$
So, the fraction of milk will be-
$1 - \dfrac{1}{4} = \dfrac{3}{4}$
The amount of milk in x litres of mixture will be-
${{\text{m}}_{\text{A}}} = \dfrac{3}{4}{\text{x}}$
For container B-
Percentage of water is 50%, so the fraction of water will be-
 $\dfrac{{50}}{{100}} = \dfrac{1}{2}$
The amount of water in y litres of mixture will be-
${{\text{w}}_{\text{B}}} = \dfrac{1}{2}{\text{y}}$
So, the fraction of milk will be-
$1 - \dfrac{1}{2} = \dfrac{1}{2}$
The amount of milk in y litres of mixture will be-
${{\text{m}}_{\text{B}}} = \dfrac{1}{2}{\text{y}}$
It is given that the final ratio of water to milk is $3 : 5$. So we can write that-
$\text{Total water} : \text{Total milk} = 3 : 5$
$ \left( {\dfrac{1}{4}{\text{x}} + \dfrac{1}{2}{\text{y}}} \right):\left( {\dfrac{3}{4}{\text{x}} + \dfrac{1}{2}{\text{y}}} \right) = 3:5 \\ $
  Multiplying and dividing LHS by 4 -
$\dfrac{{{\text{x}} + 2{\text{y}}}}{{3{\text{x}} + 2{\text{y}}}} = \dfrac{3}{5} \\ $
$5(x + 2y) = 3(3x + 2y)$
$5x + 10y = 9x + 6y$
$4x = 4y$
$x = y……...…(1)$
This means that an equal amount of milk is added from both the containers to get 12 liters of milk. Therefore, the amount of milk from each mixture is 6 liters each.
This is the required answer.

Note: In this question, we require detailed knowledge of ratios as well as linear equations. We need to understand the language of the question carefully, understand what is asked, assume the variables, and then form an equation, which can be easily solved using algebraic formulas.