Question

Two barrels contain a mixture of ethanol and gasoline. The content of the ethanol is 60% in the first barrel and 30% in the second barrel. In what ratio must the mixtures from the first and the second barrels be taken to form a mixture containing 50% ethanol?

Answer 1

Hint – In this let the ratio of the mixtures from the first and the second barrels be taken to form a mixture containing 50% ethanol be $1:x$. Now use the concept that 1 multiplied by percentage of ethanol of the first barrel add to x multiplied by the percentage of ethanol of second barrel = (1 + x) percentage of ethanol of the resulting mixture. This will give the required value of x.

Complete step-by-step answer:
Mixtures of ethanol and gasoline are given.
In first barrel the content of the ethanol (${E_1}$) is 60%
In second barrel the content of the ethanol (${E_2}$) is 30%
Now we have to find the ratio of first and second mixture so that the resulting mixture has 50% ethanol.
Let the ratio be $1:x$
So 1 multiplied by percentage of ethanol of the first barrel add to x multiplied by the percentage of ethanol of the second barrel = (1 + x) percentage of ethanol of the resulting mixture.
$ \Rightarrow 1 \times \dfrac{{60}}{{100}} + x \times \dfrac{{30}}{{100}} = \left( {1 + x} \right)\dfrac{{50}}{{100}}$
Now simplify this we have,
$ \Rightarrow 1 \times 60 + x \times 30 = \left( {1 + x} \right)50$
$ \Rightarrow 6 + 3x = \left( {1 + x} \right)5$
$ \Rightarrow 6 + 3x = 5 + 5x$
$ \Rightarrow 5x - 3x = 6 - 5$
$ \Rightarrow 2x = 1$
$ \Rightarrow x = \dfrac{1}{2}$
So the required ratio is $\dfrac{1}{x} = \dfrac{1}{{\dfrac{1}{2}}} = \dfrac{2}{1}$
So the required ratio is (2:1) so that the resulting mixture has 50% ethanol.

Note – The trick point here is that the concept of 1 multiplied by percentage of ethanol of the first barrel add to x multiplied by the percentage of ethanol of second barrel = (1 + x) percentage of ethanol of the resulting mixture, the right hand side entity of percentage of ethanol of the resulting mixture varies as per question requirement, in this question it was asked about 50%, if it would have asked about 70% or say 10% then the value of this resulting % would have changed while rest concept remains the same. So this is rather a standard approach to solve problems of this kind.