Question

In what ratio should a $20\% $ methyl alcohol solution be mixed with a $50\% $ methyl alcohol solution so that the resultant solution has $40\% $ methyl alcohol in it?
(A) $2:1$
(B) $1:2$
(C) $3:1$
(D) $2:3$

Answer 1

Hint: First we must identify the known ratio and unknown ratio, and set up the proportion, and then go for a cross multiplication, and check the value by plugging the result into an unknown ratio.

Formula used: Mixing of two solution will get added into third solution,
Thus $a + b\,\,\, = \,\,c$
$a = $Value of first solution,
$b = $Value of second solution,
$c = $Value of mixed solution.

Complete step-by-step answer:
We are mixing two solution to get grid,
So $a + b\,\,\, = \,\,c$
We don't know how much to mix nor how much we will end up with.
So we call the amount to mix $x$ and $y$ and the total will of course be $x + y$.
The given first solution is $20\% $ of methyl alcohol we take this as
$2x \to \left( 1 \right)$
In above $2x$ means $20\% $ of alcohol is converted into $2$ and $x$ be a variable to find the ratio of the percentage.
The given second solution is $50\% $ of methyl alcohol we take this as
$5y \to \left( 2 \right)$
In above $5x$ means $50\% $ of alcohol is converted into $5$ and $y$ be a variable to find the ratio of the percentage.
If we add the both solutions we get $40\% $ of methyl alcohol.
Thus we take it as,
$4\left( {x + y} \right) \to \left( 3 \right)$
If we add $\left( 1 \right)$ and $\left( 2 \right)$ we get $\left( 3 \right)$,
$2x + 5y = 4\left( {x + y} \right)$
If we multiply $4\left( {x + y} \right)$ we get,
$2x + 5y = 4x + 4y$,
If we subtract $4y$ and $5y$ then the equation becomes
$2x + y = 4x$
Thus $y = 2x$
So we will mix $2x$ for every $y$
$
  \dfrac{y}{x} = 2 \\
    \\
 $
And $\dfrac{x}{y} = \dfrac{1}{2}$
These division forms are get converted into rational form as $1:2$

So, the correct answer is “Option B”.

Note: To compare ratios, you need to have a common second number. By multiplying each ratio by the second number of the other ratio, you can determine if they are equivalent. Multiply both numbers in the first ratio by the second number of the second ratio.