Question

A fraction is such that if the numerator is multiplied by $3$ and the denominator is reduced by $3$, we get $18/11$, but if the numerator is increased by $8$ and the denominator is doubled, we get $2/5$. Find the fraction which satisfies the above conditions.

Answer 1

Hint:Firstly, we need to assume some random variables for the numerator and the denominator. Later on according to the conditions given in the problem, we need to write the mathematical equations. After simplifying them we will get the values of assumed variables.

Complete step-by-step answer:
Given that,
A fraction is such that if the numerator is multiplied by $3$ and the denominator is reduced by $3$, we get $18/11$.
But if the numerator is increased by $8$ and the denominator is doubled, we get $2/5$.
So, let us consider the fraction be $\dfrac{x}{y}$
Now, if the numerator is multiplied by $3$ and the denominator is reduced by $3$
$ \Rightarrow \dfrac{{3x}}{{y - 3}} = \dfrac{{18}}{{11}}$
Now, do cross multiplication for further simplification. We get,
$
   \Rightarrow 33x = 18y - 54 \\
   \Rightarrow 11x - 6y + 18 = 0 \\
 $
Multiply the above equation with two so we can use that later,
$
   \Rightarrow 22x - 12y + 36 = 0 \\
   \Rightarrow 12y = 22x + 36 \\
 $
It is also given that,
If the numerator is increased by $8$ and the denominator is doubled, we get $2/5$.
So, In the similar way,
$ \Rightarrow \dfrac{{x + 8}}{{2y}} = \dfrac{2}{5}$
Now, do cross multiplication for further simplification. We get,
$
   \Rightarrow 5x + 40 = 4y \\
   \Rightarrow 5x - 4y + 40 = 0 \\
 $
Multiply the above equation with three to make the $y$ term in both equations similar.
$
   \Rightarrow 15x - 12y + 120 = 0 \\
   \Rightarrow 12y = 15x + 120 \\
 $
So, as per now we have two equations with the same left-hand value. So, right-hand values alo must be equal.
$
   \Rightarrow 22x + 36 = 15x + 120 \\
   \Rightarrow 7x = 84 \\
   \Rightarrow x = 12 \\
 $
Substitute this $x$ value in one of the derived equations to get the $y$
Equation we use is $12y = 15x + 120$
Substituting $x = 12$ in the above equation,
$
   \Rightarrow 12y = 15 \times 12 + 120 \\
   \Rightarrow 12y = 180 + 120 \\
   \Rightarrow 12y = 300 \\
   \Rightarrow y = \dfrac{{300}}{{12}} \\
   \Rightarrow y = 25 \\
 $
As we succeeded in getting the values of $x$ and $y$. Substitute them in the fraction that we assumed as $\dfrac{x}{y}$ we get $\dfrac{{12}}{{25}}$.
So, the required fraction that satisfies the above conditions is $\dfrac{{12}}{{25}}$.

Note:When we look at the above kind of problems, it’s better to structure the mathematical equations according to the statements that are given in the problem. Later on, by doing simple mathematical operation and simplification we are able to find out the unknowns. Almost this kind of problem can be done in the same way.