Question

# The sum of a numerator and denominator of a fraction is 18. If the denominator increased by 2, the fraction reduces to $\frac{1}{3}$. Find the fraction.

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Hint: Use Substitution method to find the value of x and y.

Let the numerator be x and denominator be y.
And it is given to us that the sum of numerator and denominator of a fraction is 18.
Therefore we have, x + y = 18.................(i)
And also it is given to us that the denominator increased by 2 and fraction reduces to $\frac{1}{3}$ and hence we have,
$\frac{x}{{y + 2}} = \frac{1}{3}$ ..................(ii)
Now from equation (i) , we have
y= 18 – x
Again on simplifying the equation (ii) we have
3x = y + 2
Therefore on rearranging, we have
3x – y – 2 =0...................(iii)
So on substituting the value of y on equation (iii), we have
$\Rightarrow$ 3x –(18-x) -2=0
And hence on simplification, we have
$\Rightarrow$4x – 20=0
$\Rightarrow$4x = 20
Now 4 will cancel out 20 in 5 times therefore we have
$\Rightarrow$x =5
Now we have the value of x and hence on putting the value of x in equation(i) we have,
$\Rightarrow$5 + y=18
$\Rightarrow$y = 18 – 5
$\Rightarrow$y = 13
Thus the original fraction is $\frac{x}{y} = \frac{5}{{13}}$ .

Note: In this type of question we have to find the value of x and y. So in order to find the value of x and y we’ll use a substitution method after forming two equations with the help of given data and hence after substituting the value of either x or y on those equations, we’ll have our answer.