If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes $\frac{6}{5}$. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes $\frac{2}{5}$. Find the fraction.

Given:

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes $\frac{6}{5}$. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes $\frac{2}{5}$. 

 To do:

We have to find the original fraction.

Solution:

Let the numerator and denominator of the original fraction be $x$ and $y$ respectively.

The original fraction$=\frac{x}{y}$

The fraction becomes $\frac{6}{5}$ when the numerator is multiplied by 2 and the denominator is reduced by 5.

This implies,

New fraction$=\frac{2\times x}{y-5}=\frac{2x}{y-5}$

According to the question,

$\frac{2x}{y-5}=\frac{6}{5}$

$5(2x)=6(y-5)$    (On cross multiplication)

$10x=6y-30$

$6y=10x+30$

$6y=2(5x+15)$

$y=\frac{5x+15}{3}$.....(i)

When the denominator is doubled and the numerator is increased by 8, the fraction becomes $\frac{2}{5}$.

This implies,

$\frac{x+8}{2\times y}=\frac{2}{5}$

$5(x+8)=2(2y)$    (On cross multiplication)

$5x+40=4y$

$5x-4y+40=0$

$5x-4(\frac{5x+15}{3})+40=0$     (From (i))

$\frac{3(5x)-4(5x+15)+3(40)}{3}=0$

$15x-20x-60+120=3(0)$

$-5x+60=0$

$5x=60$

$x=\frac{60}{5}$

$x=12$

$\Rightarrow y=\frac{5(12)+15}{3}$

$y=\frac{60+15}{3}$

$y=\frac{75}{3}$

$y=25$

Therefore, the original fraction is $\frac{12}{25}$.  

Tutorialspoint

Updated on: 10-Oct-2022

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