The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio $2 : 3$. Determine the fraction

Given:

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio $2 : 3$.

 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$=\frac{x}{y}$

According to the question,

$x+y=2x+4$

$y=2x-x+4$

$y=x+4$.....(i)

When the numerator and denominator are increased by 3, they are in the ratio $2 : 3$.

The new numerator $=x+3$

The new denominator $=y+3$

$\Rightarrow x+3 : y+3=2 : 3$

$\frac{x+3}{y+3}=\frac{2}{3}$

$3(x+3)=2(y+3)$     (On cross multiplication)

$3x+9=2y+6$

$3x-2y+9-6=0$

$3x-2y+3=0$

$3x-2(x+4)+3=0$    (From (i))

$3x-2x-8+3=0$

$x-5=0$

$x=5$

$\Rightarrow y=x+4=5+4=9$.

Therefore, the original fraction is $\frac{5}{9}$.  

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Updated on: 10-Oct-2022

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