# The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original fraction is 29/20. Find the original fraction. - Mathematics

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original fraction is 29/20. Find the original fraction.

#### Solution

Let the numerator and the denominator of the fraction be x and x + 3, respectively.

Original fraction = x/(x+3)

Now, 2 is added to both the numerator and the denominator.

New fraction = (x+2)/(x+5)

According to the question,

x/(x+3)+(x+2)/(x+5)=29/20

=>(x(x+5)+(x+3)(x+2))/((x+3)(x+5))=29/20

=>(2x^2+10x+6)/(x^2+8x+15)=29/20

40x2+200x+120=29x2+232x+435

11x232x315=0

11x277x+45x315=0

(11x+45)(x7)=0

=>x = 7

Now x!=-45/11 as it is a fraction.

So, the original fraction becomes 7/10

Concept: Solutions of Quadratic Equations by Factorization
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