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.

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#### 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`

⇒40x^{2}+200x+120=29x^{2}+232x+435

⇒11x^{2}−32x−315=0

⇒11x^{2}−77x+45x−315=0

⇒(11x+45)(x−7)=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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