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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

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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`

 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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