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Question
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
⇒11x2−32x−315=0
⇒11x2−77x+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`
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