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.

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and denominator, the sum of the new fraction and the original fraction is \[1\frac{9}{20}\]. Find the fraction.

Answer 1

Let the denominator of the required fraction be x.

Then, its numerator = x – 3

So, the original fraction is `(x - 3)/x`.

Given,

`((x - 3) + 2)/(x + 2) + ((x - 3))/x = 29/20`

`((x - 1))/(x + 2) + ((x - 3))/x = 29/20`

`((x - 1)x + (x - 3)(x + 2))/((x + 2)x) = 29/20`

`(x^2 - x + x^2 - x - 6)/(x^2 + 2x) = 29/20`

20(2x² – 2x – 6) = 29(x² + 2x)

11x² – 98x – 120 = 0

11x² – 110x + 12x – 120 = 0

11x(x – 10) + 12(x – 10) = 0

(11x + 12)(x – 10) = 0

x = 10 or x = `-12/11`

Negative denominator not possible.

∴ x = 10

Now substitute to get the original fraction:

`(x - 3)/x`

= `(10 - 3)/10`

= `7/10`

Hence, the required fraction is `7/10`.