The numerator of a fraction is 2 less than the denominator. If 1 is added to both numerator and denominator the sum of the new and original fraction is . Find the original fraction.

Given: Numerator of a fraction is 2 less than the denominator

To find: The fraction

Assumption: Let the denominator be x

Numerator = x – 2

Therefore, the fraction

If one is added to the numerator and denominator, fraction becomes

Sum of the fractions

Sum of the fractions

Therefore,

Taking L.C.M we get,

Cross-multiplying we get,

30x² – 30x – 30 = 19x² + 19x

30x² – 19x² – 30x – 19x – 30 = 0

11x² – 49x – 30 = 0

Now we need to factorise such that, on multiplication we get 330 and on substraction we get 49.

So, equation becomes,

11x² – (55x – 6x) – 30 = 0

11x² – 55x + 6x – 30 = 0

11x(x – 5) + 6(x – 5) = 0

(11x + 6)(x – 5) = 0

So, 11x + 5 = 0 or x – 5 = 0

Putting these values in fraction

Hence, the possible fractions are,