Question

the denominator of a fraction exceeds the numerator by 1. if 2 is to taken from each the sum of the reciprocal of the new fraction and 4 times the original fraction is 5. find the original fraction

Answer 1

Original fraction: x / (x+1).
New fraction: (x-2) / ((x+1)-2) = (x-2) / (x-1).

(x-1) / (x-2) + 4x / (x+1) = 5.
Common denominator: (x-2)*(x+1).
(x+1)(x-1) / (x-2)(x+1) + (x-2)4x / (x-2)(x+1) = 5,

Answer 2

(x^2-x+x-1) / (x-2)(x+1) + (4x^2-8x) / (x-2)(x+1) = 5,
(x^2-1) / (x-2)(x+1) + (4x^2-8x) / (x-2)(x+1) = 5,
(5x^2-8x-1) / (x-2)(x+1) = 5,
Cross-multiply:
5x^2 - 8x - 1 = 5x^2 + 5x - 10x - 10,
-8x + 5x = -9,
X = 3.
Original fraction = x / (x+1) = 3/(3+1) = 3/4.