Question

the numerator of a fraction is 2 less than its denominator. when both numerator and denominator are increased by 3, the fraction is increased by 3/20. find the original fraction

Answer 1

The original fraction will be 3/5 and I am positive. :D

Answer 2

first denominator -- x
first numerator ---- x-2

new denominator -- x+3
new numerator ---- x+1

(x-2)/x + 3/20 = (x+1)/(x+3)
times 20x(x+3) , the LCD
20(x+3)(x+1) + 3x(x+3) = 20x(x+1)
expanding and simplifying gave me
x^2 + 3x - 40 = 0
(x-5)(x+8) = 0
x = 5 or x=-8

if x=5, the original fraction was 3/5
if x=-8 the original fraction was -10/-8 or 5/4

check for 3/5 , new fraction would be 6/8 or 3/4
3/5 + 3/20 = 12/20 + 3/20 = 15/20 = 3/4

but for 5/4, new fraction would be 8/7
5/4 + 3/20 = 28/20 = 7/5 ≠ 8/7
BUT, if we take the unsimplified fraction -10/-8 , new fraction would be -7/-5 = 7/5

So the original fraction would be 3/5 for sure, but
also the unsimplified fraction -10/-8