Question

A fraction becomes 1/2 if 5 is subtracted from its numerator and 3 is subtracted from its denominator. However,if we divide the numerator by 2 and add 7 to the denominator,the fraction becomes 1/4. Find the fraction.

Answer 1

original fraction is n/d

(n-5)/(d-3) = 1/2
(n/2)/(d+7) = 1/4

Solving these gives n = (d+7)/2, not a single unique solution.

Naturally, d cannot be 3, but it is odd, and n must be even, so we can have original fractions of

6/5, 8/9, 10/13, ...

or, in general,

(2k+4)/(4k+1) for k=1,2,3,...

check:
(2k+4-5)/(4k+1-3) = (2k-1)/(4k-2) = 1/2
((2k+4)/2)/(4k+1+7) = (k+2)/(4k+8) = 1/4