The denominator of a fraction in its simplest form is greater than the numerator by 3. If 7 is added to the numerator, and 5 added to the denominator, then the fraction itself is increased by 1/2. Find the original fraction.

let numerator be x and denominator be y

x = y-3 ... eqn 1

{ (x + 7) / (y+5) } - x/y = 1/2 ... eqn 2

expandin eqn 2 ...

2y (x+7) - 2x (y+5) = y (y+5)

=" 2xy + 14y - 2xy - 10x = y^2 + 5y

=" 14y - 10x = y^2 + 5y ... eqn 3

subst x = y-3 in eqn 3

=" y^2 + 5y - 14y + 10y - 30 = 0

=" y^2 + y - 30 = 0

solving the eqn

y = 5 thus x = 2

subst y = x+3 into eqn 2

2x + 14 = x + 8

x = - 6