x / y + 1/10 = ( x + 1) / (y + 1)
[ 10 x + y ] / [ 10y ] = ( x + 1) /( y + 1 )
[10x + y] (y + 1) = 10y ( x + 1)
10xy + y^2 + 10x + y = 10xy + 10y
y^2 - 9y = -10x
y^2 - 9y + 10x = 0 set this up as a quadratic in y
Using the Q Formula
9 ± sqrt [ 9^2 - 4 * 1 * 10x ] 9 ± sqrt [ 81 - 40x ]
_____________________________ = ______________________ = y
2 2
We see that since x is positive......then the radical will only be an integer when x = 2 which produces a perfect square of 1
And y becomes [ 9 + 1 ] / 2 = 5 or y = [9 -1] / 2 = 4
Check
(2/4) + 1/10 = 3 / 5 (2 / 5 ) + 1/10 = 3/6
(1/2) + 1/10 = 3/5 25 / 50 = 1/2
(12) / 20 = 3/5 = true 1/2 = 1/2 = true
The orignal fractions are (2/4) = 1/2 and (2/5)