If the numerator of a certain fraction is increased by 2 and the denominator is increased by 1, then the resulting fraction is equal to the 1/2. If how ever the numerator is increased by 1, then denominator is decreased by 2, then the resulting fraction is equal to 3/5. Find the original fraction?

Let the fraction be x / y.
According to the question,
( x+2)/(y+1) = 1/2
⇒ 2x + 4 = y+1
⇒ 2x - y = -3 .........................(i)
Again according to the question,
(x+1)/(y-2) = 3/5
⇒ 5x +5 = 3y - 6
∴ 5x - 3y = -11..........................(ii)
on multiplying Eq. (i) by 3 and then subtracting from Eq. (ii) we get,
multiplying Eq. (i) by 3
6x - 3y = -9
subtracting from Eq. (ii)
5x- 3y - ( 6x - 3y ) = -11 - (-9)
5x- 3y - 6x + 3y = -11 + 9
- x = - 2
x= 2
On putting the value of x in Eq. (i) we get
2x- y = -3
2 x 2 - y = -3
⇒ 2 x 2 -y = -3
⇒ -y = -3 - 4
∴ y = 7
∴ Original fraction = x/y =2/7