Question

If 2 is added to the numerator of a fraction it reduces to 1/2 and if 1 is subtracted from the denominator it reduces to 1/3. Find the fraction.

Answer 1

Let the fraction be x/y

Given that
x+2/y =1/2 -----> equ i
x/y-1 =1/3 -----> equ ii

From i and ii
2(x+2) =1(y) -------> equ i
3(x) =1(y-1) -------> equ ii

=> 2x+4=y
3x=y-1

Subtract i and ii. we get

2x - y = -4
3x - y = -1
(-) (+) = (+)
______________
-x = -3
_____________

=> x=3
subtitute x=3 in equ i. we get

2(3+2)=y
=> 2(5) =y
=> y=10

Therefore the value of x and y are 3 and 10 respectively.

Answer 2

Given,
$\frac{x+2}{y}$ = $\frac{1}{2}$ ...............(1)
$\frac{x}{y-1}$ = $\frac{1}{3}$ .................(2)
(1) ⇒ 2x+4 = y
      ⇒2x-y+4 = 0
(2) ⇒ 3x = y-1
      ⇒ 3x-y+1 = 0

solving the equations
3x-y+1 = 0
2x-y+4 = 0
---------------
x-3 = 0
x = 3

substitute x value in any of the above equation,
6-y+4 = 0
y = 10