Question

A fraction becomes ` (1)/(3)`, if 2 is added to both of its numerator and denominator. If 3 is added to both of its numerator and denominator then it becomes ` ( 2)/(5)`. Find the fraction.

Answer 1

Let the required fraction be `(x)/(y) `. Then,
` (x+ 2 )/( y + 2 ) = (1)/(3) rArr 3x + 6 = y + 2 `
` " " rArr 3x - y = - 4 " " `… (i)
Also, ` ( x + 3 ) /( y + 3 ) = ( 2) /( 5) rArr 5x + 15 = 2y + 6 `
`" " rArr 5x - 2y = - 9 " " `... (ii)
Multiplying (i) by 2 and subtracting (ii) from the result, we get
` 6 x - 5 x = - 8 + 9 rArr x = 1 `
Putting x = 1 in (i), we get
` 3- y = - 4 rArr y = 7`.
Thus, ` x= 1 and y = 7`
Hence, the required fraction is ` (1)/(7)`