Question: If the numerator of a fraction is multiples by 4 and the denominator is reduced by 2, the result is 2. If the numerator of the fraction is increased by 15 and 2 is subtracted from double of the denominator, the result is 9/7. Find the fraction.

Solution: 
Given,

Let the numerator of the required fraction be 'x' and the denominator be'y'. So, the required fraction would be \frac{x}{y}

According to the question,

Condition I,
If the numerator of a fraction is multiples by 4 and the denominator is reduced by 2, the result is 2
or, \dfrac{4x}{y-2} = 2

or, 4x = 2(y-2)

or, x = \dfrac{2(y-2)}{4}

or, x = \dfrac{y-2}{2} - (i)

Condition II,
If the numerator of the fraction is increased by 15 and 2 is subtracted from double of the denominator, the result is 9/7
or, \dfrac{x +15}{2y-2} = \dfrac{9}{7}

or, 7(x +15) = 9(2y -2)

or, 7x + 105 = 18y -18 

or, 7x + 105 +18 = 18y 

or, 7x + 123 = 18y

or, 7x = 18y -123- (ii)

Put value of x from equation (i) in equation (ii), we get,

or, 7 \left ( \dfrac{y-2}{2} \right )  = 18y -123

or, \dfrac{7(y-2) }{2} = 18y -123

or, 7y - 14 = 2(18y -123)

or, 7y -14  = 36y -246

or, 246 -14 = 36y -7y

or, 232 = 29y

or, y = \dfrac{232}{29}

\therefore y = 8

Put value of y in equation (i), we get,

or, x = \dfrac{8-2}{2}

or, x = \dfrac{6}{2}

\therefore x = 3

So, (x,y) = (3,8)

Therefore, the required fraction is \frac{3}{8}.