# A fraction becomes 2/3 if its numerator is increased by 1 and denominator by 2.it becomes 3/4 if its numerator is increased by 4 and the denominator by 5. find the fraction.

Let the fraction

$\frac{x}{y}\\$

According to 1st condition

$\frac{x+1}{y + 2} = \frac{2}{3} \\ \Rightarrow 3(x +1) = 2( y+ 2) \\ \Rightarrow 3x -2y = 1 .....(1) \\$

Now 2nd condition

$\frac{x+4}{y + 5} = \frac{3}{4} \\ \Rightarrow 4(x +4) = 3( y+ 5) \\ \Rightarrow 4x -3y = -1 .....(2) \\$

multiply 3 in Equation (1) and 2 in equation (2)

$9x -6y = 3 ....(3) \\ 8x -6y = -2 .....(4) \\$

Solve equation (3) and equation (4) we get

$x = 5 and, y = 7 \\$

Hence, the fraction

$= \frac{5}{7} \\$

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