If the numerator of a fraction is increased by 2 and the denominator is increased by 3, the fraction becomes $$\frac{7}{9}$$ and if both the numerator as well as he denominator are decreased by 1, the fraction becomes $$\frac{4}{5}$$. What is the original fraction ?
Correct Answer: $$\frac{5}{6}$$
Let the fraction be $$\frac{x}{y}$$
Then,
$$\eqalign{ & \Leftrightarrow \frac{{x + 2}}{{y + 3}} = \frac{7}{9} \cr & \Leftrightarrow 9x - 7y = 3.....(i) \cr & {\text{And,}} \cr & \Leftrightarrow \frac{{x - 1}}{{y - 1}} = \frac{4}{5} \cr & \Leftrightarrow 5x - 4y = 1.....(ii) \cr} $$
Solving (i) and (ii), we get :
x = 5 and y = 6
Hence, the original fraction is $$\frac{5}{6}$$
Then,
$$\eqalign{ & \Leftrightarrow \frac{{x + 2}}{{y + 3}} = \frac{7}{9} \cr & \Leftrightarrow 9x - 7y = 3.....(i) \cr & {\text{And,}} \cr & \Leftrightarrow \frac{{x - 1}}{{y - 1}} = \frac{4}{5} \cr & \Leftrightarrow 5x - 4y = 1.....(ii) \cr} $$
Solving (i) and (ii), we get :
x = 5 and y = 6
Hence, the original fraction is $$\frac{5}{6}$$