The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is $\frac{3}{2}$. Find the rational number. 

Let the numerator be x then the denominator = x+8 

∴ The rational number formed = $\frac{x}{x+8}$ 

According to the question, by adding 17 to numerator and subtracting 1 from denominator we get, 

$$\Rightarrow$$ $$\frac{x+17}{(x+8)-1}=\frac{3}{2}$$        (It is given that the new rational number is equal to $\frac{3}{2}$) 

$$\Rightarrow$$ $$\frac{x+17}{(x+7)}=\frac{3}{2}$$ 

$$\Rightarrow 2(x+17)=3(x+7)$$    (by cross multiplying the equation) 

$$\Rightarrow 2x+34=3x+21$$ 

$$\Rightarrow 2x-3x=21-34$$ 

$$\Rightarrow -x=-13$$ 

$$\Rightarrow x=13$$ 

And, $$x+8=13+8=21$$ 

Therefore, numerator = 13 and denominator = 21 

Hence, the required rational number is $\frac{13}{21}$.