7.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.
and denominator be y

As given y= (x + 8)

Also as per the conditions given, we have

\(\frac{x+17}{x+8-1}=\frac{3}2\)

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

\(\Rightarrow 2(x + 17) = 3(x + 7) \quad\)[Cross-multiplication]

\(\Rightarrow 2x + 34 = 3x + 21\quad \)[Solving the bracket]

\(\Rightarrow 2x – 3x = 21 – 34 \quad \)[Transposing 3x to LHS and 34 to RHS]

\(\Rightarrow -x = -13\)

\(\Rightarrow x = 13\)

Hence, numerator = 13

and denominator = 13 + 8 = 21

Thus, the rational number is \(\frac { 13 }{ 21 }\).