Question

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.

Answer 1

Step 1:Assuming the rational number

Let the numerator of the rational number be $x$.

Therefore by the given condition denominator $=x+8$

Hence the rational number is $\frac{x}{x+8}$

Step 2.Finding the rational number.

According to question If the numerator is increased by $17$ and the denominator is decreased by $1$, the number obtained is $\frac{3}{2}$

Therefore we have

,$\frac{x+17}{x+8-1}=\frac{3}{2}$

$$\begin{gathered} \Rightarrow \frac{x+17}{x+7}=\frac{3}{2} \\ \Rightarrow 2(x+17)=3(x+7) \\ \Rightarrow 2x+34=3x+21 \\ \Rightarrow 2x-3x=21-34 \\ \Rightarrow -x=-13 \\ \Rightarrow x=13 \end{gathered}$$

Hence the numerator is $13$.

Now the denominator $=x+8$

$$\begin{gathered} =13+8 \\ =21 \end{gathered}$$

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