Question

The denominator of a fraction is 1 more than its numerator. The sum of the fraction and its reciprocal is $\frac{5}{2}$. Find the fraction.

Answer 1

Step 1: Formation of quadratic equation

Let the numerator be $\mathit{x}$.

So, as the denominator is 1 more than numerator, it will be $x+1$.

Thus, the number is $\frac{x}{x+1}$. Also, it is given that $\frac{x}{x+1}+\frac{x+1}{x}=\frac{5}{2}$

Solve $\frac{x}{x+1}+\frac{x+1}{x}=\frac{5}{2}$, we get,

$$\begin{gathered} \Rightarrow \frac{x}{x+1}+\frac{x+1}{x}=\frac{5}{2} \\ \Rightarrow \frac{x^2+{(x+1)}^2}{x(x+1)}=\frac{5}{2} \\ \Rightarrow \frac{x^2+x^2+1+2x}{x^2+x}=\frac{5}{2} \\ \Rightarrow 2(2x^2+1+2x)=5(x^2+x) \\ \Rightarrow 4x^2+2+4x=5x^2+5x \\ \Rightarrow 5x^2-4x^2+5x-4x-2=0 \\ \Rightarrow x^2+x-2=0 \\ \Rightarrow x^2+2x-x-2=0 \\ \Rightarrow x(x+2)-1(x+2)=0 \\ \Rightarrow (x-1)(x+2)=0 \\ \Rightarrow x=1,-2 \end{gathered}$$

As the number cannot be negative, $x=1$

Step 2: Calculation for required fraction

Replace $\mathit{x}\mathbf{=}\mathbf{1}$ in $\frac{\mathbf{x}}{\mathbf{x}\mathbf{+}\mathbf{1}}$

$$\begin{gathered} \frac{x}{x+1} \\ =\frac{1}{1+1} \\ =\frac{1}{2} \end{gathered}$$

Thus, the number is $\frac{1}{2}$.