Question

The difference of two numbers is 5 and the difference of their reciprocals is $\left(\frac{1}{10}\right)$. Find the numbers.

Answer 1

$$\begin{gathered} \text{Let the numbers be}x\text{and}(x+5). \\ \text{According to the question:} \\ \frac{1}{x}\text{-}\frac{1}{x+5}=\frac{1}{10} \\ \Rightarrow \frac{x+5-x}{x(x+5)}=\frac{1}{10} \\ \Rightarrow \frac{5}{x^2+5x}=\frac{1}{10} \\ \Rightarrow x^2+5x=50 \\ \Rightarrow x^2+5x-50=0 \\ \Rightarrow x(x+10)-5(x+10)=0 \\ \Rightarrow (x+10)(x-5)=0 \\ \Rightarrow x+10=0\text{or}x-5=0 \\ \Rightarrow x=-10\text{or}x\text{= 5} \\ \mathrm{If}x=5,\text{the numbers are 5 and 10. (∵ 5 + 5 = 10)} \\ \mathrm{If}x=-10,\text{the numbers are -10 and -5. (∵ -10 + 5 = -5)} \\ \text{Hence, the numbers are (5, 10) and (-10, -5).} \end{gathered}$$