Question

The sum of two numbers is 8. Determine the numbers, if the sum of their reciprocals is $\frac{8}{15}.$

Answer 1

$$\begin{gathered} \text{Let the numbers be}x\text{and}(8-x). \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}: \\ \frac{1}{x}+\frac{1}{8-x}=\frac{8}{15} \\ \begin{array}{l}\Rightarrow \frac{8-\cancel{x}+\cancel{x}}{x(8-x)}=\frac{8}{15}\\ \Rightarrow \frac{{\cancel{8}}^1}{8x-x^2}=\frac{{\cancel{8}}^1}{15} \\ \Rightarrow \frac{1}{8x-x^2}=\frac{1}{15} \\ \Rightarrow 8x-x^2=15\end{array} \\ \Rightarrow x^2-8x+15=0 \\ \Rightarrow x^2-(5+3)x+15=0 \\ \Rightarrow x^2-5x-3x+15=0 \\ \Rightarrow x(x-5)-3(x-5)=0 \\ \Rightarrow (x-5)(x-3)=0 \\ \Rightarrow x-5=0\text{or}x-3=0 \\ \Rightarrow x=5\text{or}x\text{= 3} \\ \mathrm{If}x=5,\text{the numbers are 5 and 3. {∵}(8-5)=3\} \\ \mathrm{If}x=3,\text{the numbers are 3 and 5. {∵}(8-3)=5\} \\ \text{Hence, the numbers are 5 and 3.} \end{gathered}$$