Question

The sum of the two natural numbers is 8 and the sum of their reciprocals is 8 /15 . Find the numbers by eliminatiom method

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{natural}\mathrm{numbers}\mathrm{be}\mathrm{x}\mathrm{and}\mathrm{y}. \\ \mathrm{x}+\mathrm{y}=8....\left(1\right) \\ \frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=\frac{8}{15}....\left(2\right) \\ \mathrm{Consider}\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=\frac{8}{15} \\ \Rightarrow \frac{\mathrm{y}+\mathrm{x}}{\mathrm{xy}}=\frac{8}{15} \\ \Rightarrow \frac{8}{\mathrm{xy}}=\frac{8}{15}\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow \mathrm{xy}=15 \\ \mathrm{Now},{\left(\mathrm{x}-\mathrm{y}\right)}^2={\left(\mathrm{x}+\mathrm{y}\right)}^2-4\mathrm{xy}={\left(8\right)}^2-4\times 15=64-60=4 \\ \mathrm{Now},\mathrm{x}-\mathrm{y}=2.....\left(3\right)\mathrm{or}\mathrm{x}-\mathrm{y}=-2.....\left(4\right) \\ \mathrm{Adding}\left(1\right)\mathrm{and}\left(3\right),\mathrm{we}\mathrm{get} \\ 2\mathrm{x}=10\Rightarrow \mathrm{x}=5 \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get}\mathrm{y}=3 \\ \mathrm{Adding}\left(1\right)\mathrm{and}\left(4\right),\mathrm{we}\mathrm{get} \\ 2\mathrm{x}=6 \\ \Rightarrow \mathrm{x}=3 \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get}\mathrm{y}=5 \\ \mathrm{So},\mathrm{required}\mathrm{natural}\mathrm{numbers}\mathrm{are}3\mathrm{and}5. \end{gathered}$$