Question

The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.

Answer 1

Let one numbers be x then other (9 − x).

Then according to question
$$\begin{gathered} \frac{1}{x}+\frac{1}{\left(9-x\right)}=\frac{1}{2} \\ \Rightarrow \frac{\left(9-x\right)+x}{x\left(9-x\right)}=\frac{1}{2} \\ \Rightarrow \frac{9}{x\left(9-x\right)}=\frac{1}{2} \end{gathered}$$

By cross multiplication

$$\begin{gathered} \Rightarrow 18=x\left(9-x\right) \\ \Rightarrow x^2-9x+18=0 \\ \Rightarrow x^2-6x-3x+18=0 \\ \Rightarrow \left(x-6\right)x-3\left(x-6\right)=0 \\ \Rightarrow \left(x-6\right)\left(x-3\right)=0 \\ \Rightarrow x=6,3 \end{gathered}$$

Since, x being a number,

Therefore,

When x = 6 then

(9 − x) = (9 − 6) = 3

When x = 3 then

(9 − x) = (9 − 3) = 6

Thus, two consecutive number be either 3, 6 or 6, 3.