Question

The difference of two numbers is the same as their product, and the sum of their reciprocals is $5$. Find the numbers.

Short answer

The two numbers are $\frac{1}{2}and\frac{1}{3}$.

Step-by-step solution

Step 1. Given information.

Assume the two numbers to be xand y.

From the question,

$$\begin{gathered} x-y=xy......\left(i\right) \\ \frac{1}{x}+\frac{1}{y}=5......\left(ii\right) \end{gathered}$$

Multiply equation (ii) with xy,

role="math" localid="1646838166713" $$\begin{gathered} \frac{xy}{x}+\frac{xy}{y}=5xy \\ y+x=5xy......\left(iii\right) \end{gathered}$$

Step 2. Find the two numbers.

Add equations (i) and (iii),

$$\begin{gathered} \left(x+y\right)+\left(x-y\right)=5xy+xy \\ 2x=6xy \\ y=\frac{1}{3} \end{gathered}$$

Substitute the value ofy equation (i),

$$\begin{gathered} x-\frac{1}{3}=\frac{x}{3} \\ x-\frac{x}{3}=\frac{1}{3} \\ \frac{2x}{3}=\frac{1}{3} \\ x=\frac{1}{2} \end{gathered}$$