Maths Problem of the Week

the Mean of Two Unknown Numbers

If

$$\frac{1}{x}+\frac{1}{y}=10$$

and

$$xy=10$$

What is the mean of $x$ and $y$ ?

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Solution

The mean $$\stackrel{-}{x}=\frac{x+y}{2}$$
Multiplying both parts of the quotient by $xy$ gives
$$\stackrel{-}{x}=\frac{\left(x+y\right)xy}{2xy}=\frac{1}{2}(\frac{1}{x}+\frac{1}{y})xy$$
Substituting the values given we get $$\stackrel{-}{x}=\frac{10\times 10}{2}=50$$

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