# What fraction is halfway between 1/3 and 1/5?

Sep 11, 2015

$\frac{4}{15}$

#### Explanation:

A trick question: The obvious, but wrong answer is $\frac{1}{4}$.

Actually $\frac{1}{4}$ is called the harmonic mean of $\frac{1}{3}$ and $\frac{1}{5}$ - being the reciprocal of the average of the reciprocals.

To find the number halfway between $a$ and $b$, calculate $\frac{1}{2} \left(a + b\right)$ ...

$\frac{1}{2} \left(\frac{1}{3} + \frac{1}{5}\right) = \frac{1}{2} \left(\frac{5}{15} + \frac{3}{15}\right) = \frac{1}{2} \left(\frac{8}{15}\right) = \frac{4}{15}$

Note that to add the fractions $\frac{1}{3}$ and $\frac{1}{5}$ we first give them a common denominator $15$ by multiplying them by $\frac{5}{5}$ and $\frac{3}{3}$ respectively. Once we have a common denominator, then we can just add the numerators.