Practice Puzzle: Summing to 1

The Problem

It is easy to add the same fractional unit to get 1: $\frac{1}{2} + \frac{1}{2} = 1$ $\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$

Challenge: Can you write 1 as the sum of three different fractional units?

The Solution

We need three unit fractions: $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$

Let’s start with the biggest unit fraction: $\frac{1}{2}$. Remaining needed: $\frac{1}{2}$. Now we need two different fractions that add up to $\frac{1}{2}$. We know $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$, but they must be different.

Let’s try $\frac{1}{3}$. $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. We are short by $\frac{1}{6}$. Wait! That’s a unit fraction!

So: $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$

Try This: Can you find four different fractional units that add up to 1?