Egyptian fractions with the greedy algorithm

Text

The ancient Egyptians wrote every fraction as a sum of distinct unit fractions (numerator $1$) using the greedy algorithm: at each step you subtract the largest possible unit fraction. Apply the method to $\frac{5}{11}$.

Solution

Step 1. $\frac{5}{11}\approx 0.4545$; the largest unit fraction not exceeding it is $\frac13$ (since $\frac13\le\frac{5}{11}<\frac12$). Subtracting: $\frac{5}{11}-\frac13 = \frac{15-11}{33} = \frac{4}{33}.$ Step 2. $\frac{4}{33}\approx 0.1212$; the largest unit fraction not exceeding it is $\frac19$ (since $\frac18=0.125>\frac{4}{33}$). Subtracting, with common denominator $99$: $\frac{4}{33}-\frac19 = \frac{12-11}{99} = \frac{1}{99}.$ The numerator is now $1$, so we stop: $\frac{5}{11} = \boxed{\frac13+\frac19+\frac{1}{99}}.$