You may also like

problem icon

Egyptian Fractions

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

problem icon

Fractions of Fractions

Weekly Problem 44 - 2013
If you know that a fraction of X is the same as a different fraction of Y, can you work out X/Y?

problem icon

Keep it Simple

Can all unit fractions be written as the sum of two unit fractions?

Harmonic Triangle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

$$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$$
$$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$$
$$\frac{1}{12} = \frac{1}{20} + \frac{1}{30}$$

Look at the diagonal lines running from the right down to the left. The fractions in the first one are $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and so on.

If $\frac{1}{n}$ is at the end of the nth row, the fraction above it must be $\frac{1}{n-1}$ and the fraction below it must be $\frac{1}{n+1}$.

Have a look at the second diagonal (the one formed by taking the second number in each row: it starts $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}$.

Can you find a pattern for these numbers so that you can work them out easily (without having to subtract fractions)?

Can you explain why the pattern works?