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# Keep it Simple

#### Unit fractions (fractions which have numerators of 1) can be written as the sum of two __different__ unit fractions.

The denominator of the last fraction is the product of the denominators of the first two fractions.
#### Alison started playing around with $\frac{1}{6}$ and was surprised to find that there wasn't just one way of doing this.

#### Charlie tried to do the same with $\frac{1}{8}$. Can you finish Charlie's calculations to see which ones work?

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Age 11 to 14

Challenge Level

*Keep it Simple printable sheet*

For example

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

Charlie thought he'd spotted a rule and made up some more examples.

$\frac{1}{2} = \frac{1}{10} + \frac{1}{20}$

$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$

$\frac{1}{3} = \frac{1}{7} + \frac{1}{21}$

$\frac{1}{4} = \frac{1}{5} + \frac{1}{20}$

Can you describe Charlie's rule?

The denominator of the last fraction is the product of the denominators of the first two fractions.

Are all his examples correct?

What do you notice about the sums that are correct?

Find some other correct examples..

How would you explain to Charlie how to generate lots of correct examples?

She found:

$\frac{1}{6} = \frac{1}{7} + \frac{1}{42}$

$\frac{1}{6} = \frac{1}{8} + \frac{1}{24}$

$\frac{1}{6} = \frac{1}{9} + \frac{1}{18}$

$\frac{1}{6} = \frac{1}{10} + \frac{1}{15}$

$\frac{1}{6} = \frac{1}{12} + \frac{1}{12}$ (BUT she realised this one didn't count because they were not different.)

$\frac{1}{8} = \frac{1}{9} + ?$

$\frac{1}{8} = \frac{1}{10} + ?$

$\frac{1}{8} = \frac{1}{11} + ?$

..........

Can all unit fractions be made in more than one way like this?

Choose different unit fractions of your own to test out your theories.

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.