### Tweedle Dum and Tweedle Dee

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...

### Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

### Sum Equals Product

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 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

# Unit Fractions

##### Stage: 3 Challenge Level:

Consider the equation ${1\over a} +{1\over b}+{1\over c} = 1$ where $a$, $b$ and $c$ are natural numbers and $0 < a < b < c$. Prove that $a< 3$ and also that $b< 4$ and hence that there is only one set of values which satisfy this equation.

Find the six sets of values which satisfy the equation ${1\over a} +{1\over b}+{1\over c} +{1\over d}= 1$ where $a$, $b$, $c$ and $d$ are natural numbers and $0 < a < b < c < d$.