### 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...

### 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 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

### Special Sums and Products

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.

# Mathematical Swimmer

### Why do this problem?

This problem involves calculating with fractions and also uses knowledge of factors and multiples. It is a difficult problem that requires clear and logical thinking.

### Key questions

How do you think you can start on this problem?
Would it help to make a list of some prime numbers?
Why not think of numbers which have many factors and try them?
Would it help to make a list of numbers which have many factors?
Why not start with $12$? What are its factors?
Are all the numbers up to $12$ either a prime number or a fraction that will simplify?
Now do you think you can try with larger numbers?

### Possible extension

Learners could on to Peaches Today, Peaches Tomorrow.

### Possible support

If learners are finding coping with fractions are very difficult suggest trying this Stage 2 problem, Fractions Made Faster or even Fractional Wall.