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...
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
This problem is taken from the UKMT Mathematical Challenges.