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?
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.
They were standing under a tree, each with an arm round the
other's neck, and Alice knew which was which in a moment, because
one of them had "DUM" embroidered on his collar, and the other
"I suppose they've each got 'TWEEDLE' round at the back of the
collar," she said to herself.
Two brothers were left some money, amounting to an exact number
of pounds all in coins, to divide between them. DEE undertook the
"But your heap is larger than mine!" cried DUM.
"True," said DEE. "Allow me to present you with one-third of my
DUM added it to his heap, and after looking thoughtfully at the
now gigantic pyramid, he suddenly exclaimed "I am well off now!
Here is half of the heap for you."
"You are generous," said DEE, as he swept up the money.
"Two-thirds of this heap is the least I can offer you."
"I will not be outdone in generosity!" cried DUM, hastily
handing over three-quarters of his property.
"Prudence is a virtue," remarked DEE. "Content yourself with
two-thirds of my present wealth."
"One-third of mine is all I can now afford!" retorted DUM.
"And now, if I give you one pound," remarked DEE, "we shall, I
think, be square?"
He was right. How much money was shared between them?