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. . . .

It would be nice to have a strategy for disentangling any tangled ropes...

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Can all unit fractions be written as the sum of two unit fractions?

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.

Take a look at the video and try to find a sequence of moves that will take you back to zero.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

How much of the square is coloured blue? How will the pattern continue?

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Here is a chance to play a fractions version of the classic Countdown Game.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

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...

Look carefully at the video of a tangle and explain what's happening.

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?

Sometime during every hour the minute hand lies directly above the hour hand. At what time between 4 and 5 o'clock does this happen?

Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?

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

A jigsaw where pieces only go together if the fractions are equivalent.

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

My recipe is for 12 cakes - how do I change it if I want to make a different number of cakes?

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Which dilutions can you make using only 10ml pipettes?

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions