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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?

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

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

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

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What fractions can you find between the square roots of 65 and 67?

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Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

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

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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. . . .

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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.

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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.

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Take a look at the video and try to find a sequence of moves that will untangle the ropes.

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What fractions of the largest circle are the two shaded regions?

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

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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.

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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.

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Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

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There are lots of ideas to explore in these sequences of ordered fractions.

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When I type a sequence of letters my calculator gives the product of all the numbers in the corresponding memories. What numbers should I store so that when I type 'ONE' it returns 1, and when I type. . . .

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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?

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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?

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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?

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The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

This article extends and investigates the ideas in the problem "Stretching Fractions".

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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.

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Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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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.

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Pick two rods of different colours. Given an unlimited supply of rods of each of the two colours, how can we work out what fraction the shorter rod is of the longer one?

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

Written for teachers, this article describes four basic approaches children use in understanding fractions as equal parts of a whole.

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Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

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There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

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A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

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I need a figure for the fish population in a lake. How does it help to catch and mark 40 fish?

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Can you work out the parentage of the ancient hero Gilgamesh?

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The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

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

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Can you find the pairs that represent the same amount of money?

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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?

Who first used fractions? Were they always written in the same way? How did fractions reach us here? These are the sorts of questions which this article will answer for you.

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Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you find its length?

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Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

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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...

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What do you notice about these families of recurring decimals?

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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the. . . .