Fractions

problem
F'arc'tion

Age

14 to 16

Challenge level

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 paper.
problem
Fair shares?

Age

14 to 16

Challenge level

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?
problem
As easy as 1,2,3

Age

11 to 14

Challenge level

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 'TWO' it returns 2, and so on.
problem
Special sums and products

Age

11 to 14

Challenge level

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.
problem
Sum equals product

Age

11 to 14

Challenge level

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?
problem
Lower bound

Age

14 to 16

Challenge level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
problem
Not continued fractions

Age

14 to 18

Challenge level

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

Age

11 to 14

Challenge level

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...
problem
There's a limit

Age

14 to 18

Challenge level

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?
problem
Rationals between...

Age

14 to 16

Challenge level

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