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Farey Sequences

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

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Do Unto Caesar

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 assets of the players at the beginning of the evening?

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

Peaches Today, Peaches Tomorrow...

Age 11 to 14 Challenge Level:

The monkey will always keep a whole number of peaches.

For Part (ii), why not try a starting value and see what happens? Then decide whether you need to start with a bigger or smaller starting value next time.

For part (iii), you could use a tree diagram: at each stage, branch off the fractions it would be possible to try next so that all possibilities are checked.