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How Big?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

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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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Oh for the Mathematics of Yesteryear

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45 days against the siege of the enemy?

Punky Fish

Stage: 3 Short Challenge Level: Challenge Level:1

If there are f females and m males then $86=8f+9m=8(f+m)+m$. Now $86=8\times 10+6=8 \times 9+14...$ Only the first of these is feasible so m=6 and f=4. Therefore the ratio of male to female fish is $3:2$.

Alternatively, dividing the stripes into groups of $8$, there are $6$ left over. One left over is produced by each male, so there are $6$ or $14$ or... males. However, as $14 \times 9 = 126 > 86$, there must be six males. Then, this leaves $32$ stripes remaining, so there are $4$ females. Therefore the ratio of male to female fish is $3:2$.

This problem is taken from the UKMT Mathematical Challenges.