How Big?

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

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? 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: Dividing the stripes into groups of$8$, we could have$10$groups of$8$with$6$left over. If we add a stripe to$6$groups of$8$we will have$6$males. So there are$6$males and$4$females. Therefore the ratio of male to female fish is$3:2$. Alternatively, 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\$.

This problem is taken from the UKMT Mathematical Challenges.