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? Mixing Paints Stage: 3 Challenge Level: A decorator can buy pink paint from two manufacturers. • Paint A is made up from red and white paint in the ratio$1:3$• Paint B is made up from red and white paint in the ratio$1:7$He can mix the paints to produce a different shade of pink. If Paint A and Paint B come in same size cans, what is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios: •$1:4$•$1:5$•$1:6$Another decorator buys pink paint from two different manufacturers: • Paint C is made up from red and white paint in the ratio$1:4$• Paint D is made up from red and white paint in the ratio$1:9$What is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios: •$1:5$•$1:6$•$1:7$•$1:8$Is it always possible to combine two paints made up in the ratios$1:x$and$1:y$and turn them into paint made up in the ratio$1:z$? (where$x < z < y\$)

Experiment with a few more examples.

Can you describe an efficient way of doing this?

Mixing More Paints is a follow-up question to this one.