### 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:

They were based on the misconception that the 'parts' in the ratios were of equal size and that you could add the ratios to work out the necessary combinations.

However, one can of paint A (ratio 1:3) and one can of paint B (ratio 1:7) does not produce paint in the ratio 2:10 (or 1:5), since that would require that the one part red in can A has the same volume as the one part red in can B. This can't be the case since there are 4 parts in can A and 8 parts in can B, so 1/4 of can A is red and 1/8 of can B is red.

To compare equal quantitities we will need to express the ratio of the colours in can A as 2:6, so we have:

in can A: 2/8 red and 6/8 white,
in can B: 1/8 red and 7/8 white.

Combining one can of each paint will now give us

3/16 red and 13/16 white,

that is, paint in the ratio 3:13.

We did received two correct solutions from students at St Albans High School:

Click here to see Anjali's solution and here to see Lydia's solution.

We also received this algebraic approach from Jennifer, from NLCS Jeju in Korea.