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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? # Cereal Mix ##### Age 11 to 14 Challenge Level: Jack from Stoke by Nayland Middle School sent in a solution to the problem that makes, in his words, "the World's most bland cereal bar": Use$500$grams of seeds and$500$grams of nuts this will give you a mixture that weighs$1000$grams and costs$£5$Anja from Stoke by Nayland Middle School, John and Andrew from Lazonby C of E School and Esther (school not given) managed to combine all 3 ingredients as follows:$60\%$of seeds$= £2.40 = 600 \, \text{g}20\%$of nuts$= £1.20 = 200 \, \text{g}20\%$of apricots$= £1.40 = 200 \,\text{g} 100%$of mix$£5.00 = 1000\,\text{g}$or$1\,\text{kg}$Anja says: I started out by experimenting with different mixtures until I found this one which adds up to exactly$£5.00$. Esther observes that: The ratio of apricots : nuts : seeds is$1 : 1 : 3$Taylor & Nia from Llandaff City Church in Wales Primary School also combined the 3 ingredients, but in a different way:$1/2 \, \text{kg}$seeds$£2.001/8 \, \text{kg}$seeds$£0.50 1/8 \, \text{kg}$nuts$£0.751/4 \, \text{kg}$dried apricots$£1.75$We solved this problem through the process of trial and error The ratio that Taylor and Nia have worked out is: Apricots : nuts : seeds =$2 : 1 : 5$The two solutions above make sense since you need seeds to match the nuts (they cost$£6$and$£4$, so equal amounts of each will average out to £5), plus twice as many seeds as apricots (they cost$£7$and$£4$so two$4$s and one$7$will average out to £5). So altogether the seeds must amount to the nuts plus double the apricots. The problem can also be defined in one equation (where$a =$weight of apricots,$n =$weight of nuts,$s =$weight of seeds): $$7a + 6n + 4s = 5(a + n + s)$$ $$7a + 6n + 4s = 5a + 5n + 5s$$ $$2a+ n = s$$ Any combination that fits the relationship$s = 2a + n\$ will satisfy the criteria (see the examples above). Notice that this is a more general solution than the ratios offered above.

You can now use the equation to find many more combinations.