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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?

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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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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 $£2.00$
$1/8 \, \text{kg}$ seeds $£0.50 $
$1/8 \, \text{kg}$ nuts $£0.75$
$1/4 \, \text{kg}$ 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.