Cereal mix

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:

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:

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.