Cereal mix
Problem
A farmer is supplying a mix of seeds, nuts and dried apricots to a manufacturer of crunchy cereal bars.
The ingredients cost:

dried apricots $ £7$ per kg

nuts $ £6$ per kg

seeds $ £4$ per kg
He has been asked to supply a mix which costs $ £5$ per kg.
What combination of ingredients could he supply?
Is there a relationship between the amounts of each ingredient that he could supply?
Getting Started
Dried apricots and nuts cost more than £5 per kg so any mix will need to contain seeds.
How does the quantity of nuts affect the quantity of seeds that will be required?
How does the quantity of dried apricots affect the quantity of seeds that will be required?
Student Solutions
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:
(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):
$$7a + 6n + 4s = 5(a + n + s)$$
$$2a+ n = s$$
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.
Teachers' Resources
Why do this problem:
A simple problem to get students starting to think about ratio.Possible approach :
Ask students to suggest possible combinations and their suggestions could be written up for everyone in the group to see.Key questions :

Do all these combinations cost $ £5$ per kg? How do you find out?

Do any of these combinations give identical mixes? How do you find out?

Are there any relationships between the quantities of the different ingredients?