GCSE Maths — Ratio, Proportion & Rates of Change

Taught through feeding, dosing and running costs

Approx. 60 minutes · KS4 · Foundation and Higher content flagged separately

Before you start

Every number in this lesson is the kind a keeper actually works out, usually standing up, usually before doing something that matters. Get a ratio wrong when mixing a disinfectant and you either fail to kill anything or burn an animal's feet.

Nothing here is timed. There is no score. You can look at any answer whenever you like — that is not cheating, it is how anyone learns a method.

What you'll be able to do by the end

• Divide a quantity in a given ratio, and work backwards from one part.
• Use direct and inverse proportion, and tell which one a situation is.
• Convert between units of mass, volume and concentration without losing a factor of ten.
• Calculate a percentage change and reverse it.
• Work with a rate — cost per day, growth per week, dose per kilogram.

Part 1 — Ratio in the real setting

Dividing a quantity in a ratio

A dry food mix is made in the ratio 5 : 3 : 2 — insect protein to fruit to supplement. You want to make 2 kg.

Check: 1000 + 600 + 400 = 2000 g. Always check. It takes four seconds and catches most errors.

Working backwards from one part

Same 5 : 3 : 2 mix. You have exactly 750 g of fruit and want to use all of it.

Fruit is 3 parts, so one part = 750 ÷ 3 = 250 g.

Check yourself

A disinfectant is diluted 1 : 49 with water — one part concentrate to forty-nine parts water. You need 5 litres of solution.

How much concentrate?

Part 2 — Direct and inverse proportion

Direct proportion: as one goes up, the other goes up by the same factor. Double one, double the other. y = kx.

Inverse proportion: as one goes up, the other goes down by the same factor. Double one, halve the other. y = k / x.

The exam tests whether you can tell which is which. The animal room gives clean examples of both.

Direct. Food required and number of animals. Twenty hedgehogs eat twice what ten hedgehogs eat.

Inverse. Food supply and how long it lasts. A sack that feeds ten hedgehogs for twelve days feeds twenty for six.

Worked example — direct

Twelve hedgehogs eat 900 g of food per day. How much do 20 eat?

Worked example — inverse

A sack of food lasts 12 hedgehogs 15 days. How long for 20?

Check yourself

Three misting nozzles fill a reservoir in 40 minutes. How long would five nozzles take, assuming they work at the same rate?

Part 3 — Units, concentration and dosing

This is where marks are lost to arithmetic rather than method, and where a real error does real harm.

Mass: 1 kg = 1000 g. 1 g = 1000 mg.
Volume: 1 litre = 1000 ml.
Concentration: mg per ml, or g per litre. They are the same relationship at different scales.

Worked example — dosing by body mass

A medication is given at 5 mg per kg of body mass. The animal weighs 380 g. The solution is 10 mg per ml.

Three conversions, three chances to be out by a factor of ten. Write the units next to every number as you go — that alone catches most of these.

Check yourself

A supplement is dosed at 2 mg per kg. The animal weighs 1.4 kg. The solution supplies 4 mg per ml. What volume is given?

Part 4 — Percentages and rates of change

Percentage change

A hoglet weighs 62 g on Monday and 74 g the following Monday.

Reverse percentages — the one people get wrong

Feed costs rose by 12% and a sack now costs £28. What did it cost before?

The wrong method is subtracting 12% of £28. That gives £24.64, which is wrong.

The right method: £28 represents 112% of the original.

Check: £25 × 1.12 = £28. Correct.

Rate of change

A tortoise gains mass at roughly 40 g per month.

Rates are just division with units attached. Keep the units visible and the method is obvious.

Check yourself

Electricity for the heating cost £180 last quarter and £207 this quarter. Calculate the percentage increase.

Part 5 — Exam-style questions

1. A food mix is made in the ratio 7 : 4 : 1. A batch contains 240 g of the middle ingredient. Calculate the total mass of the batch. (3 marks)

2. A disinfectant is diluted 1 : 24 with water. Calculate the volume of concentrate needed to make 3 litres of solution. (3 marks)

3. Eight animals consume a sack of food in 21 days. Assuming the same rate per animal, calculate how long the same sack would last 14 animals. (3 marks)

4. A medication is given at 8 mg per kg of body mass. An animal weighs 450 g. The solution contains 20 mg per ml. Calculate the volume to be given, in ml. (4 marks)

5. After a 15% increase, a sack of food costs £36.80. Calculate the price before the increase. (3 marks)

6. (Higher) The mass of a growing animal is directly proportional to the square root of its age in weeks. At 16 weeks it has a mass of 240 g. Calculate its mass at 25 weeks. (4 marks)

If you want to go further

• Take a real care guide and work out the true annual cost of one animal — food, heating, lighting, substrate and an annual vet check. Compare it to the purchase price.
• Related lessons: Compound Measures · Percentages & Financial Maths · Direct & Inverse Proportion (Higher).

Notes for the adult

Curriculum: GCSE Maths — ratio and proportion, compound measures, percentage change and reverse percentages, and rates of change. Question 6 is Higher tier only and is flagged as such; everything else is accessible at Foundation.

The misconceptions this lesson targets: reading "1 : 49" as one in forty-nine, and reversing a percentage by subtracting it. Both are addressed explicitly with the wrong answer shown alongside the right one, because students who have made the error need to see it named.

On the dosing section. The arithmetic is standard compound-measures content and the veterinary framing is what makes it stick. The lesson states plainly that it is not veterinary guidance. If you are working with a student who might take it as such, say so aloud as well.

For a student who finds this hard. Parts 1 and 4 stand alone as a shorter session on ratio and percentages, without proportion or dosing. That is a complete and useful piece of maths on its own.

Access: read-aloud, dyslexia-friendly font, enlarged text, high contrast and calm mode on every page. Worked examples are laid out one step per line rather than as continuous prose.

No timing, no scoring. Nothing here is marked, logged or reported. Answers are available immediately by design.

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