Overview
What you’ll learn
- Write ratios in the form a : b and simplify them using common factors.
- Express one quantity as a fraction of another, including fractions greater than 1.
- Divide an amount in a part:part ratio, and write a part-to-whole comparison as a ratio.
- Solve recipe, mixing and best-buy problems with proportional reasoning.
- Use scale factors and 1:n map scales to convert between drawings and real lengths.
Specification checklist (AQA 8300 · Edexcel 1MA1 · OCR J560)
Statements are paraphrased from the shared DfE subject content and grouped by tier. Foundation content is studied by every student; Higher statements are examined on Higher-tier papers only. QuizQuip is not affiliated with AQA, Pearson Edexcel or OCR.
Foundation — every student
- 12.1Use ratio notation, reduce ratios to simplest form using common factors, and express one quantity as a fraction of another where the fraction is less than or greater than 1.
- 12.2Divide a given quantity into two parts in a given part:part ratio, and express the division of a quantity into two parts as a ratio.
- 12.3Apply ratio to real contexts and problems, such as those involving conversion, comparison, scaling, mixing and concentrations.
- 12.4Use scale factors, scale diagrams and maps, including 1:n scales, converting between diagram measurements and real lengths with correct units.
Higher — Higher tier only
- 12.5Relate the language of ratio to fractions and to linear functions: the multiplier linking two proportional quantities appears as the gradient of the line y = kx.
Key points
- 1A ratio compares parts: 24 : 36 = 2 : 3 after dividing by the common factor 12.
- 2Ratio units must be the same kind before simplifying: 50 cm : 2 m = 50 : 200 = 1 : 4.
- 3To share £120 in 3 : 5, find one part (£120 ÷ 8 = £15), then multiply: £45 and £75.
- 4One quantity as a fraction of another: 20 g of 60 g is ⅓ — first quantity over the second.
- 5Recipes scale by a single multiplier: doubling every ingredient keeps the taste; a half-batch halves them.
- 6Best buys compare unit prices: price per gram (or per item), not total price.
- 7Map scale 1 : 50 000 means 1 cm on the map is 50 000 cm = 500 m on the ground.
- 8Ratio and fraction describe the same comparison in different clothes: 3 of 7 parts = 3⁄7.
Notes · 1 of 4
Ratio notation and simplifying
Compare like with like, then cancel.
A ratio compares two (or more) quantities in the same units. 24 : 36 simplifies by dividing both parts by 12 to 2 : 3 — the ratio keeps its meaning because both parts shrank by the same factor. Units must match first: 50 cm against 2 m is 50 : 200 (since 2 m = 200 cm), which simplifies to 1 : 4. Simplify fully — 10 : 15 is not finished; 2 : 3 is.
One quantity as a fraction of another puts the first over the second: 20 g of 60 g is 20⁄60 = ⅓; 60 g of 20 g is 60⁄20 = 3, a fraction greater than 1 — perfectly legal, it just says “three times as much”. Ratios and fractions are two clothes for one comparison: in a 3 : 5 mix, one part is 3⁄8 of the total.
Notes · 2 of 4
Dividing a quantity in a ratio
Find one part, then multiply.
To share £120 in the ratio 3 : 5: add the parts (3 + 5 = 8), find one part (120 ÷ 8 = £15), then multiply each side (3 × 15 = £45, 5 × 15 = £75). The check is instant: £45 + £75 = £120 ✓. With three-way ratios the method is identical — one part at a time.
Part : whole questions run the other way: “30 of the 50 counters are red — what is the ratio red : blue?” Red = 30, blue = 20, so 30 : 20 = 3 : 2. And fractions reappear: red counters are 30⁄50 = ⅗ of the tube. Exam questions love flipping between the ratio, the fraction and the actual counts.
Notes · 3 of 4
Ratio in context: recipes, mixing and best buys
Scale with a multiplier; compare with unit prices.
Recipes scale every ingredient by one multiplier. A pancake mix for 8 uses 2 eggs; for 14 pancakes the multiplier is 14⁄8 = 1.75, so eggs = 2 × 1.75 = 3.5. Working out how many pancakes the available ingredients allow runs backwards: 6 eggs at 2 per 8 pancakes feeds 6 ÷ 2 × 8 = 24 pancakes — the limiting ingredient decides.
Mixing keeps the ratio as ingredients change: paint mixed green : white in 3 : 4 needs 12 green with 16 white; if only 9 green is left, the white scales to 9 × 4⁄3 = 12. Best buys compare unit prices: 500 g for £2.40 costs 0.48p per gram... more neatly £2.40 ÷ 500 = £0.0048 per gram vs £3.60 ÷ 800 = £0.0045 per gram — the 800 g pack wins despite costing more.
Notes · 4 of 4
Scale factors, diagrams and maps
1 : n means 1 of those units equals n of them.
A scale factor multiplies every length. A scale model at 1 : 20 with a 15 cm car body represents 15 × 20 = 300 cm = 3 m of real car. Maps use 1 : n scales: at 1 : 50 000, each 1 cm of map is 50 000 cm of ground — and unit conversion finishes the job: 50 000 cm = 500 m, so 4 cm of map is 2 km.
Working from real life to the drawing divides: a 5 m room drawn at 1 : 100 is 500 cm ÷ 100 = 5 cm on paper. Keep the units of the scale — a scale in cm against real km needs two conversions, and writing the units at every line is what keeps 1 : 25 000 questions honest.
Revise
Key terms
The words examiners expect you to use precisely.
- Ratio
- A comparison of two or more quantities of the same kind.Flour : sugar = 2 : 1.
- Simplest form
- The ratio with the largest common factor cancelled.24 : 36 = 2 : 3.
- Part
- One share unit in a ratio.In 3 : 5 the total is 8 parts.
- Unit price
- The price per single unit of measure, used to compare buys.£1.35 per litre.
- Scale factor
- The multiplier from diagram lengths to real lengths.Model 1 : 20.
- Scale (of a map)
- The ratio 1 : n between map distance and real distance.1 : 25 000.
Show 1 more key termShow fewer key terms
- Limiting ingredient
- The ingredient that runs out first and caps the recipe.6 eggs cap 24 pancakes.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Simplifying 45 : 2 (minutes against hours).
The quantities are in different units.
Instead: Convert first: 45 min : 120 min = 3 : 8.
Mistake: Sharing £120 in 3 : 5 as £40 and £60 (dividing by 2).
The ratio has 3 + 5 = 8 parts, not 2.
Instead: One part is £15: £45 and £75.
Show 10 more mistakesShow fewer mistakes
Mistake: Writing the ratio red : blue as 30 : 50 when 30 of 50 counters are red.
50 counts both colours, so blue is 20.
Instead: 30 : 20 = 3 : 2.
Mistake: Adding the same amount to both sides of a ratio to keep it equal: 2 : 3 becomes 4 : 5.
Ratios scale by multiplication, not addition.
Instead: Doubling gives 4 : 6.
Mistake: Comparing pack prices instead of unit prices.
Bigger packs cost more in total by definition.
Instead: Compute price per gram or per litre for each.
Mistake: Scaling only one ingredient of a recipe.
Every ingredient scales by the same multiplier.
Instead: Flour, milk and eggs all × 1.75 for a 14-pancake batch.
Mistake: Reading a 1 : 50 000 map scale as 1 cm = 5 km.
50 000 cm is 500 m, not 5 km.
Instead: 1 cm = 0.5 km, so 4 cm = 2 km.
Mistake: Multiplying when converting real life to the drawing.
Drawings are smaller: divide by the scale factor.
Instead: 5 m at 1 : 100 is 500 ÷ 100 = 5 cm.
Mistake: Writing the fraction greater than 1 as impossible.
“60 g of 20 g” is genuinely 3, a fraction over 1.
Instead: One quantity as a fraction of another can exceed 1.
Mistake: Cancelling a ratio by subtracting the common amount: 10 : 15 as 5 : 10.
Simplification divides.
Instead: 10 : 15 = 2 : 3 (divide by 5).
Mistake: Mixing units in scale answers: writing 300 cm as 300 m.
The conversion is part of the answer.
Instead: 300 cm = 3 m — state it.
Mistake: Dividing the total by the largest ratio number.
The total divides by the sum of the parts.
Instead: £120 in 3 : 5 → one part = 120 ÷ 8.
Check yourself
Quick quiz
30 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Simplify a ratio
Written in its simplest form, 24 : 36 is
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Practise
Topical questions
18 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this chapter, in three tiers, each with a model answer, mark scheme and examiner tip. Higher-tier questions are tagged.
Practise topical questionsExam practice
Practice paper
A 30-mark paper in the style of a GCSE paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 12 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 12: ratio notation and simplifying, fractions of quantities, sharing in part:part and part:whole ratios, recipes, mixing and best buys, and scale factors with maps and models.
Mark scheme
PROChapter 12 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
How is this chapter examined?
Paper 1
non-calculator · 1 h 30
Exact fractions, surds and multiples of π expected
Papers 2 and 3
calculator · 1 h 30 each
Non-exact answers to 3 s.f., angles to 1 d.p.
Tiers
Foundation (grades 1–5) · Higher (grades 4–9)
Higher assumes all Foundation content
Want a full paper? Try the Paper 1-style (Foundation), Papers 2 & 3 style (Foundation), Paper 1-style (Higher), Papers 2 & 3 style (Higher) sets.
Questions
Frequently asked questions
Is ratio on both GCSE tiers?
Yes — ratio, proportion and rates of change is heavily weighted at both Foundation and Higher. Everything in this chapter can appear on either tier; the Higher papers simply ask longer, multi-step questions with the same skills.
How do I remember the method for sharing in a ratio?
Three words: parts, one, multiply. Add the parts, divide the total by that sum to price one part, then multiply each side of the ratio. Finish by checking the shares add back to the original amount.
What is the fastest way to compare two deals?
Unit price. Divide each deal’s price by its size (per 100 g, per litre, per item) and compare the two numbers. It also gives you the cost of any other amount at the same rate.
Do I need to memorise the map scale conversions?
You need the conversions: 1 m = 100 cm and 1 km = 1000 m (so 1 km = 100 000 cm). Combined with the given scale, every map question is just careful unit work — write the units on each line.
Up next
Chapter 13: Proportion and rates of change