Ratio Problems in Context

SPEC R3, R6, R7, R8 • RATIO, PROPORTION & RATES OF CHANGE

Scaling recipes, best-buy comparisons, ratio → fraction conversions, and proportion as equality of ratios — the applied half of ratio, and a reliable source of 3–5 mark questions on every paper.

Where This Lesson Fits

Lesson 3.01 taught you how to share a quantity in a given ratio. This lesson takes the same ratio skills and applies them to real-world problems:

What You'll Master

Expressing one quantity as a fraction of another (R3)

Converting between ratio and fraction (R8)

Finding the total or another share when given just one part

Best-buy comparisons — unit pricing

Scaling recipes and other direct-proportion problems

Expressing a multiplicative relationship as a ratio or fraction (R6)

Using proportion as equality of ratios (R7)

Exam Signal

These questions carry 2–5 marks each on Papers 1, 2 and 3. They blend with percentages, fractions and algebra — this is where ratio shows up in longer, multi-step problems.

Express One Quantity as a Fraction of Another (R3)

Spec R3 requires you to express one quantity as a fraction of another, where the fraction may be less than 1 or greater than 1.

Method

Fraction = numerator quantity ÷ denominator quantity (keep the same unit on both). Simplify.

Two Directions

QuestionWorkingAnswer15 g as a fraction of 25 g15/253/5 (less than 1)25 g as a fraction of 15 g25/155/3 (greater than 1)£4 as a fraction of £64/62/3 (less than 1)

Units Matter

As with ratios, both quantities must be in the same unit before you form the fraction. "50 cm as a fraction of 2 m" means 50 cm over 200 cm = 1/4.

Ratio → Fraction: Part Over Total

Key Rule — Ratio to Fraction

In a ratio a : b, the first part is a / (a + b) of the total, not a / b. The denominator is the total number of parts.

Worked Examples

3 : 5 → first part = 3/8 of total | second = 5/82 : 3 : 4 → middle part = 3/9 = 1/3 of total

Classic Error

Students write "in ratio 3 : 5, the first person gets 3/5 of the total." That's wrong — 3/5 is the ratio of first person to the other person, not to the whole. The correct fraction is 3/8.

Multiplicative Relationship as Ratio or Fraction

Spec R6 requires you to express a multiplicative relationship between two quantities as a ratio or a fraction. In plain terms: if one quantity is always a fixed multiple of another, that link is a ratio and a fraction.

Example — Recipe Ingredients

A recipe uses 200 g flour and 50 g butter. Write the ratio and the fraction that links them.flour : butter = 200 : 50 = 4 : 1flour = 4 × butter or butter = ¼ × flourThe ratio and the multiplier are two views of the same relationship.

Why This Matters

Once you write the relationship as flour = 4 × butter, you can scale the recipe for any amount. This is also the bridge to direct proportion (Lesson 3.05): the "4" is the constant of proportionality.

Scaling Recipes & Direct Proportion

If the quantities are in direct proportion, doubling the people doubles every ingredient. Scale using a single multiplier.

1

Find the scale factor — new amount ÷ old amount (e.g. new people ÷ old people).

2

Multiply every ingredient by that scale factor.

EX

Scaling a recipe

A recipe for 4 people uses 200 g of flour. How much flour is needed for 7 people?

Scale factor = 7 ÷ 4 = 1.75.

Flour needed = 200 × 1.75 = 350 g.

Unit-rate check: 200 ÷ 4 = 50 g per person, so 50 × 7 = 350 g ✓

Avoid This

Don't add a fixed amount ("50 g per extra person"). That only works if the ingredient has a fixed baseline — which is a different, non-proportional situation.

Best-Buy Comparisons: Unit Pricing

Key Rule — Best Buy

To compare value, find the unit price — price per gram, per ml, per item — and compare. The lowest unit price is the best value.

Example — Coffee

PackWorkingUnit price500 g for £3.20320p ÷ 500g0.64 p/g750 g for £4.65465p ÷ 750g0.62 p/gThe 750 g pack is better value — cheaper by 0.02 p per gram.

Pence or Pounds?

Either unit works — just be consistent across both packs. Keeping everything in pence often avoids tiny decimals like 0.0062.

Worked Example — Best-Buy Apples

EX

4 Marks — Paper 2 or 3

Apples cost £1.80 for 2 kg at supermarket A, or £2.70 for 3 kg at supermarket B. Which is better value, and by how much per kg?

M

Find price per kg at each supermarket, then compare.

C

A: 180p ÷ 2 kg = 90 p/kg.

C

B: 270p ÷ 3 kg = 90 p/kg.

E

Same value — both 90 p per kg. Neither is "better". State this explicitly: the key mark is for the correct comparison, not for picking a winner.

Examiner Alert — "No Difference" Answers

When unit prices match, don't invent a winner. Write: "Both cost the same per kg, so neither is better value." That earns the evaluation mark.

Currency Conversion

Currency conversion is direct proportion: the exchange rate is a multiplier. Apply it consistently.

Converting £ → €

£200 at €1.18 per £1 = 200 × 1.18 = €236

Converting Back € → £

€236 ÷ 1.18 = £200 ✓To convert the other way, divide by the same rate.

Read the Rate Carefully

Rates can be written two ways: "£1 = €1.18" or "€1 ≈ £0.85" (the reciprocal, rounded). Check which direction you're converting before you multiply or divide.

Proportion: Equality of Ratios (R7)

Definition — Proportion

Two ratios are in proportion when they are equivalent. If a : b = c : d, then a × d = b × c (the "cross-multiplication" rule).

Equivalent Ratios

3 : 5 = 6 : 10 = 12 : 20 = 15 : 25Multiply (or divide) both parts by the same number and the ratio stays the same.

Find the Missing Value

4 : 6 = x : 15. Find x.

Cross-multiply: 4 × 15 = 6 × x → 60 = 6x → x = 10.

Check: 4 : 6 = 2 : 3 and 10 : 15 = 2 : 3 ✓

In Context — Pens

"If 3 pens cost £1.20, how much do 7 pens cost?"

pens : cost in proportion: 3 : 1.20 = 7 : x

Cross-multiply: 3x = 7 × 1.20 = 8.40 → x = £2.80.

Ratios Link to Linear Functions (R8)

Spec R8 requires you to relate ratios to fractions and to linear functions. A fixed ratio between two variables describes a straight line through the origin.

Example

If y : x = 2 : 1 always, then y = 2x.

The points (1, 2), (2, 4), (3, 6) all fit — a straight line through the origin with gradient 2.

Equivalently, y/x = 2 for every (x, y) on the line — the fraction is constant.

Every ratio of the form y : x = k : 1 graphs as a straight line through the origin with gradient k .

Worked Example — Part:Whole in Context

EX

Part : part in two related ratios

Two storage boxes each hold books and DVDs only. Box X has a total capacity of 240 items, with books : DVDs = 5 : 3. Box Y has 168 books, with books : DVDs = 4 : 3. Which box holds more DVDs, and by how many?

M

For X, use the total with a part:part ratio. For Y, use the known part (books) to find one part, then multiply for DVDs.

C

Box X: total parts = 5 + 3 = 8. One part = 240 ÷ 8 = 30. DVDs = 3 × 30 = 90.

C

Box Y: books = 168 = 4 parts, so one part = 168 ÷ 4 = 42. DVDs = 3 × 42 = 126.

C

Difference: 126 − 90 = 36.

E

Box Y holds 36 more DVDs than Box X. Notice the two ratios were used differently — X needed the total, Y needed just the known share.

Examiner Alert — Best Buy by Total Price

Examiner Alert — Fraction of Total

The 3/5 vs 3/8 Trap

In ratio 3 : 5, students write "first person gets 3/5 of the total". That's the classic error.

The correct fraction is 3/8 — the denominator is the total parts (3 + 5 = 8), not the other share.

Quick Sanity Check

If first person got 3/5 of the total, second would need 2/5 — but 2/5 doesn't match a ratio of 5!

Check: 3/8 + 5/8 = 8/8 = 1 whole ✓ — fractions must add to 1.

Habit to Build

Any time you convert a ratio to a fraction, ask: "Does my first fraction + second fraction equal 1?" If not, check the denominator.

Practice Question

A fruit bowl contains apples and oranges only, in the ratio 5 : 3. There are 15 apples.

(a) How many oranges are in the bowl? [2 marks]

(b) What fraction of the fruit in the bowl is oranges? Give your answer in simplest form. [1 mark]

M

(a) Use the 15 apples = 5 parts to find one part, then multiply by 3 for oranges. (b) Fraction = oranges / total fruit.

C

(a) One part = 15 ÷ 5 = 3. Oranges = 3 × 3 = 9.

C

(b) Total fruit parts = 5 + 3 = 8. Fraction of oranges = 3/8. Already in simplest form (HCF(3,8) = 1).

E

(a) 9 oranges. (b) 3/8. Check: 15 apples + 9 oranges = 24 fruit; 9/24 = 3/8 ✓

Context Pitfalls: Quick Recap

TrapFix

Writing 3/5 instead of 3/8 for ratio 3 : 5Denominator = total parts , not the other part

Best buy by total priceAlways compute unit price , same unit both sides

Adding a fixed amount to scale a recipeUse a multiplier (new ÷ old)

Multiplying instead of dividing when converting currency backBack-conversion: divide by the rate

Declaring a winner when unit prices are equalState " same value " explicitly — that earns the evaluation mark

Mixing units when finding unit price (p per g vs £ per kg)Pick one unit and stay consistent across both options

Lesson Summary

✓ Ratio ↔ Fraction (R8)

First part of a : b is a/(a+b) of the total. For three-part ratios, the same rule: part/sum-of-all-parts.

✓ Multiplicative Relationships (R6)

Express as a ratio and a fraction. Flour : butter = 4 : 1 means flour = 4 × butter.

✓ Best Buy

Compute unit price (p per g, p per ml, £ per item). Lowest unit price wins. If equal, state "same value".

✓ Scaling & Direct Proportion

Find the scale factor (new ÷ old), multiply every quantity by it. Recipes, currency conversion, map scales all work the same way.

✓ Proportion as Equality of Ratios (R7)

If a : b = c : d, then ad = bc. Use cross-multiplication to find a missing value.