OxfordAQA International GCSE Mathematics: Ratio, rates, best buys and direct and inverse proportion (9260) – Revision Notes

These notes condense section 3.1.3 (Ratio and proportion), statements N16 to N20, in the OxfordAQA International GCSE Mathematics specification (9260), Version 5.1. Everything here is Core content, which can also be assessed at Extension. For full explanations and longer worked examples, read the study guide. When you are ready, use the practice questions, the course hub, the checklist and the free diagnostics.

Key ideas at a glance

Ref	Idea	Remember
N16	Ratio a : b, simplest form	Same unit first. Whole numbers, no common factor.
N16	Ratio to fraction	3 : 7 gives 3/10 and 7/10 of the whole
N17	Divide in a ratio	Add parts, find one part, multiply
N18	Ratio problems	Use k for one part, then form an equation
N19	Rates and best buys	Compare per one unit, same units throughout
N20	Direct proportion	Ratio y ÷ x is constant
N20	Inverse proportion	Product x × y is constant
N20	Repeated change	Multiply the multipliers

N16: ratio and simplest form

Method in steps:

1. Express all the parts in one unit, such as grams or centimetres.
2. Remove any decimals or fractions by scaling every part by one common multiplier.
3. Divide through by the highest common factor.
4. Leave the units off the answer.

Examples:

1.5 m : 60 cm = 150 : 60 = 5 : 2
36 : 48 : 60 = 3 : 4 : 5   (divide by 12)
3/4 : 1/6 = 9 : 2          (multiply by 12)
5 : 8 = 1 : 1.6            (divide by 5, form 1 : n)

Ratio and fraction:

• For parts a : b, the first part is a out of (a + b) equal slices of the whole, and the second is b out of (a + b).
• Fractions 4/15 and 11/15 mean the ratio 4 : 11.
• If 5/9 are girls, boys to girls is 4 : 5.

N17: dividing in a ratio

Share 1350 in the ratio 4 : 5 : 6
parts = 15, one part = 1350 ÷ 15 = 90
shares: 360, 450, 540     (sum 1350)

When a share is known, find one part from it:

boys : girls = 7 : 5, 45 girls
1 part = 45 ÷ 5 = 9
boys = 7 × 9 = 63

When a difference is known:

shares 3 : 8 differ by 40
5 parts = 40, 1 part = 8, total = 11 × 8 = 88

N18: ratio problems

Combine two ratios by making the shared term equal:

P : Q = 3 : 5 and Q : R = 10 : 7
make Q = 10 in both: P : Q = 6 : 10, Q : R = 10 : 7
P : Q : R = 6 : 10 : 7

Change in quantities: write the start as multiples such as 3k and 5k, apply the change, then use the new ratio or equality to form an equation. Solve for k and answer the question that was asked, which is often the original amounts and not k.

Mixtures and recipes: scale every part by the same factor. A 3 : 5 mix of syrup to water makes 8 parts. For 2.4 litres, one part is 0.3 litres, so 0.9 litres of syrup and 1.5 litres of water.

N19: rates, pay and best buys

Rate	How to find it	Typical unit
Unit price	cost ÷ quantity	money per kg, per litre
Pay	hours × rate per hour	money per hour
Exchange	quantity × rate or ÷ rate	one currency per unit of another
Speed	distance ÷ time	kilometres per hour, metres per second
Density	mass ÷ volume	grams per cubic centimetre

Time conversion: 2 h 20 min = 2 1/3 h, not 2.2 h. Minutes ÷ 60.

Overtime: the overtime rate is the basic rate × the multiplier. Time and a half is × 1.5. Double time is × 2.

Best-buy method in steps:

1. Convert to the same unit.
2. Find the cost per one unit (or the amount per one unit of money) for every option.
3. Compare, and write the answer in words with the figures that support it.

Worked reminder: 3 kg for 7.50 or 5 kg for 12.00. Per kg: 7.50 ÷ 3 = 2.50 and 12 ÷ 5 = 2.40. The 5 kg pack is the better buy.

Foreign currency: if 1 euro = 1.08 dollars, then dollars to euros divide by 1.08, and euros to dollars multiply by 1.08. A sense check helps: the currency that is worth more gives the smaller number.

N20: direct and inverse proportion

	Direct	Inverse
Behaviour	Both increase together	One increases, the other decreases
Constant	y ÷ x	x × y
Method	Find the value of 1, then scale	Find the total, then divide
Graph	Straight line through the origin	Curve falling towards the axes
Test from table	Divide each pair	Multiply each pair

Direct: 20 pens cost 14. One pen costs 0.70, so 35 pens cost 24.50.

Inverse: 4 machines take 18 hours. Total 72 machine-hours, so 6 machines take 72 ÷ 6 = 12 hours.

Working in inverse proportion questions assumes every worker, pump or machine works at the same constant rate. Say so if the question asks you to state an assumption.

Writing these relationships with a letter k is algebraic variation, which is a separate part of the specification (A18).

Repeated proportional change: multiply the multipliers. An increase of 10% then a decrease of 10% gives 1.10 × 0.90 = 0.99, which is a fall of 1%, not zero.

Worked reminders

Speed with a mixed time: a train covers 210 km in 2 h 20 min.

2 h 20 min = 2 1/3 h = 7/3 h
speed = 210 ÷ (7/3) = 210 × 3/7 = 90 km/h

A ratio change problem: two lengths are in the ratio 3 : 5. After 8 cm is cut from the longer one, they are in the ratio 3 : 4.

3k : (5k - 8) = 3 : 4
4 × 3k = 3 × (5k - 8)
12k = 15k - 24, so k = 8
lengths: 24 cm and 40 cm
check: 24 : 32 = 3 : 4

A repeated change: a value of 400 rises by 10% and then rises by 20%.

1.10 × 1.20 = 1.32
400 × 1.32 = 528, an overall rise of 32%

A direct proportion from a rate: a tap fills 15 litres in 4 minutes. In 10 minutes it fills 15 ÷ 4 × 10 = 37.5 litres, if the flow is steady.

Must-know distinctions

• Ratio versus fraction: 3 : 7 is a comparison of parts. 3/10 is a part of the whole.
• Ratio versus rate: a ratio of like quantities has no unit. A rate of unlike quantities has a unit such as km/h.
• Direct versus inverse: ask “if one doubles, what happens to the other?”
• Total price versus unit price: only unit price decides the best buy.
• Adding versus multiplying percentage changes: multiply.

Quick self-test

1. Simplify 36 : 48 : 60.
2. Write 1.5 m : 60 cm in simplest form.
3. Divide 252 in the ratio 2 : 5.
4. Boys to girls is 7 : 5. There are 45 girls. How many boys?
5. A ratio is 4 : 11. What fraction of the total is the first part?
6. Write 5 : 8 in the form 1 : n.
7. 20 pens cost 14. What do 35 pens cost?
8. 4 machines take 18 hours. How long do 6 machines take?
9. Which is the better buy, 3 kg for 7.50 or 5 kg for 12.00?
10. A worker is paid 9.20 an hour. In a week she works 40 hours at the basic rate and 6 hours at double time. Find her pay.
11. A value rises by 10% and then falls by 10%. Find the overall percentage change.
12. A job takes 5 people 12 days. How many people finish it in 4 days?

Answers

1. 3 : 4 : 5
2. 5 : 2
3. 72 and 180
4. 63
5. 4/15
6. 1 : 1.6
7. 24.50
8. 12 hours
9. 5 kg for 12.00 (2.40 per kg against 2.50 per kg)
10. 40 × 9.20 = 368.00, then 6 × 18.40 = 110.40, total 478.40
11. 1.10 × 0.90 = 0.99, a fall of 1%
12. 60 person-days, so 15 people

Where marks are usually lost

• Not converting units before simplifying a ratio such as kilograms to grams.
• Dividing by the number of shares instead of the total number of parts.
• Giving the answer as the value of one part, k, instead of the quantity asked for.
• Using 3/7 as the fraction for a 3 : 7 ratio.
• Working out cost per unit for one option only, or comparing a total price with a price per unit.
• Giving a best buy with no figures. Show the unit cost of every option and name the better one.
• Converting 1 h 45 min to 1.45 h instead of 1.75 h.
• Treating an inverse proportion as direct, so that more workers take longer.
• Adding percentage changes instead of multiplying the multipliers.
• Rounding the unit price too early, which can make two close options look equal.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams in May/June 2018 onwards, published by OxfordAQA. Section 3.1.3, Ratio and proportion, statements N16 to N20.