AQA GCSE Mathematics 8300: Ratio, proportion and rates of change – Revision Notes

These notes cover Topic 3, Ratio, proportion and rates of change (references R1–R16), of the AQA GCSE Mathematics (8300) specification, Version 1.0, for exams in May/June 2017 onwards and every May/June and November series after that. Most of this topic is on both the Foundation and Higher tiers. Statements the specification places in its “Higher content only” column are marked Higher tier only below. Any of it can appear on Paper 1 (non-calculator) or on Papers 2 and 3 (calculator).

For full explanations, read the ratio, proportion and rates of change study guide first. Then test yourself with the practice questions. The course hub is AQA GCSE Mathematics, and the printable checklist lists every specification statement. To find your gaps quickly, try one of the free diagnostics.

Tier map

Reference	What you must do	Tier
R1, R11	Convert units; use speed, rates of pay, unit pricing	Foundation and Higher
R1, R11	Use density and pressure, in numbers and in algebra	Foundation and Higher (additional foundation)
R2	Scale factors, scale diagrams, maps	Foundation and Higher
R3–R8	Fractions of amounts, ratio notation, sharing, proportion as equal ratios, ratios as fractions and linear functions	Foundation and Higher
R9	Percentages, percentage change, original value, simple interest	Foundation and Higher
R10	Direct and inverse proportion problems, with graphs and algebra	Foundation and Higher
R12	Compare lengths, areas and volumes with ratios; link to similarity and trig ratios	Foundation and Higher
R13	Interpret proportion equations; X ∝ 1/Y means X is inversely proportional to Y	Foundation and Higher
R13	Construct equations for direct and inverse proportion	Higher tier only
R14	Gradient of a straight line as a rate of change; proportion graphs	Foundation and Higher
R15	Gradient at a point on a curve; average and instantaneous rates of change	Higher tier only
R16	Growth and decay, including compound interest	Foundation and Higher
R16	General iterative processes	Higher tier only

Units and compound units (R1, R11)

Conversion	Rule
Length	1 m = 100 cm, 1 km = 1000 m
Area	1 m² = 100² cm² = 10 000 cm²
Volume	1 m³ = 100³ cm³ = 1 000 000 cm³; 1 litre = 1000 cm³
Speed	km/h to m/s: × 1000 ÷ 3600 (so ÷ 3.6)

Compound unit	Formula	Typical units
Speed	distance ÷ time	m/s, km/h
Density	mass ÷ volume	g/cm³, kg/m³
Pressure	force ÷ area	N/m²
Rate of pay	pay ÷ time	£ per hour
Unit price	cost ÷ quantity	pence per 100 g

Worked reminder: 72 km/h = 72 000 m ÷ 3600 s = 20 m/s. Squared and cubed units need the conversion factor squared or cubed: 3.5 m² = 35 000 cm², not 350 cm².

Ratio (R3–R8)

Method in steps: sharing in a ratio

1. Add the parts of the ratio.
2. Divide the total by that number to get one part.
3. Multiply by each share of the ratio.
4. Check that the shares add back to the total.

Method in steps: a difference is given

1. Find the difference in parts (ratio 4 : 7 has a difference of 3 parts).
2. Divide the actual difference by it: 18 ÷ 3 = 6 per part.
3. Scale up: 4 × 6 = 24 and 7 × 6 = 42.

• Simplest form (R4): convert to the same units first, then divide by the highest common factor.
• Fraction of another (R3): “A as a fraction of B” is A/B. It can be greater than 1.
• Proportion as equal ratios (R7): if 5 : 8 = x : 20, then x = 5 × 20 ÷ 8 = 12.5.
• Ratio to fraction (R8): in a : b the first quantity is a/(a + b) of the total and a/b of the second.
• Ratio as a linear function (R8): if y : x = 3 : 2 then y = 1.5x, a straight line through the origin.
• Combining ratios: if a : b = 2 : 3 and b : c = 4 : 5, make b the same (12) to get a : b : c = 8 : 12 : 15.
• Best buy (R5): compare cost per equal quantity. 400 g for £1.56 is 39p per 100 g; 650 g for £2.47 is 38p per 100 g, so the 650 g pack is better value.
• Maps and scales (R2): on a 1 : 25 000 map, 8.6 cm represents 8.6 × 25 000 = 215 000 cm = 2.15 km.

Percentages (R9)

Change	Multiplier
Increase by 15%	× 1.15
Decrease by 7%	× 0.93
140% of the amount	× 1.4
Original value after a 15% increase	÷ 1.15

Method in steps: original value

1. Write the new value as a percentage of the original (after a 15% rise it is 115%).
2. Divide by the matching multiplier.
3. Example: £89.70 after a 15% rise came from 89.70 ÷ 1.15 = £78.

• Percentage change = (change ÷ original) × 100.
• Simple interest is the same amount each year: £2400 at 3.5% for 4 years earns 2400 × 0.035 × 4 = £336.

Direct and inverse proportion (R10, R13, R14)

Statement	Equation	Graph
y is directly proportional to x	y = kx	Straight line through the origin
y is inversely proportional to x	y = k/x	Curve; both variables positive, never meets the axes
y ∝ x²	y = kx²	Curve through the origin
y ∝ 1/√x	y = k/√x	Decreasing curve

“X is inversely proportional to Y” means the same as “X is proportional to 1/Y”.

Method in steps: constructing the equation (Higher tier only)

1. Write the relationship with a constant k.
2. Substitute the given pair of values and solve for k.
3. Rewrite the equation with the value of k.
4. Substitute to answer the question.

Worked reminders: if y ∝ x and y = 21 when x = 6, then k = 3.5 and y = 35 when x = 10. If y ∝ 1/x and y = 8 when x = 5, then k = 40 and y = 2.5 when x = 16.

Similarity links (R12)

Lengths in similar shapes are in a single ratio, the scale factor. In a right-angled triangle, the trigonometric ratios (for example opposite ÷ hypotenuse) stay the same when the triangle is enlarged. That is why sin, cos and tan depend only on the angle. You can also compare any two lengths, areas or volumes with a ratio (both tiers): use the same units, then simplify, so 0.6 m² : 1500 cm² = 6000 : 1500 = 4 : 1. How areas and volumes scale in similar figures is covered in the Geometry and measures topic.

Rates of change (R14, R15)

• Straight line (R14): the gradient is the rate of change. On a distance–time graph it is speed; on a cost–quantity graph it is price per unit.
• Average rate of change (Higher tier only, R15): the gradient of the chord joining two points on a curve.
• Instantaneous rate of change (Higher tier only, R15): the gradient of the tangent at a point. Draw the tangent with a ruler, pick two points far apart on it, and find rise ÷ run.
• Always give units: for depth in cm against time in minutes, the rate is in cm per minute.

Growth and decay (R16)

Compound interest: Total accrued = P(1 + r/100)ⁿ. You must know this formula; it is not given in the exam.

• Growth uses a multiplier above 1; depreciation uses one below 1.
• £5000 at 4% compound interest for 3 years: 5000 × 1.04³ = £5624.32.
• “How many years until…” questions: work out successive powers until the value first passes the target.

General iterative processes (Higher tier only): a rule such as uₙ₊₁ = 0.5uₙ + 12 with u₀ = 30 gives u₁ = 27, u₂ = 25.5, u₃ = 24.75. Feed each answer back into the rule, and keep full values.

Must-know distinctions

• Part : part vs part : whole. In 3 : 5 the first share is 3/8 of the total, not 3/5.
• Simple vs compound interest. Simple adds the same interest each year; compound multiplies by the same factor each year.
• “Of” vs “change”. 140% of £50 is £70; an increase of 140% on £50 gives £120.
• Direct vs inverse. Direct: doubling x doubles y. Inverse: doubling x halves y.
• Chord vs tangent. A chord gives an average rate; a tangent gives the rate at one instant.

Quick self-test

1. Write 250 g : 2 kg in its simplest form.
2. Share 72 in the ratio 5 : 3.
3. Write 48 as a percentage of 60.
4. What is the multiplier for a 7% decrease?
5. A price after 30% off is £42. Find the original price.
6. A car travels 120 km in 1 hour 36 minutes. Find its average speed in km/h.
7. A block has mass 3.6 kg and volume 450 cm³. Find its density in g/cm³.
8. y is inversely proportional to x, and y = 6 when x = 4. Find y when x = 3.
9. £1500 earns 2% compound interest per year. Find the value after 2 years.
10. On a 1 : 200 plan a wall is 4.5 cm long. How long is the real wall in metres?

Answers

1. 250 : 2000 = 1 : 8
2. 72 ÷ 8 = 9 per part, so 45 and 27
3. 48 ÷ 60 × 100 = 80%
4. 0.93
5. 42 ÷ 0.7 = £60
6. 1 hour 36 minutes = 1.6 hours; 120 ÷ 1.6 = 75 km/h
7. 3600 g ÷ 450 cm³ = 8 g/cm³
8. k = 24, so y = 24 ÷ 3 = 8
9. 1500 × 1.02² = £1560.60
10. 4.5 × 200 = 900 cm = 9 m

Where marks are usually lost

• Writing 1 hour 36 minutes as 1.36 hours in a speed calculation.
• Converting m² or m³ with a factor of 100 instead of 100² or 100³.
• Leaving a ratio unsimplified, or simplifying before making the units match.
• Finding a reverse percentage by taking the percentage off the new value instead of dividing by the multiplier.
• Using n × r% for compound interest, which is the simple interest method.
• Giving money answers to one decimal place (£1560.6) instead of two (£1560.60).
• Writing y = k/x for “inversely proportional to x²”.
• Reading a tangent gradient from two points very close together, which makes the estimate inaccurate.
• Leaving out units on a rate of change or a density.

Official syllabus

AQA GCSE Mathematics (8300) specification, Version 1.0 (12 September 2014), for teaching from September 2015 and exams in May/June 2017 onwards, published by AQA. Section 3.3, Ratio, proportion and rates of change, R1–R16.