Ratio Proportion and Rates of Change

What you'll learn

This topic covers how quantities relate to each other through ratios, direct and inverse proportion, and calculations involving rates. You'll work with percentages, scaling problems, compound measures like speed and density, and conversions between units. These skills appear across both Foundation and Higher tier papers, particularly in Paper 2 and Paper 3.

Key terms and definitions

Ratio — a comparison of two or more quantities showing how many times one value contains another, written in the form a:b

Direct proportion — when two quantities increase or decrease at the same rate, so that their ratio remains constant (as one doubles, the other doubles)

Inverse proportion — when one quantity increases at the same rate as another decreases (as one doubles, the other halves)

Compound measure — a unit formed from two or more other units, such as speed (distance/time) or density (mass/volume)

Unitary method — finding the value of one unit first, then multiplying to find the required amount

Scale factor — the multiplier used to enlarge or reduce a quantity in direct proportion

Percentage multiplier — the decimal used to calculate percentage changes in one step (e.g. 1.15 for a 15% increase)

Best value — the product offering the lowest cost per unit when comparing prices

Core concepts

Writing and simplifying ratios

Ratios compare quantities and must use the same units before simplification.

To simplify a ratio, divide all parts by their highest common factor (HCF):

• 12:18 = 6:9 = 2:3

To convert ratios with different units:

• 2 kg:500 g = 2000 g:500 g = 4:1

Part-to-part ratios compare different components (boys:girls = 3:2)

Part-to-whole ratios compare one part to the total (boys:total students = 3:5)

For three-part ratios, apply the same principles:

• Simplify 6:9:15 by dividing by 3 to get 2:3:5

Sharing in a ratio

To divide a quantity in a given ratio:

1. Add the ratio parts to find the total number of shares
2. Divide the quantity by this total to find one share
3. Multiply each ratio part by the value of one share

Example: Share £360 in the ratio 5:4

• Total shares = 5 + 4 = 9
• One share = £360 ÷ 9 = £40
• First amount = 5 × £40 = £200
• Second amount = 4 × £40 = £160

For recipe or mixture problems, use the same method to scale ingredients proportionally.

Direct and inverse proportion

Direct proportion uses the formula y = kx where k is the constant of proportionality.

To solve direct proportion problems:

1. Find k using the given values (k = y/x)
2. Use k to calculate the unknown value

Example: If 5 kg of sugar costs £4.50, find the cost of 8 kg

• Cost per kg (k) = £4.50 ÷ 5 = £0.90
• Cost of 8 kg = £0.90 × 8 = £7.20

Inverse proportion uses the formula y = k/x or xy = k

When one quantity doubles, the other halves. To solve:

1. Find k by multiplying the given x and y values
2. Use k to find unknowns

Example: 6 workers complete a job in 10 days. How long for 8 workers?

• k = 6 × 10 = 60
• Time = 60 ÷ 8 = 7.5 days

Graphical recognition:

• Direct proportion graphs are straight lines through the origin
• Inverse proportion graphs are curves (hyperbolas)

Percentages

Calculate percentages using multipliers for efficiency:

Percentage increases:

• 20% increase: multiply by 1.20
• 7% increase: multiply by 1.07

Percentage decreases:

• 30% decrease: multiply by 0.70
• 12% decrease: multiply by 0.88

Reverse percentages — finding the original amount: If £84 is 70% of the original price, divide by 0.70:

• Original = £84 ÷ 0.70 = £120

Compound interest formula: Final amount = P × (1 + r/100)^n

Where P = principal, r = rate, n = number of years

Percentage change formula: Percentage change = (change ÷ original) × 100

Compound measures and conversions

Speed:

• Speed = distance ÷ time
• Distance = speed × time
• Time = distance ÷ speed

Units include km/h, m/s, mph. Convert between units carefully:

• To convert km/h to m/s: divide by 3.6
• To convert m/s to km/h: multiply by 3.6

Density:

• Density = mass ÷ volume
• Mass = density × volume
• Volume = mass ÷ density

Common units: g/cm³, kg/m³

Pressure:

• Pressure = force ÷ area

Units: N/m² (Pascals), N/cm²

Currency conversion: Use exchange rates as scale factors. If £1 = €1.18:

• Convert £50 to euros: 50 × 1.18 = €59
• Convert €70.80 to pounds: 70.80 ÷ 1.18 = £60

Unit conversions:

• 1 km = 1000 m
• 1 m = 100 cm
• 1 kg = 1000 g
• 1 litre = 1000 ml = 1000 cm³
• 1 m³ = 1000 litres

When converting area units, square the conversion factor (1 m² = 10,000 cm²)

When converting volume units, cube the conversion factor (1 m³ = 1,000,000 cm³)

Growth and decay

Simple interest adds the same amount each time period: Total = P + (P × r/100 × n)

Compound growth/decay multiplies by a multiplier each time:

• Population growth of 3% annually over 5 years: P × 1.03⁵
• Depreciation of 15% annually over 4 years: P × 0.85⁴

Exponential graphs:

• Growth: curves upward steeply
• Decay: curves downward approaching zero

Comparing rates and best value

To find best value, calculate the cost per unit (or amount per penny):

Example: Which is better value?

• 500 g for £2.40: £2.40 ÷ 500 = £0.0048 per gram
• 750 g for £3.50: £3.50 ÷ 750 = £0.00467 per gram

The 750 g option is better value (lower cost per gram)

For rates like typing speeds or production rates, compare units completed per hour/minute.

Worked examples

Example 1: Ratio and proportion (Foundation/Higher)

Question: A fruit juice is made by mixing orange, pineapple and mango in the ratio 5:3:2. How much of each juice is needed to make 2.5 litres?

Solution:

• Total parts = 5 + 3 + 2 = 10
• 2.5 litres = 2500 ml
• One part = 2500 ÷ 10 = 250 ml
• Orange = 5 × 250 = 1250 ml = 1.25 litres (1 mark)
• Pineapple = 3 × 250 = 750 ml = 0.75 litres (1 mark)
• Mango = 2 × 250 = 500 ml = 0.5 litres (1 mark)

Example 2: Compound measures (Higher)

Question: A car travels 150 km in 1 hour 45 minutes. Calculate the average speed in km/h.

Solution:

• Convert time to hours: 1 hour 45 minutes = 1.75 hours (1 mark)
• Speed = distance ÷ time
• Speed = 150 ÷ 1.75 (1 mark)
• Speed = 85.7 km/h (to 1 d.p.) (1 mark)

Example 3: Reverse percentages (Higher)

Question: After a 12% increase, a mobile phone costs £268.80. What was the original price?

Solution:

• 112% = £268.80 (1 mark)
• 1% = £268.80 ÷ 112 = £2.40
• 100% = £2.40 × 100 = £240 (1 mark)

Alternative method:

• Multiplier for 12% increase = 1.12
• Original price = £268.80 ÷ 1.12 = £240 (2 marks)

Example 4: Inverse proportion (Higher)

Question: It takes 4 machines 6 hours to pack a warehouse order. How long would it take 3 machines working at the same rate?

Solution:

• This is inverse proportion (fewer machines take longer) (1 mark)
• Total machine-hours = 4 × 6 = 24 (1 mark)
• Time for 3 machines = 24 ÷ 3 = 8 hours (1 mark)

Common mistakes and how to avoid them

• Not converting units before simplifying ratios — always express quantities in the same units first (e.g. convert kg to g)

• Adding ratios incorrectly — remember that 3:2 does not mean 3 and 2 (total 5), but rather 3 parts out of 5 and 2 parts out of 5

• Confusing direct and inverse proportion — check whether quantities increase together (direct) or oppositely (inverse); time and workers is usually inverse

• Using the wrong multiplier for percentages — for a 30% decrease, multiply by 0.7 not 0.3; the multiplier is what remains, not what's removed

• Forgetting to square or cube conversion factors — when converting area multiply by the conversion factor squared; for volume cube it (1 m² = 100² = 10,000 cm²)

• Mixing up speed formula components — use a triangle memory aid or remember "distance = speed × time" as the base formula, then rearrange

Exam technique for "Ratio Proportion and Rates of Change"

• Show your method clearly — ratio and proportion questions award method marks even if your final answer is wrong; write down the multiplier or scaling factor you're using

• Check your calculator mode — compound interest and growth questions require power functions; verify you get sensible answers (populations shouldn't become negative, prices usually increase)

• Units matter — always give units in compound measure answers; check the question asks for km/h not m/s before finalizing; examiners penalize missing or incorrect units

• "Inverse proportion" or "proportional" are trigger words — these signal specific mathematical relationships; don't just assume any two quantities that change together are proportional

Quick revision summary

Ratios compare quantities in the same units and simplify using HCF. Share amounts by finding one part then multiplying. Direct proportion means y = kx; inverse proportion means xy = k. Use percentage multipliers for efficient calculations and divide to reverse percentages. Compound measures like speed (distance/time) and density (mass/volume) follow formulae triangles. Convert units carefully, squaring for area and cubing for volume. Compound growth uses multipliers raised to a power. Compare rates by finding cost per unit for best value decisions.