🔢 Big numbers
3,500,000 = 3.5 × 106. Move the decimal point 6 places left, so the index is +6.
Mathematical Skills » Maths for Geography: Number, Scale and Percentages
What you'll learn this session
Study time: 30 minutes
Cambridge spec: Geographical skills, Mathematical skills
- Add, subtract, multiply and divide with geography data, and use map scale and area
- Write numbers in standard notation and round to significant figures
- Use proportion, ratio, magnitude and frequency
- Work with decimals, fractions and percentages, including percentage change, and complete tables
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Unlock This CourseNumber work, scale and area
Maths is tested on every geography paper, and you can use a calculator. Always write your working and put the units in your answer. Marks are often given for the method.
Key terms:
- Map scale: how real distance is shrunk on a map. On a 1:50 000 map, 1 cm stands for 50,000 cm, which is 0.5 km.
- Area: the amount of surface a place covers. For a rectangle, length × width. On maps it is often given in square kilometres (km2).
Worked example: number work
A made-up farm grows 3.5 tonnes of wheat on each hectare and has 12 hectares. Total: 3.5 × 12 = 42 tonnes. Shared equally by 6 families: 42 ÷ 6 = 7 tonnes each.
Worked example: scale and area
A park measures 4 cm by 3 cm on a 1:25 000 map. First, 1 cm = 250 m. Length = 4 × 250 = 1,000 m = 1 km. Width = 3 × 250 = 750 m = 0.75 km. Area = 1 × 0.75 = 0.75 km2. Convert each side to real distance first, then multiply.
You met scale on maps in Grid References, Directions and Distance. Gradient and pie chart angles are in Contours, Gradient and Cross-Sections and Drawing and Reading Graphs, and discharge is in Rivers and Drainage Basins: Key Features. Use their methods here.
Standard notation and significant figures
Standard notation (standard form) writes very big or very small numbers as a number between 1 and 10, multiplied by a power of 10. The power is called the index. A positive index means a big number. A negative index means a small number.
📏 Small numbers
0.00042 = 4.2 × 10−4. Move the decimal point 4 places right, so the index is −4.
Significant figures (s.f.) count digits from the first non-zero digit. Round the same way as decimal places, then fill any gap with zeros.
Worked example: rounding
A made-up city has 48,672 people. To 1 s.f. it is 50,000. To 2 s.f. it is 49,000. To 3 s.f. it is 48,700. For 0.004573 to 2 s.f., the first significant digit is 4, so the answer is 0.0046.
Proportion, ratio, magnitude and frequency
A ratio compares two or more amounts, such as urban to rural people. A proportion is one part of the whole, written as a fraction or a percentage.
Worked example: ratio
A made-up country of 50 million people has an urban to rural ratio of 3 : 2. There are 3 + 2 = 5 parts, so one part = 50 ÷ 5 = 10 million. Urban = 3 × 10 = 30 million. Rural = 2 × 10 = 20 million. The urban proportion is 3/5 = 60%. To simplify 120 : 80, divide both by 40 to get 3 : 2.
- Magnitude: how big an event is, such as the height of a flood.
- Frequency: how often an event happens, such as once every 20 years.
Hazards usually follow a pattern: high magnitude events are rare (low frequency) and low magnitude events are common (high frequency). If a made-up river has a large flood about once every 20 years, expect about 100 ÷ 20 = 5 large floods in 100 years.
Decimals, fractions and percentages
These are three ways of writing the same amount, so 3/4 = 0.75 = 75%. To turn a fraction into a percentage, divide the top by the bottom, then multiply by 100.
Worked example: percentages
In a made-up survey of 40 people, 14 used the bus. 14 ÷ 40 = 0.35, which is 35%. As a simple fraction, 14/40 = 7/20. To find 35% of 200 people: 0.35 × 200 = 70 people.
Percentage change
Percentage change tells you how much something has grown or shrunk compared with where it started.
Formula
Percentage change = (change ÷ original value) × 100. A rise is a percentage increase. A fall is a percentage decrease.
📈 Increase
A made-up town grows from 80,000 to 92,000. Change = 12,000. 12,000 ÷ 80,000 = 0.15, so a 15% increase.
📉 Decrease
Rainfall falls from 250 mm to 200 mm. Change = 50 mm. 50 ÷ 250 = 0.2, so a 20% decrease.
Completing and interpreting tables
Use the working above to fill the gaps, then say what the table shows.
| Country | Total population (millions) | Urban population (millions) | Percentage urban |
|---|---|---|---|
| A | 40 | 28 | 70% |
| B | 25 | ? | 60% |
| C | 50 | 20 | ? |
For B, 60% of 25 = 0.6 × 25 = 15 million. For C, 20 ÷ 50 = 0.4, which is 40%. To interpret the table: Country A has the highest percentage urban (70%), while Country C has the largest total population but the lowest percentage urban. Quote figures, and use units and the words in the column headings.
Common mistakes
Dividing by the new value instead of the original when finding percentage change. Forgetting to convert map centimetres into real distance before finding area. Writing 45 × 105 as standard notation (the first number must be between 1 and 10). Counting leading zeros as significant figures. Leaving off units.
Exam-style question
Fig. 1.1 shows made-up population data for two islands.
| Island | Population in 2010 | Population in 2020 |
|---|---|---|
| K | 48,000 | 60,000 |
| L | 5,000 | 7,000 |
(a) Calculate the percentage increase in the population of Island K from 2010 to 2020. [2 marks]
(b) A road on Island K measures 7 cm on a map with a scale of 1:50 000. Calculate the real length of the road in kilometres. [2 marks]
(c) Write 60,000 in standard notation. [1 mark]
(d) Assess which island's population grew more between 2010 and 2020. [6 marks]
Model answer
(a) Change = 60,000 − 48,000 = 12,000 (1). 12,000 ÷ 48,000 × 100 = 25% increase (1).
(b) 1 cm = 0.5 km (1), so 7 × 0.5 = 3.5 km (1).
(c) 6 × 104 (1).
(d) Level 3 (5-6 marks): Island K grew by more people, 12,000 compared with 2,000 for Island L, so in actual numbers K grew more. But the starting populations were very different. Island K grew by 25%, while Island L grew by 2,000 ÷ 5,000 × 100 = 40%. So in percentage terms L grew faster. Percentage change is fairer for comparing islands of different size, so L's population grew more in relation to its size, although K added far more people. A clear judgement backed by both calculations reaches the top level.
Exam tip
For an "Assess" question that gives two sets of numbers, calculate both the actual change and the percentage change, then say which is more useful and why.