Scale and Proportion in ContextEdexcel GCSE Maths: Revision notes
Section 1
Scale Factors, Maps and Scale Drawings
A scale factor links a real-life length to its representation on a map or scale drawing. Scales are usually written as a ratio, e.g. 1 : 25000, meaning 1 cm on the drawing represents 25000 cm (250 m) in real life.
To find a real-life distance: measured length × scale factor. To find a drawing length: real-life length ÷ scale factor.
Always convert both parts of the ratio to the same unit before using it, and convert your final answer to a sensible unit (e.g. cm to km for large distances).
On a map with scale 1 : 50000, a distance of 4 cm represents 4 × 50000 = 200000 cm = 2 km in real life.
Section 2
Ratio in Real Contexts: Conversion, Mixing and Concentration
Ratio and proportion apply to many real-world settings:
- Conversion: changing between units or currencies using a fixed ratio
- Comparison: expressing how much bigger or smaller one quantity is than another
- Scaling: enlarging or reducing a recipe, mixture or model in proportion
- Mixing and concentration: combining substances in a given ratio (e.g. paint colours, squash and water) and finding the amount of each ingredient needed for a different total quantity
A squash is mixed in the ratio 1 : 4 (squash : water). To make 2.5 litres in total, there are 5 parts, so each part is 0.5 litres — 0.5 litres of squash and 2 litres of water.
Section 3
Comparing Lengths, Areas and Volumes Using Ratio
When two shapes are similar (identical shape, different size), their lengths, areas and volumes are all connected by ratio, and this links directly to scale factors:
- If the length scale factor between two similar shapes is k, then their areas are in the ratio k² : 1
- Their volumes (for 3D similar solids) are in the ratio k³ : 1
This is why enlarging a shape's dimensions has a much bigger effect on its area or volume than on its length.
Do not use the length scale factor directly for area or volume — you must square it for area, or cube it for volume.
Section 4
Proportion as Equality of Ratios
Two quantities are in proportion when their ratio stays the same as both quantities change together — this is the same idea as equal fractions. If a/b = c/d, then a, b, c, d are in proportion.
This links ratio directly to linear functions: a proportional relationship between x and y can always be written as y = kx, a straight line through the origin.
Section 5
Recognising Direct and Inverse Proportion Graphs
- A direct proportion graph (y = kx) is a straight line passing through the origin (0, 0). The steeper the line, the larger the value of k.
- An inverse proportion graph (y = k/x) is a curve that decreases as x increases, approaching but never touching either axis.
Recognising the shape of these graphs (or their equations) tells you immediately what type of relationship is being described, without needing extra calculation.
If a graph is a straight line NOT through the origin, it is neither direct nor inverse proportion — it is simply a linear relationship y = mx + c with c ≠ 0.
Must Know
- A scale factor (e.g. 1 : 25000) converts between a drawing/map length and the real-life length
- Real-life distance = drawing length × scale factor; drawing length = real-life distance ÷ scale factor
- Mixing/concentration problems: split the total into the number of parts given by the ratio
- For similar shapes with length scale factor k: area scale factor = k², volume scale factor = k³
- Proportion means two ratios are equal (a/b = c/d); this links to the straight line y = kx through the origin
- Direct proportion graphs are straight lines through the origin; inverse proportion graphs are curves approaching both axes
That's the notes covered.
Carry on to the next subtopic.