Curriculum fit: GCSE Maths Ratio, proportion and rates of change (Foundation and Higher; proportional to squares is Higher). US: Grade 7-8 and Algebra I variation.
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What Is It?
Two quantities are in direct proportion when one is a constant multiple of the other: if one doubles, so does the other. Two quantities are in inverse proportion when their product is constant: if one doubles, the other halves.
The constant is called the constant of proportionality, k. Finding k from one pair of values lets you answer any other question about the relationship.
Key Formulas and Ideas
- Direct proportion: y = kx (graph is a straight line through the origin).
- Inverse proportion: y = k/x.
- Proportional to a square: y = kx² (Higher).
Worked Examples
Direct
y is directly proportional to x. When x = 3, y = 12. Find y when x = 7.
- Write y = kx.
- Find k: 12 = k × 3, so k = 4.
- Then y = 4 × 7 = 28.
✓ Answer 28
Inverse
4 workers take 6 days to finish a job. How long would 8 workers take?
- Workers × days is constant: k = 4 × 6 = 24.
- With 8 workers, days = 24 ÷ 8 = 3.
✓ Answer 3 days
Proportional to a Square (Higher)
y is proportional to x². When x = 2, y = 20. Find y when x = 4.
- y = kx². From 20 = k × 4, k = 5.
- When x = 4, y = 5 × 16 = 80.
✓ Answer 80
Common Mistakes to Avoid
- Using direct proportion for an inverse problem, such as more workers taking more days.
- Not finding k first.
- Squaring the wrong quantity in y = kx².
- Mixing up the roles of the variables.
Practice Questions
Frequently Asked Questions
How can I tell which kind of proportion it is?
Ask what happens when one quantity increases. If the other increases in the same ratio it is direct; if it decreases, inverse.
What does 'y varies directly with x' mean?
It means the same as 'y is directly proportional to x'.
Keep Going With These Guides
Ratio and Proportion: How to Simplify, Share and Scale
Simplify ratios, share an amount in a ratio and solve scaling problems with the unit method, with worked examples for GCSE and middle school.
KS3GCSEMiddle SchoolHigh SchoolStraight Line Graphs: Gradient and y = mx + c
Plot straight lines, work out the gradient and read the equation from a graph. Covers y = mx + c (slope-intercept form), with worked examples.
GCSEHigh SchoolSimilar Shapes: Scale Factors, Area and Volume
Use scale factors to find missing lengths, and learn how area and volume scale with k squared and k cubed, with worked examples.
GCSEHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.