Direct and Inverse Proportions
Focused last-minute revision with key concepts, formulas, worked examples, exam traps and practice.
Chapter Snapshot
- Direct proportion means both quantities change in the same ratio.
- For direct proportion y/x is constant.
- Inverse proportion means one increases while the other decreases so the product stays constant.
- For inverse proportion xy is constant.
- Identify the relationship before calculating.
What You Will Learn
- Recognise direct proportion
- Recognise inverse proportion
- Use constant ratios
- Use constant products
- Solve real-life proportion problems
Last-Minute Chapter Map
Use this chapter in three passes:
- Understand the core ideas.
- Memorise formulas and exam traps.
- Attempt the compact practice set without looking at the answers.
Direct proportion
If one quantity doubles and the other also doubles under unchanged conditions, the relationship is direct. Examples include number of identical items and total cost.
Inverse proportion
If one quantity doubles and the other halves while the overall job or distance remains fixed, the relationship is inverse. Examples include workers and time for the same work.
How to recognise
Check whether a ratio or a product remains constant. Do not decide only because one quantity increases or decreases.
Unitary method
For direct proportion, find the value for one unit and scale up. For inverse proportion, the constant-product method is usually faster.
Formula Sheet
Formula / Key Relations
Direct proportion
Equivalent to y=kx.
Inverse proportion
Product remains constant.
Worked Examples
More workers means fewer days: inverse proportion.
Cost is directly proportional to quantity.
Exam Traps
Quick Revision
Quick Revision
- Direct proportion means both quantities change in the same ratio.
- For direct proportion y/x is constant.
- Inverse proportion means one increases while the other decreases so the product stays constant.
- For inverse proportion xy is constant.
- Identify the relationship before calculating.
- Revise the formula sheet once more before attempting questions.
Compact Practice
5 pens cost ₹60; 10 pens?
View Solution
₹120
If y/x = 4 and x=7, y?
View Solution
28
8 workers need 12 days; 16 workers?
View Solution
6 days
Speed and time for fixed distance are what type?
View Solution
Inverse proportion
Final Chapter Summary
Chapter Summary
- Direct proportion means both quantities change in the same ratio.
- For direct proportion y/x is constant.
- Inverse proportion means one increases while the other decreases so the product stays constant.
- For inverse proportion xy is constant.
- Identify the relationship before calculating.
- Prioritise correct method, clean working and final units where applicable.