Class 8 — Mathematics (NCERT)
Direct and Inverse Proportions
Class 8
- ✓By the end of this lesson students will be able to define and identify situations involving direct proportion.
- ✓By the end of this lesson students will be able to solve problems related to direct proportion using appropriate methods.
- ✓By the end of this lesson students will be able to define and identify situations involving inverse proportion.
- ✓By the end of this lesson students will be able to solve problems related to inverse proportion using appropriate methods.
- ✓By the end of this lesson students will be able to distinguish between direct and inverse proportion in various real-life scenarios.
Key concepts
Two quantities are said to be in direct proportion if an increase (or decrease) in one quantity leads to a corresponding increase (or decrease) in the other quantity, such that their ratio remains constant. In simpler terms, as one quantity changes, the other quantity changes in the same direction proportionally.
Two quantities are said to be in inverse proportion if an increase in one quantity leads to a corresponding decrease in the other quantity, or vice-versa, such that their product remains constant. In simpler terms, as one quantity increases, the other quantity decreases proportionally.
Key facts to remember
- 1In direct proportion, the ratio of the two quantities (x/y) remains constant.
- 2In inverse proportion, the product of the two quantities (xy) remains constant.
- 3Direct proportion means that if one quantity doubles, the other quantity also doubles.
- 4Inverse proportion means that if one quantity doubles, the other quantity halves.
- 5The unitary method is a useful technique for solving problems involving direct and inverse proportions.
- 6Examples of direct proportion: cost of items and quantity, distance covered and time taken (at constant speed).
- 7Examples of inverse proportion: speed and time taken (for a fixed distance), number of workers and time taken for a job.
Worked examples
Example 1
If 8 kg of sugar costs ₹240, what would be the cost of 15 kg of sugar?
Answer
The cost of 15 kg of sugar is ₹450.
Alternatively, you can use the unitary method: Cost of 1 kg sugar = ₹240 / 8 = ₹30. Cost of 15 kg sugar = 15 × ₹30 = ₹450.
Example 2
A car takes 3 hours to cover a certain distance at a speed of 60 km/h. How long will it take to cover the same distance if its speed is 45 km/h?
Answer
It will take 4 hours to cover the same distance at a speed of 45 km/h.
Example 3
12 workers can complete a piece of work in 20 days. How many workers would be required to complete the same work in 15 days?
Answer
16 workers would be required to complete the same work in 15 days.
Common mistakes
- ✗Confusing direct proportion with inverse proportion, leading to incorrect setup of equations.
- ✗Incorrectly applying the formula (e.g., using x₁y₁ = x₂y₂ for direct proportion or x₁/y₁ = x₂/y₂ for inverse proportion).
- ✗Making calculation errors during cross-multiplication or division.
- ✗Not identifying the constant of proportionality correctly or using it inconsistently.
- ✗Failing to include appropriate units in the final answer, where applicable.
Exam tips
- ★Always read the problem statement carefully to determine whether it is a case of direct or inverse proportion before attempting to solve.
- ★Clearly write down the given quantities and the unknown quantity, assigning variables if necessary.
- ★Show all steps of your working clearly and logically, as method marks are often awarded in examinations.
- ★Double-check your calculations to avoid errors. A quick mental check can often catch obvious mistakes.
- ★Ensure your final answer is reasonable and makes sense in the context of the problem.
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