↔️ Class 8 · Commercial maths

Direct and Inverse Proportion for Class 8: notes, worked examples and practice

Some quantities rise together (more pens, more cost) and some move in opposite directions (more workers, fewer days). Deciding which kind it is is the key step.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

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Notes

Direct and Inverse Proportion: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

Direct proportion

Two quantities are in direct proportion when one increases or decreases and the other does so at the same rate. Their ratio stays the same: x₁ ÷ y₁ = x₂ ÷ y₂.

Inverse proportion

Two quantities are in inverse proportion when one increases and the other decreases so that their product stays the same: x₁ × y₁ = x₂ × y₂.

Deciding which one

Ask: if one doubles, what happens to the other? If it doubles too, it is direct. If it halves, it is inverse.

The unitary method

For direct proportion find the value of one unit and multiply. For inverse proportion find the total work first, then divide.

Everyday examples

Direct: cost and quantity, distance and time at constant speed. Inverse: workers and days, speed and time for a fixed distance.

Formulas and rules to remember

Direct: x₁ ÷ y₁ = x₂ ÷ y₂
Inverse: x₁ × y₁ = x₂ × y₂
Distance = speed × time

Solved examples

Example 1

8 pens cost ₹96. How much do 5 pens cost?

One pen costs 96 ÷ 8 = ₹12.
5 pens cost 5 × 12 = ₹60.

Answer: ₹60

Example 2

12 men can build a wall in 10 days. How many days will 8 men take?

Total work = 12 × 10 = 120 man-days.
Days for 8 men = 120 ÷ 8 = 15.

Answer: 15 days

Example 3

A car travelling at 60 km/h takes 4 hours for a journey. How long at 80 km/h?

Distance = 60 × 4 = 240 km.
Time = 240 ÷ 80 = 3 hours.

Answer: 3 hours

Example 4

On a map 2 cm stands for 50 km. How far is a distance of 7 cm on the map?

1 cm stands for 25 km.
7 cm stands for 7 × 25 = 175 km.

Answer: 175 km

Avoid these

Common mistakes

Using direct proportion for an inverse problem

More workers means fewer days, not more. Check the direction before you calculate.

Mismatched units

Convert all quantities to the same units first, such as minutes and hours.

Forgetting that the total work stays fixed

In inverse proportion the product of workers and days is constant.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

If 2 pens cost ₹30, the cost of 18 pens is (in ₹)
Answer: 270
If 5 oranges cost ₹20, the cost of 13 oranges is (in ₹)
Answer: 52
If 4 notebooks cost ₹44, the cost of 6 notebooks is (in ₹)
Answer: 66
If 8 pens cost ₹80, the cost of 17 pens is (in ₹)
Answer: 170
If 3 kg of rice cost ₹54, the cost of 15 kg of rice is (in ₹)
Answer: 270
12 workers can finish a job in 15 days. How many days will 3 workers take (working at the same rate)?
Answer: 60
15 workers can finish a job in 18 days. How many days will 3 workers take (working at the same rate)?
Answer: 90
9 workers can finish a job in 8 days. How many days will 2 workers take (working at the same rate)?
Answer: 36
4 workers can finish a job in 12 days. How many days will 3 workers take (working at the same rate)?
Answer: 16
4 workers can finish a job in 8 days. How many days will 8 workers take (working at the same rate)?
Answer: 4

More practice

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FAQ

Direct and Inverse Proportion Class 8: questions students ask

How do I know if it is direct or inverse proportion?

If both quantities increase together, it is direct. If one increases while the other decreases, it is inverse.

What is the formula for inverse proportion?

The product of the two quantities stays the same: x₁ × y₁ = x₂ × y₂.

What is the unitary method?

Find the value of one unit and then multiply to get the required number of units.

Is time and work direct or inverse?

The number of workers and the days needed are in inverse proportion.

Which board is this direct and inverse proportion page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

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