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Tamil Nadu (TNBSE)Class 7 Mathematics★ 6-10 marks in the TN Class 7 Term 1 exam through proportion and unitary-method word problems

Direct and Inverse Proportion

Samacheer Kalvi (TN State Board) Class 7 Maths — Term 1 Chapter 4, Direct and Inverse Proportion. Ratio and proportion, direct proportion (quantities rise and fall together) and inverse proportion (one rises as the other falls), and the unitary method, with worked examples, exercises and quick revision.

⏱ 15 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Define ratio and proportion
2Identify direct proportion and solve such problems
3Identify inverse proportion and solve such problems
4Apply the unitary method
5Classify real situations as direct or inverse

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Why this chapter matters

Direct and inverse proportion model everyday relationships between quantities — cost, speed, work and time. Identifying the type of proportion and solving by the unitary method are directly tested in the TN Class 7 Term 1 exam.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Class 6 — Ratio and Proportion

Comparing quantities.

Direct and Inverse Proportion — Class 7 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 7 Mathematics, Term 1 — Chapter 4. How quantities change together.

1. About this chapter

This chapter covers ratio and proportion, direct proportion, inverse proportion, and the unitary method.

2. Ratio and proportion

A ratio compares two quantities of the same kind by division, written a : b (or a/b).
A proportion states that two ratios are equal, a : b = c : d, i.e. a × d = b × c (product of extremes = product of means).

3. Direct proportion

Two quantities are in direct proportion if they increase or decrease together so that their ratio stays constant: a/b = constant.
Example: more articles cost more money. If 4 pens cost ₹20, then 10 pens cost (20 ÷ 4) × 10 = ₹50.

4. Inverse proportion

Two quantities are in inverse proportion if one increases as the other decreases so that their product stays constant: a × b = constant.
Example: more workers take less time. If 6 workers finish a job in 10 days, then 12 workers finish it in (6 × 10) ÷ 12 = 5 days.

5. The unitary method

First find the value of one unit, then multiply for the required number of units.
Example: 5 kg of rice cost ₹250 → 1 kg costs ₹50 → 8 kg cost ₹400.

6. Worked examples

Example 1. If 3 books cost ₹120, find the cost of 7 books. 1 book = 120 ÷ 3 = ₹40 → 7 books = 40 × 7 = ₹280 (direct proportion).

Example 2. 8 taps fill a tank in 9 hours. How long will 12 taps take? Inverse: 8 × 9 = 12 × t → t = 72 ÷ 12 = 6 hours.

Example 3. Are speed and time (for a fixed distance) directly or inversely proportional? Inversely — higher speed means less time.

7. Exercises (Samacheer Kalvi)

If 6 m of cloth cost ₹450, find the cost of 10 m.
A car travels 240 km on 16 litres of petrol. How far on 6 litres?
15 workers build a wall in 8 days. How many days for 10 workers?
If 9 pipes fill a tank in 20 minutes, how long for 12 pipes?
State whether each is direct or inverse: (a) number of pens and total cost (b) number of workers and time taken.

8. Common mistakes

Mistake: Multiplying when you should divide in inverse proportion. Fix: In inverse proportion, the product is constant (a₁b₁ = a₂b₂).
Mistake: Confusing direct and inverse. Fix: Direct → both rise/fall together; inverse → one rises as the other falls.
Mistake: Cross-multiplying a proportion wrongly. Fix: a : b = c : d means a × d = b × c.

9. Quick revision

Term 1 · Ch 4 · proportion.
Ratio a : b; proportion a : b = c : d → ad = bc.
Direct: a/b constant (rise/fall together) — use the unitary method.
Inverse: a × b constant (one up, other down).

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Proportion

a : b = c : d → a×d = b×c

Extremes × = means ×.

Direct proportion

a/b = constant

Rise and fall together.

Inverse proportion

a × b = constant

One up, the other down.

Unitary method

find one unit, then multiply

Used for direct proportion.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Multiplying when you should divide in inverse proportion

✓ In inverse proportion the product is constant (a₁b₁ = a₂b₂).

WATCH OUT

✗ Confusing direct and inverse

✓ Direct → both rise/fall together; inverse → one rises as the other falls.

WATCH OUT

✗ Cross-multiplying a proportion wrongly

✓ a : b = c : d means a × d = b × c.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Direct proportion

Cost and quantity word problems.

Questions

Inverse proportion

Workers, time and pipes problems.

Questions

Classify

Decide direct or inverse for given situations.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Direct

If 6 m of cloth cost ₹450, find the cost of 10 m.

▶ Show solution

1 m = 75; 10 m = ₹750.

Q2MEDIUM· Direct

A car travels 240 km on 16 litres. How far on 6 litres?

▶ Show solution

1 litre = 15 km; 6 litres = 90 km.

Q3MEDIUM· Inverse

15 workers build a wall in 8 days. How many days for 10 workers?

▶ Show solution

15 × 8 = 10 × d → d = 12 days.

Q4MEDIUM· Inverse

If 9 pipes fill a tank in 20 minutes, how long for 12 pipes?

▶ Show solution

9 × 20 = 12 × t → t = 15 minutes.

Q5EASY· Classify

Is the number of workers and the time to finish a job direct or inverse?

▶ Show solution

Inverse proportion.

Q6EASY· Proportion

Find x: 4 : 6 = x : 18.

▶ Show solution

4 × 18 = 6x → x = 12.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Term 1 Chapter 4 of Samacheer Kalvi Class 7 Maths.
•Ratio a : b; proportion a : b = c : d gives a×d = b×c.
•Direct proportion: ratio stays constant; quantities rise and fall together.
•Inverse proportion: product stays constant; one rises as the other falls.
•Unitary method: find the value of one unit, then multiply.
•Cost-quantity is direct; workers-time and speed-time are inverse.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-10 marks across proportion word problems

Question type	Marks each	Typical count	What it tests
Direct proportion	2	2	Cost/quantity sums
Inverse proportion	2	2	Workers/time/pipes
Classify / proportion	1	1-2	Type and solving a proportion

Prep strategy

First decide direct or inverse
For direct, use the unitary method
For inverse, keep the product constant
Use ad = bc for proportions

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Shopping

Working out the cost of any quantity from a unit price.

Travel

Relating speed, time and distance.

Work planning

Estimating time when the number of workers changes.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

State the type of proportion first
Show the unitary-method steps for direct problems
Use a₁b₁ = a₂b₂ for inverse problems
Check the answer makes real-world sense

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

If 12 men dig a trench in 9 days working 8 hours a day, find the days for 18 men working 6 hours a day.
Explain why speed and time are inversely proportional for a fixed distance.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN Class 7 Term 1 ExamHigh

NMMS / Foundation MathsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Ask what happens to the second quantity when the first increases: if it also increases (more pens, more cost) it is direct; if it decreases (more workers, less time) it is inverse.

A way to solve direct-proportion problems by first finding the value of a single unit and then multiplying by the number of units required.

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