Proportional Reasoning For Class 8 by Shubham Maheshwari

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Chapter 1

Direct Proportion and Unit Rate

Key Concepts

Direct proportion means two quantities change at the same constant rate. In exams, unit rates, tables, graphs, and equations help you find missing values accurately.

• Unit rate = amount for 1 unit.

• Direct proportion rule: y = kx, where k is the constant of proportionality.

• Find k using: k = y ÷ x.

• A proportional table has a constant value of y ÷ x.

• Its graph is a straight line passing through (0, 0).

• If one quantity doubles, the other also doubles.

• Use matching units before calculating.

Before you continue: How can you check from a table whether two quantities are directly proportional?

Key Terms

Direct proportion - A relationship where one quantity changes by the same factor as another.

Unit rate - The amount corresponding to 1 unit.

Constant of proportionality - The fixed number k in y = kx.

Ratio - A comparison of two quantities.

Proportional table - A table in which y ÷ x remains constant.

Origin - The point (0, 0) on a graph.

Active Recall

Direct proportion

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Unit rate

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Constant of proportionality

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Ratio

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Proportional table

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Origin

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Worked Examples

Example 1: Unit rate

A car travels 180 km in 3 hours. Find its speed.

• Speed = distance ÷ time.

• 180 ÷ 3 = 60.

• The unit rate is 60 km/h.

Now you try: A cyclist travels 84 km in 4 hours. Find the unit rate.

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Example 2: Using direct proportion

Five notebooks cost ₹150. Find the cost of 8 notebooks.

• Cost of 1 notebook = 150 ÷ 5 = ₹30.

• Cost of 8 notebooks = 8 × 30 = ₹240.

• Therefore, 8 notebooks cost ₹240.

Now you try: Seven pens cost ₹105. Find the cost of 12 pens.

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Example 3: Table and equation

The table shows hours worked and wages earned.

Hours, x

Wages, y (₹)

360

900

1440

• Calculate y ÷ x: 360 ÷ 2 = 180.

• Check: 900 ÷ 5 = 180 and 1440 ÷ 8 = 180.

• Thus, k = 180, so the equation is y = 180x.

• For 7 hours: y = 180 × 7 = ₹1260.

Now you try: If y = 75x, find y when x = 6.

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Practice Questions

• (Easy) A tap fills 24 litres in 3 minutes. Find the unit rate.

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• (Easy) Complete the proportional table: 4 kg costs ₹280; find the cost of 7 kg.

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• (Medium) Is this table directly proportional? Give a reason: x = 2, 4, 6; y = 10, 20, 31.

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• (Medium) A map uses 1 cm for 8 km. How far does 6.5 cm represent?

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• (Hard) A machine produces 420 parts in 7 hours. How many parts will it produce in 12.5 hours?

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• (Hard) The graph of a relationship is a straight line through (0, 0) and (4, 28). Find the equation and the value of y when x = 9.

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Answer Key

• 8 litres per minute

• ₹490

• No; 31 ÷ 6 is not 5

• 52 km

• 750 parts

• y = 7x; y = 63

Exam Tips & Common Mistakes

• Do not divide in the wrong order. Unit rate is always “amount ÷ number of units.”

• Check that all units match, such as minutes with minutes or metres with metres.

• Do not assume every straight line shows direct proportion; it must pass through the origin.

• For tables, calculate y ÷ x in at least two columns.

• Show the unit rate and the final calculation; markers award method marks.

• Write the equation before substituting values in harder questions.

• Use a quick estimate to check whether your answer is sensible.

Quick Reference

• unit rate = total amount ÷ number of units

• y = kx

• k = y ÷ x

• new value = unit rate × new number of units

• Direct proportion graph: straight line through (0, 0)

Mnemonic: U-T-G-E

U nit rate, T able check, G raph through origin,

E quation, then substitute.

• Direct proportion in a table: y ÷ x stays constant.

• Direct proportion in an equation: y = kx.

• Direct proportion in a graph: a straight line through (0, 0).

Confidence Check

• I can find a unit rate. [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5

• I can test whether two quantities are directly proportional. [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5

• I can complete a proportional table. [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5

• I can use y = kx to solve problems. [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5

• I can identify direct proportion from a graph. [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5

End of chapter one. 4 more chapters in the full book.

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What's inside: 5 chapters

1. Direct Proportion and Unit Rate
2. Inverse Proportion and Product Constant
3. Ratio, Proportion, and Comparison
4. Percent Proportion with Increase/Decrease
5. Mixed Proportion Problems and Graphs

About this book

"Proportional Reasoning For Class 8" is a study guide book by Shubham Maheshwari with 5 chapters and approximately 4,115 words. Chapter module on proportional reasoning for Class 8.

This book was created using Inkfluence AI, an AI-powered book generation platform that helps authors write, design, and publish complete books. It was made with the Study Guide Generator.

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What is "Proportional Reasoning For Class 8" about?

Chapter module on proportional reasoning for Class 8

How many chapters are in "Proportional Reasoning For Class 8"?

The book contains 5 chapters and approximately 4,115 words. Topics covered include Direct Proportion and Unit Rate, Inverse Proportion and Product Constant, Ratio, Proportion, and Comparison, Percent Proportion with Increase/Decrease, and more.

Who wrote "Proportional Reasoning For Class 8"?

This book was written by Shubham Maheshwari and created using Inkfluence AI, an AI book generation platform that helps authors write, design, and publish books.

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