Class 8 → Mathematics → Chapter 7: Proportional Reasoning–1 (New Course)

I. Chapter Summary

This chapter introduces the concept of proportional reasoning, which helps students understand relationships between quantities. It focuses on ratios, equivalent ratios, and direct proportion. Students learn how two quantities change together and how to solve real-life problems using proportional thinking. The chapter builds a strong foundation for topics like percentage, profit & loss, and algebra in higher classes.

II. Key Concepts Covered

1. Ratio

• A comparison of two quantities of the same kind.
• Example: Ratio of 2:3 means 2 parts to 3 parts.

2. Equivalent Ratios

• Ratios that represent the same relationship.
• Example:
  $2:3 = 4:6 \ne 6:9$

3. Proportion

• Equality of two ratios:
  $\frac{a}{b} = \frac{c}{d}$​

4. Direct Proportion

• Two quantities are in direct proportion if one increases, the other also increases.
• Example: More workers → more work done.

5. Unitary Method

• Finding the value of one unit first and then scaling up.
• Example: If 5 pens cost ₹50, then 1 pen costs ₹10.

6. Cross Multiplication

• Used to solve proportions:
• If $\frac{a}{b} = \frac{c}{d} \quad \Rightarrow \quad ad = bc$, then
  $a \times d = b \times c$

Visual Understanding of Proportional Reasoning

4

III. Important Questions

(A) MCQs (1 Mark)

1. The ratio 4:6 is equal to:
   (a) 2:3 (b) 3:2 (c) 4:5 (d) 6:4
   Answer: (a) 2:3
2. If $2:5 = x:15$, then x =
   (a) 3 (b) 6 (c) 5 (d) 10
   Answer: (b) 6
3. Which of the following shows direct proportion?
   (a) Speed and time
   (b) Distance and time (constant speed)
   (c) Workers and time
   (d) Height and age
   Answer: (b)
4. If 10 apples cost ₹100, cost of 1 apple is:
   (a) ₹5 (b) ₹10 (c) ₹20 (d) ₹15
   Answer: (b) ₹10

(B) Short Answer Questions (2/3 Marks)

1. Define ratio with an example.
2. Find equivalent ratios of 3:5.
3. Solve: $4:6 = x:12$
4. Explain direct proportion with example.

(C) Long Answer Questions (5 Marks)

1. Explain proportion and cross multiplication method.
2. Solve problems using unitary method.
3. Show that distance and time are in direct proportion (constant speed).
4. Solve real-life problems involving ratios and proportions.

(D) HOTS Questions

1. If the cost of 8 books is ₹240, find the cost of 15 books using proportional reasoning.
2. A car travels 120 km in 2 hours. How far will it travel in 5 hours at the same speed?

IV. Key Formulas / Concepts

• Ratio: a:b
• Proportion: $\frac{a}{b} = \frac{c}{d}$​
• Cross multiplication: $a \times d = b \times c$
• Direct proportion: $y \propto x$

Example:

• If 2 pens cost ₹20, then 1 pen = ₹10

V. Deleted Portions (CBSE 2025–2026)

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Note: This is an estimate. Actual marks distribution may vary.

VII. Previous Year Questions (PYQs)

1 Mark

• Find an equivalent ratio of 2:3. (CBSE 2020)

2/3 Marks

• Solve proportion using cross multiplication. (CBSE 2019)

5 Marks

• Solve real-life problems using unitary method. (CBSE 2018)

VIII. Real-World Application Examples

• Shopping: Cost and quantity relationship.
• Travel: Distance and time calculation.
• Cooking: Ingredient ratios.
• Construction: Material proportions.

IX. Student Tips & Strategies for Success

Time Management

• Practice ratio problems daily.
• Revise formulas regularly.

Exam Preparation

• Learn unitary method properly.
• Solve word problems carefully.

Stress Management

• Break problems into steps.
• Use diagrams or tables.

X. Career Guidance & Exploration (Class 8 Level)

• Important for:
  • Economics & Commerce
  • Engineering
  • Data Analysis
• Helps in logical reasoning and decision-making.

XI. Important Notes

• Always simplify ratios.
• Check units before comparing quantities.
• Practice real-life problems.
• Refer to NCERT and CBSE updates regularly.