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CBSE • Class 8 • Mathematics • Ch 10
Estimated Time: 45 Mins
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Proportional Reasoning-2

In Class 8 Mathematics, "Proportional Reasoning-2" delivers rigorous mathematical reasoning, algebraic precision, and geometric mastery aligned with the 2026–27 NCERT curriculum.

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Overview Core Concepts Summary Self Check Take MCQ Test

📈 Have You Ever Wondered?

Why did Albert Einstein call Compound Interest the 'Eighth Wonder of the World'? In advanced commercial mathematics, interest earns interest, compound...

Why did Albert Einstein call Compound Interest the 'Eighth Wonder of the World'? In advanced commercial mathematics, interest earns interest, compounding savings into exponential growth over time.

Why This Chapter Matters

In Class 8 Mathematics, "Proportional Reasoning-2" delivers rigorous mathematical reasoning, algebraic precision, and geometric mastery aligned with the 2026–27 NCERT curriculum.

Before You Begin (Prerequisites)

  • Simple interest formula from Chapter 8.
  • Percentage increase and decrease.
  • Exponents and powers from Chapter 2.

What You Will Learn (Core Objectives)

  • Distinguish between Simple Interest (linear) and Compound Interest (exponential).
  • Apply the Compound Interest formula: $A = P\left(1 + \frac{R}{100}\right)^n$.
  • Calculate compound interest compounded annually and semi-annually.
  • Apply compounding formulas to population growth and asset depreciation.
  • Compare financial investment options using compound growth.

Chapter Roadmap & Progression

1 1. Simple vs. Compound Interest
2 2. The Compound Interest Formula
3 3. Depreciation and Population Grow...

Complete Concept Guide (100% Curriculum Coverage)

1. Simple vs. Compound Interest

  • Simple Interest: Interest is calculated strictly on the original principal every year; interest amount remains constant.
  • Compound Interest (CI): Interest earned in the first year is added to the principal to form the new principal for the second year. Interest earns interest!

2. The Compound Interest Formula

The total amount $A$ after $n$ years at rate $R\%$ per annum compounded annually is: $$\mathbf{A = P\left(1 + \frac{R}{100}\right)^n} \quad \text{and} \quad \mathbf{\text{CI} = A - P}$$

3. Depreciation and Population Growth

The same formula models real-world rates: population increasing at rate $R\%$ grows as $P\left(1 + \frac{R}{100}\right)^n$, while vehicle value depreciating (decreasing) over time uses: $A = P\left(1 - \frac{R}{100}\right)^n$.

Visual Learning & Conceptual Map

Proportional Reasoning-2 Mathematical Matrix

Core formulas, geometric proofs, and computational algorithms
Mathematical Framework

1. Simple vs. Compound Interest • 2. The Compound Interest Formula • 3. Depreciation and Population Growth

Chapter Summary & 10 Key Takeaways

Takeaway 1
Simple Interest: Constant linear return on original principal.
Takeaway 2
Compound Interest: Exponential growth where interest earns interest.
Takeaway 3
Formula: $A = P\left(1 + \frac{R}{100}\right)^n$.
Takeaway 4
Semi-Annual Compounding: Rate is halved ($\frac{R}{2}$), time periods doubled ($2n$).
Takeaway 5
Depreciation: Value decreases exponentially using $(1 - \frac{R}{100})^n$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Calculate the compound interest on ₹10,000 for 2 years at 10% per annum compounded annually.
Reveal Answer & Explanation
Answer: $A = 10,000\left(1 + \frac{10}{100}\right)^2 = 10,000 \times (1.1)^2 = 10,000 \times 1.21 = ₹ 12,100$. $\text{CI} = 12,100 - 10,000 = ₹ 2,100$.
Use A = P(1+R/100)^n.
2
Why is Compound Interest always greater than Simple Interest for periods longer than 1 year?
Reveal Answer & Explanation
Answer: Because in compound interest, the interest from each year is added to the principal, so subsequent interest is calculated on a larger base amount.
Interest earns interest on interest.
3
A motorcycle was bought at ₹80,000. Its value depreciates at 10% per annum. Find its value after 1 year.
Reveal Answer & Explanation
Answer: $\text{Value} = 80,000\left(1 - \frac{10}{100}\right) = 80,000 \times 0.9 = ₹ 72,000$.
Subtract 10% depreciation.
4
What changes in the formula if interest is compounded half-yearly?
Reveal Answer & Explanation
Answer: The annual interest rate is divided by 2 ($R/2$), and the number of compounding conversion periods is doubled ($2n$).
Half rate and double periods.
5
The population of a city was 50,000 in 2024. If it grows at 5% per annum, find the population in 2026.
Reveal Answer & Explanation
Answer: $50,000\left(1 + \frac{5}{100}\right)^2 = 50,000 \times (1.05)^2 = 50,000 \times 1.1025 = 55,125$.
Compounding population growth.
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