Introduction
This chapter discusses two important Mathematical concepts: Compound Interest and Uniform Rate of Increase or Decrease. These topics are central to understanding how quantities multiply over time—whether in finance, population growth, depreciation, or resource consumption. Through clear examples and step‑by‑step explanations, students gain confidence in calculating compound interest and modeling uniform percentage changes.
- Compound Interest
1.1 Definition & Fundamental Formula
Compound interest is the interest calculated on both the initial principal and the accumulated interest over previous periods. The formula for amount A after nnn periods at an interest rate r%per period, with principal P, is:
A=P(1+r/100)n
The compound interest (CI) earned is:
CI=A−P=P[(1+r/100)n−1]
1.2 Illustrative Examples
Example 1: Simple Compound Interest (Annual)
Principal P=₹10,000P = ₹10{,}000P=₹10,000, annual rate r=5%r = 5\%r=5%, period n=3n = 3n=3 years.
A=10000(1+5/100)3=10000×(1.05)3≈10000×1.157625=₹11,576.25
Compound interest earned:
CI=11,576.25−10,000=₹1,576.25
Example 2: Semi‑Annual Compounding (within syllabus scope)
If interest is compounded semi‑annually, rate per half‑year = 2.5% (half of 5%), and number of periods n=6n = 6n=6.
A=10000(1+0.025)6≈10000×1.159694=₹11,596.94
CI = ₹1,596.94.
1.3 Key Points for Students
- Understand difference between simple interest and compound interest.
- When compounding frequency increases, the effective interest earned increases.
- Use the correct rate and period conversion when compounding more than once annually
- Uniform Rate of Increase or Decrease
2.1 Concept & Formula
Uniform rate of change applies when a quantity increases or decreases by a fixed percentage each period. The formula resembles compound interest:
For increase at rate r%
New Value=Original Value×(1+r/100)n
For decrease at rate r%
New Value=Original Value×(1−r/100)n
2.2 Detailed Examples
Example 3: Uniform Increase
Suppose a population of 5,000 increases uniformly by 4% each year for 3 years.
Population after 3 years=5000×(1.04)3≈5000×1.124864=5,624.32
So, the population becomes approximately 5,624.
Example 4: Uniform Decrease (Depreciation)
A machine priced at ₹20,000 depreciates uniformly at 10% per year for 2 years.
Value after 2 years=20000×(1−0.10)2=20000×(0.9)2=20000×0.81=₹16,200
2.3 Practical Insights
- Uniform percentage change illustrates real‑life applications: savings growth, inflation, population studies, depreciation of assets.
- Always adjust the formula depending whether it’s an increase or a decrease.
- Encourages clarity in setting up the base value and applying the rate consistently across periods.