Simple and Compound Interest

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Simple and Compound Interest — Key Facts for NABE (Pakistan)

• Simple Interest (SI): SI = (P × R × T) / 100; Amount = P + SI
• Compound Interest (CI): Amount = P(1 + R/100)^T; CI = Amount - Principal
• Compound Interest is calculated on the previous year’s amount (principal + interest)
• Interest is always charged on Principal in Simple Interest
• ⚡ Exam tip: For short periods (1-2 years), both give similar results. For longer periods, CI gives higher returns

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Simple and Compound Interest — NABE (Pakistan) Study Guide

Key Terms

• Principal (P): The original amount borrowed or invested
• Rate (R): Interest rate per year (percentage)
• Time (T): Duration of the loan or investment in years
• Simple Interest (SI): Interest calculated only on the original principal
• Compound Interest (CI): Interest calculated on principal + accumulated interest
• Amount (A): Total amount to be paid/received (Principal + Interest)

Simple Interest Formulas

Basic Formula:

SI = (P × R × T) / 100

Amount Formula:

A = P + SI = P + (P × R × T)/100 = P(1 + RT/100)

Finding Unknown Values:

• P = (SI × 100) / (R × T)
• R = (SI × 100) / (P × T)
• T = (SI × 100) / (P × R)

Example Calculations

Example 1: Find SI on Rs. 5000 at 8% per annum for 3 years.

• SI = (5000 × 8 × 3) / 100 = Rs. 1200
• Amount = 5000 + 1200 = Rs. 6200

Example 2: Find the rate if SI on Rs. 2500 for 4 years is Rs. 800.

• R = (800 × 100) / (2500 × 4) = 8%

Compound Interest Formulas

Annual Compounding:

A = P(1 + R/100)^T
CI = A - P

Half-Yearly Compounding: Rate becomes R/2, Time becomes 2T

A = P(1 + R/200)^(2T)

Quarterly Compounding: Rate becomes R/4, Time becomes 4T

A = P(1 + R/400)^(4T)

Comparison: SI vs CI

Year	SI (10%)	CI (10%)
1	100	100
2	200	210
3	300	331

For Rs. 1000 at 10% per annum for 3 years:

• SI = (1000 × 10 × 3)/100 = Rs. 300
• CI = 1000[(1.1)³ - 1] = 1000 × 0.331 = Rs. 331

Key Difference: CI yields more because interest earns interest.

NABE Exam Pattern

Common question types:

1. Direct SI/CI calculations
2. Finding rate or time when other values are given
3. Comparing SI and CI for same principal and rate
4. Population growth problems (similar to CI)
5. Depreciation problems

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Simple and Compound Interest — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Time Value of Money

The fundamental principle behind interest is that money available now is worth more than the same amount in the future. This is because:

• Money can be invested to earn returns
• Inflation reduces purchasing power over time
• There’s uncertainty about future receipt

Interest compensates for:

1. Time value of money
2. Risk of non-repayment
3. Opportunity cost of lending

2. Simple Interest — Complete Derivation

Concept: Interest is calculated uniformly each year on the original principal.

Year-by-Year Breakdown:

• Year 1 Interest = P × R/100
• Year 2 Interest = P × R/100 (same base)
• Year 3 Interest = P × R/100 (same base)
• Total SI = 3P × R/100 = P × R × T/100

Equivalent Formula for Any Period: If time is given in months: T = months/12 If time is given in days: T = days/365 (or 360 for commercial calculations)

Commercial Month: Some problems use 30 days/month for simplicity.

3. Compound Interest — Complete Analysis

Key Difference: Each year’s interest becomes part of the principal for the next year.

Year-by-Year for P = 1000, R = 10%:

• Start: P = 1000
• End Year 1: A = 1000 × 1.1 = 1100
• End Year 2: A = 1100 × 1.1 = 1210
• End Year 3: A = 1210 × 1.1 = 1331
• CI Total = 1331 - 1000 = 331

General Formula Derivation:

A = P(1 + R/100)^T

For Fractional Years:

• 2.5 years: A = P(1 + R/100)² × (1 + 0.5R/100)

4. Different Compounding Frequencies

Formula when interest is compounded n times per year:

A = P(1 + R/(100n))^(nT)

Effective Annual Rate (EAR):

• Nominal rate R compounded annually: EAR = R
• Nominal rate R compounded semi-annually: EAR = (1 + R/200)² - 1
• Example: 12% compounded semi-annually: EAR = (1.06)² - 1 = 12.36%

5. Applications — Population Growth

Formula (similar to CI):

P = P₀(1 + r/100)^T

Example: Population of a town is 50,000 and grows at 5% per annum. Find population after 4 years.

• P = 50000(1 + 5/100)^4
• P = 50000 × (1.05)^4
• P = 50000 × 1.2155 ≈ 60,775

6. Applications — Depreciation

Formula (similar to CI but with negative rate):

V = V₀(1 - r/100)^T

Example: A machine worth Rs. 100,000 depreciates at 10% per year. Find value after 3 years.

• V = 100000(1 - 10/100)^3
• V = 100000 × (0.9)³ = Rs. 72,900

7. Comparison: SI vs CI Over Time

For Principal P, Rate R%, Time T years:

Simple Interest: A = P(1 + RT/100) Compound Interest: A = P(1 + R/100)^T

When is SI > CI?

• Only when T < 1 year and R(SI) > R(CI)… otherwise CI always ≥ SI

Difference Formula: For 2 years: Difference = P(R/100)²/10000 For 3 years: Difference = P[(R/100)²(3 + R/100)]/10000

8. Installment Problems

When a loan is repaid in equal installments:

Example: A loan of Rs. 10,000 at 10% per annum is repaid in 2 equal annual installments. Find installment amount.

Formula for installment:

Installment = P × (1 + R/100)^T × R/100 / [(1 + R/100)^T - 1]

Solution:

• Using formula: Installment = 10000 × 1.1² × 0.1 / (1.1² - 1)
• = 10000 × 1.21 × 0.1 / 0.21
• = 1210 / 0.21 = Rs. 5761.90

9. Common Mistakes to Avoid

1. Time Period Confusion: Convert months to years properly
2. Rate Per Period: Adjust rate when compounding frequency changes
3. Simple vs. Compound: Don’t mix formulas
4. Rounding Errors: Keep more decimals during calculation
5. Interpreting “Rate”: Is it per annum or per period?

Practice Questions for NABE

1. Find the compound interest on Rs. 8000 at 15% per annum for 2 years.
2. The simple interest on a sum for 3 years at 8% is Rs. 720. Find the compound interest for the same period at the same rate.
3. A sum of money doubles in 5 years at simple interest. Find the rate of interest.
4. The population of a city is 100,000 and decreases at 5% per annum. Find population after 3 years.
5. A loan of Rs. 5000 is to be repaid in 3 equal annual installments at 12% interest. Find the installment amount.

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