Simple & Compound Interest

Master the concepts of simple and compound interest with our comprehensive guide. Learn about interest calculations, applications, and problem-solving techniques.

Table of Contents

Simplified Quantitative Formulas: Simple & Compound Interest

  • Principal (P): The initial amount of money (loan or investment).
  • Simple Interest (SI): Interest calculated only on the principal each period. SI = (P × R × T) / 100
  • Amount (A): Total after interest: A = P + SI
  • Compound Interest (CI): Interest is periodically added to the principal, so future interest is earned on interest as well. A = P × (1 + R/100)^N
  • Compounding Frequency: For half-yearly, quarterly, etc., adjust rate and periods. Half-yearly: A = P × (1 + (R/2)/100)^(2N)
  • Changing Rates: If rates change each year, multiply accordingly: A = P × (1 + R₁/100) × (1 + R₂/100) × ...
  • CI vs SI: For the same P, R, and total time > 1 year, Compound interest > Simple interest.
  • Difference Between CI and SI: For two years, CI – SI ≈ P × (R/100)^2
  • Fractional Time: For fractional years, handle integer and fraction parts separately.
  • Present Worth: Present value of a future sum: K / (1 + R/100)^N
  • Installments (EMI): x = P × (r/100) × (1+r/100)^n / [ (1+r/100)^n – 1 ]
  • Special Cases: If interest is compounded more frequently, or if rates change, always adjust the formula accordingly.

What do these mean? (Super Simple Explanations & Examples)

  • Principal (P): The starting money you put in or borrow.
    Example: If you put ₹100 in a piggy bank, your principal is ₹100.
  • Simple Interest (SI): Extra money you get (or pay) just for the original amount, not on any interest earned before.
    Example: Borrow ₹100 at 5% per year for 2 years, SI = (100×5×2)/100 = ₹10.
  • Amount (A): The total you have at the end: your starting money plus all the interest.
    Example: If you earned ₹10 interest, your amount is ₹100 + ₹10 = ₹110.
  • Compound Interest (CI): You earn interest on your interest too! Each year, the interest is added to your money, and next year you get interest on the new total.
    Example: ₹100 at 5% for 2 years: Year 1: ₹5, Year 2: ₹5.25 (because now you have ₹105, so 5% of ₹105 = ₹5.25). Total = ₹110.25.
  • Compounding Frequency: If interest is added more often (like every 6 months), use half the rate but double the periods.
    Example: ₹100 at 10% per year, compounded half-yearly for 1 year: Rate = 5%, periods = 2. A = 100×(1+0.05)^2 = ₹110.25.
  • Changing Rates: If the rate changes each year, multiply for each year.
    Example: ₹100 at 10% first year, 20% second year: A = 100×1.1×1.2 = ₹132.
  • CI vs SI: Compound interest is always more than simple interest if you leave your money for more than 1 year, because you earn interest on interest.
    Example: For ₹100 at 10% for 2 years: SI = ₹20, CI = ₹21.
  • Difference CI–SI: For 2 years, the extra you get from compounding is about P×(R/100)^2.
    Example: ₹100 at 10% for 2 years: Difference = 100×(0.1)^2 = ₹1.
  • Fractional Time: If you invest for 2.5 years, do 2 years as usual, then add half a year's interest.
    Example: ₹100 at 10% for 2.5 years: First 2 years: 100×(1.1)^2 = ₹121. Half year: 10% of ₹121 = ₹6.05, so total = ₹127.05.
  • Present Worth: How much a future sum is worth today.
    Example: If you'll get ₹110 in 1 year at 10%, present worth = 110/(1.1) = ₹100.
  • Installments (EMI): If you borrow money and pay it back in equal parts, this formula tells you how much to pay each time.
    Example: Borrow ₹1000 for 2 years at 10%: EMI = 1000×0.1×(1.1)^2/[(1.1)^2–1] ≈ ₹576.20 per year.
  • Special Cases: If interest is compounded quarterly, use R/4 and 4N. If rates change, multiply for each year.
    Example: ₹100 at 10% for 1 year, compounded quarterly: A = 100×(1+0.025)^4 ≈ ₹110.38.

1. Basic Concepts

(a) Understanding Interest

Interest is the cost of borrowing money or the return on investment.

Key Terms:

  • Principal (P): The initial amount of money
  • Rate of Interest (R): The percentage charged on the principal
  • Time (T): The duration for which the money is borrowed/invested
  • Interest (I): The amount charged/earned on the principal
  • Amount (A): Principal + Interest

Example 1: Basic Interest

Principal = ₹1000

Rate = 10% per annum

Time = 2 years

Interest = ₹200

Amount = ₹1200

Interest Components

Principal: 83.33%

Interest: 16.67%

(b) Types of Interest

Different types of interest and their characteristics.

Types of Interest:

  • Simple Interest: Interest calculated only on the principal amount
  • Compound Interest: Interest calculated on principal and accumulated interest
  • Fixed Interest: Rate remains constant throughout the period
  • Variable Interest: Rate changes based on market conditions

Example 2: Interest Types

Principal = ₹1000

Rate = 10% per annum

Time = 2 years

Simple Interest = ₹200

Compound Interest = ₹210

2. Simple Interest

(a) Simple Interest Formula

Simple interest is calculated only on the principal amount.

Simple Interest Formula:

I = P × R × T

A = P + I = P(1 + RT)

where:

  • I = Interest
  • P = Principal
  • R = Rate of interest per annum
  • T = Time in years
  • A = Amount

Example 3: Simple Interest Calculation

P = ₹5000

R = 8% per annum

T = 3 years

I = 5000 × 0.08 × 3 = ₹1200

A = 5000 + 1200 = ₹6200

(b) Simple Interest Applications

Applications of simple interest in real-world scenarios.

Common Applications:

  • Short-term loans
  • Fixed deposits
  • Savings accounts
  • Personal loans

Example 4: Loan Repayment

Loan amount = ₹10000

Rate = 12% per annum

Time = 2 years

Interest = 10000 × 0.12 × 2 = ₹2400

Total repayment = ₹12400

Monthly payment = ₹516.67

3. Compound Interest

(a) Compound Interest Formula

Compound interest is calculated on principal and accumulated interest.

Compound Interest Formula:

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

I = A - P

where:

  • A = Amount
  • P = Principal
  • R = Annual interest rate
  • n = Number of times interest is compounded per year
  • T = Time in years
  • I = Interest

Example 5: Compound Interest Calculation

P = ₹10000

R = 10% per annum

n = 2 (semi-annual)

T = 2 years

A = 10000(1 + 0.1/2)^(2×2) = ₹12155.06

I = 12155.06 - 10000 = ₹2155.06

(b) Compounding Frequencies

Different compounding frequencies and their effects.

Common Compounding Frequencies:

  • Annually (n=1)
  • Semi-annually (n=2)
  • Quarterly (n=4)
  • Monthly (n=12)
  • Daily (n=365)

Example 6: Compounding Frequency

P = ₹10000

R = 12% per annum

T = 1 year

Annual: A = ₹11200

Semi-annual: A = ₹11236

Quarterly: A = ₹11255.09

Monthly: A = ₹11268.25

4. Applications

(a) Investment Planning

Using interest calculations for investment decisions.

Investment Planning:

  • Future Value Calculation
  • Present Value Calculation
  • Rate of Return Analysis
  • Investment Comparison

Example 7: Investment Planning

Investment = ₹50000

Rate = 8% per annum (compounded annually)

Time = 5 years

Future Value = 50000(1 + 0.08)^5 = ₹73466.40

Total Interest = ₹23466.40

(b) Loan Amortization

Understanding loan repayment schedules.

Loan Amortization:

  • Monthly Payment Calculation
  • Interest vs Principal Breakdown
  • Amortization Schedule
  • Early Payment Analysis

Example 8: Loan Amortization

Loan = ₹100000

Rate = 10% per annum

Term = 3 years

Monthly Payment = ₹3226.72

Total Interest = ₹16161.92

5. Advanced Concepts

(a) Effective Annual Rate

Understanding the true annual interest rate.

Effective Annual Rate (EAR):

EAR = (1 + R/n)^n - 1

where:

  • R = Nominal annual rate
  • n = Number of compounding periods per year

Example 9: EAR Calculation

Nominal Rate = 12% per annum

Compounding = Monthly

EAR = (1 + 0.12/12)^12 - 1 = 12.68%

(b) Continuous Compounding

Interest compounded continuously.

Continuous Compounding Formula:

A = Pe^(RT)

where:

  • e = Euler's number (≈ 2.71828)
  • P = Principal
  • R = Annual interest rate
  • T = Time in years

Example 10: Continuous Compounding

P = ₹10000

R = 10% per annum

T = 2 years

A = 10000 × e^(0.1×2) = ₹12214.03

(c) Important Theorems

Key Theorems:

  1. Rule of 72: Time to double = 72/Rate
  2. Rule of 114: Time to triple = 114/Rate
  3. Rule of 144: Time to quadruple = 144/Rate
  4. Present Value = Future Value/(1 + R)^T

Example 11: Rule of 72

Rate = 8% per annum

Time to double = 72/8 = 9 years

Rate = 12% per annum

Time to double = 72/12 = 6 years

Ready to Test Your Knowledge?

Put your understanding of Interest to the test with our comprehensive set of 20 practice questions, ranging from basic to advanced difficulty.

Practice Interest Questions