Compound Interest

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Use compound interest formulas for different compounding periods.
  • Interpret principal, rate, time, and number of compounding periods.
  • Compare simple and compound interest conceptually.

Key Ideas

The general compound interest formula is:

\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

where:

  • \(A\) = final amount
  • \(P\) = principal (initial amount)
  • \(r\) = annual interest rate (decimal)
  • \(n\) = number of compounding periods per year
  • \(t\) = time in years

Common compounding periods:

  • annually: \(n = 1\)
  • monthly: \(n = 12\)
  • quarterly: \(n = 4\)
  • daily: \(n = 365\) (or 360, depending on the context)

For annual compounding:

\[ A = P(1+r)^t \]

The exponent \(nt\) captures how many times interest is applied in total.

For comparison, simple interest uses

\[ A = P(1 + rt), \]

which grows linearly rather than exponentially.

Common Problem Types

1. Annual compounding

Use the simpler model \(A = P(1+r)^t\).

2. Multiple compounding periods

Use
\[A = P\left(1 + \frac{r}{n}\right)^{nt}.\]

3. Write an expression vs. compute a value

Be clear when the problem wants an expression (no rounding) versus a numerical approximation.

4. Compare Simple vs Compound Interest

Simple interest uses the formula

\[ A = P(1 + rt) \]

while compound interest (annual compounding) uses

\[ A = P(1+r)^t. \]

The key difference is that simple interest is calculated only on the original principal, while compound interest earns interest on both the principal and previously earned interest.

Example:

Invest $1000 at 5% for 3 years.

Simple interest:

\[ A = 1000(1 + 0.05 \cdot 3) = 1150 \]

Compound interest:

\[ A = 1000(1.05)^3 \approx 1157.63 \]

Because compound interest earns interest on prior interest, it grows faster over time.

Strategies

  • Convert percent rates to decimals before substituting into formulas.
  • Identify \(n\) correctly—this determines how many times interest is added.
  • Evaluate the factor \(\left(1 + \frac{r}{n}\right)\) before applying the exponent.
  • When asked for an expression, leave the answer in exact form (no rounding).
  • For conceptual comparisons, think:
    • simple interest → adds
    • compound interest → multiplies

Worked Examples

Example 1 — Annual Compounding

You invest $2,000 at 5% annual interest, compounded once per year. How much will you have after 4 years?

Solution:

  • \(P = 2000\)
  • \(r = 0.05\)
  • \(n = 1\)
  • \(t = 4\)

Use:

\[ A = P(1 + r)^t \]

Compute:

\[ A = 2000(1.05)^4 \]

Approximate (if allowed):

  • \((1.05)^4 \approx 1.21550625\)

So:

\[ A \approx 2000 \cdot 1.21550625 = 2431.01 \]

Amount after 4 years: $2,431.01

Example 2 — Monthly Compounding

Deposit $1,000 at 6% annual interest, compounded monthly. How much after 3 years?

Solution:

  • \(P = 1000\)
  • \(r = 0.06\)
  • \(n = 12\)
  • \(t = 3\)

Use:

\[ A = 1000\left(1 + \frac{0.06}{12}\right)^{36} \]

Inside:

  • \(\frac{0.06}{12} = 0.005\)
  • Exponent: \(12 \cdot 3 = 36\)

So the expression becomes:

\[ A = 1000(1.005)^{36} \]

(Leave exact unless approximation is required.)

WarningCommon Mistakes
  • Forgetting to convert interest rate to a decimal.
  • Forgetting to divide \(r\) by \(n\) for monthly, quarterly, or daily compounding.
  • Using \(t\) instead of \(nt\) in the exponent.
  • Rounding intermediate steps too early.

Practice Problems

  1. $500 invested at 4% annual interest, compounded yearly for 5 years.
    Write the expression for the amount \(A\).

  2. $1,200 invested at 3% annual interest, compounded monthly for 10 years.
    Write a formula for \(A\).

  3. $800 invested at 8% annual interest, compounded quarterly for 2 years.
    Write the expression for \(A\).

  4. Compare: which grows faster over the same time — 5% simple interest or 5% compound interest (annual) on the same principal?

  5. You invest $5,000 at 2% interest compounded annually for 3 years.
    Write the exact expression and compute a rounded value.

1.
\(P = 500\), \(r = 0.04\), \(n = 1\), \(t = 5\)

\[ A = 500(1 + 0.04)^5 = 500(1.04)^5 \]

2.
\(P = 1200\), \(r = 0.03\), \(n = 12\), \(t = 10\)

\[ A = 1200\left(1 + \frac{0.03}{12}\right)^{120} \]

Inside:

  • \(\frac{0.03}{12} = 0.0025\)
  • Exponent: \(120\)

So:

\[ A = 1200(1.0025)^{120} \]

3.
\(P = 800\), \(r = 0.08\), \(n = 4\), \(t = 2\)

\[ A = 800\left(1 + \frac{0.08}{4}\right)^{8} \]

Inside:

  • \(\frac{0.08}{4} = 0.02\)

So:

\[ A = 800(1.02)^8 \]

4.
Conceptually:

  • Simple interest adds a constant amount each year → linear.
  • Compound interest multiplies and earns interest on interest → exponential.

Thus, compound interest grows faster for the same principal and rate.

5.
\(P = 5000\), \(r = 0.02\), \(n = 1\), \(t = 3\)

Exact expression:

\[ A = 5000(1.02)^3 \]

Compute:

  • \((1.02)^2 = 1.0404\)
  • \((1.02)^3 = 1.061208\)

So:

\[ A \approx 5000 \cdot 1.061208 = 5306.04 \]

Amount after 3 years: $5,306.04

Summary

  • Compound interest grows by multiplying by
    \[1 + \frac{r}{n}\]
    each compounding period.
  • Use the exponent \(nt\) to reflect total number of compounding cycles.
  • Annual compounding simplifies to \(A = P(1+r)^t\).
  • Compound interest grows faster than simple interest over time.
  • Always keep units (years, periods per year) consistent.
  • Convert percentages to decimals early.
  • Identify \(n\) first—monthly vs quarterly changes the formula.
  • Leave expressions exact unless asked to approximate.
  • Compound interest = exponential growth.