Annuities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Annuities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A series of equal payments made at regular intervals over a fixed period of time. The future value and present value formulas calculate the total worth of these payment streams.

Imagine depositing \$100 every month into a savings account. Each deposit earns interest for a different amount of time—the first deposit earns interest for the full term, the last deposit barely earns any. An annuity formula adds up all these differently-growing deposits in one clean expression, instead of computing compound interest on each payment separately.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An annuity is a geometric series of compound interest calculations. Each payment grows at a different rate depending on when it was made. The formulas are closed-form sums of these geometric series.

Common stuck point: The interest rate i in the formula is the PERIODIC rate, not the annual rate. For monthly payments at 6% annual, use i = 0.06/12 = 0.005, and n is the total number of months, not years.

Sense of Study hint: Convert everything to the same time period first: divide the annual rate by 12 for monthly, and multiply years by 12 for total payments.

Worked Examples

Example 1

easy
You deposit \200 at the end of each month into an account earning 6\% annual interest compounded monthly. How much will you have after 1$ year?

Solution

  1. 1
    This is an ordinary annuity. The future value formula is FV = P \cdot \frac{(1+r)^n - 1}{r}.
  2. 2
    Monthly rate: r = \frac{0.06}{12} = 0.005. Number of payments: n = 12.
  3. 3
    FV = 200 \cdot \frac{(1.005)^{12} - 1}{0.005} = 200 \cdot \frac{1.06168 - 1}{0.005} = 200 \cdot \frac{0.06168}{0.005}.
  4. 4
    FV = 200 \cdot 12.336 = \2{,}467.20$.

Answer

\$2{,}467.20
An ordinary annuity involves equal payments made at the end of each period. The future value formula accounts for the fact that earlier payments earn more interest than later ones. The total deposits are \2{,}400, so \67.20 is earned in interest.

Example 2

medium
What monthly payment is needed to accumulate \50{,}000 in 10 years if the account earns 4.8\%$ annual interest compounded monthly?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
You want to receive \1{,}000 per month for 20 years from a retirement account earning 5\%$ annual interest compounded monthly. How much must be in the account at the start of retirement?

Example 2

hard
Compare an ordinary annuity and an annuity due, both with \500 monthly payments at 6\% annual interest compounded monthly for 5$ years. How much more does the annuity due accumulate?
← Back to Annuities Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

compound interest