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 adds up many regular deposits that have each been growing for a different length of time.

Common stuck point: The procedure for annuities is the easy part; the trap is using the annual rate as $i$ . Asking "Is the same amount paid repeatedly at fixed intervals, rather than once?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the same amount paid repeatedly at fixed intervals, rather than once?

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?

Answer

\$2{,}467.20

First step

This is an ordinary annuity. The future value formula is

FV = P \cdot \frac{(1+r)^n - 1}{r}

Full solution

2
Monthly rate: $r = \frac{0.06}{12} = 0.005$ . Number of payments: $n = 12$ .
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
$FV = 200 \cdot 12.336 = \$2{,}467.20$ .

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

10

years if the account earns

4.8\%

annual interest compounded monthly?

Example 3

medium

You deposit \$150 at the end of each month for 5 years into an account earning 4.8% annual interest compounded monthly. Find the future value.

Example 4

medium

An annuity due pays \$400 at the start of each month for 3 years at 6% annual interest compounded monthly. Find the future value.

Example 5

hard

How many years until \$300 monthly deposits reach \$100,000 at 7.2% annual interest compounded monthly?

Example 6

hard

What annual interest rate makes \$100 monthly deposits reach \$15,000 in 10 years? Solve to 0.1%.

Example 7

challenge

A perpetuity pays \$5,000 per year forever at 4% annual interest. Find its present value.

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?

Example 3

easy

For monthly payments at 6% annual interest, find the periodic rate

i

Example 4

easy

A 20-year monthly annuity has how many payment periods

n

Example 5

easy

Find the future value of an ordinary annuity:

PMT = 100

i = 0.05

n = 2

Example 6

easy

Identify the formula type for a mortgage with known monthly payment, solving for loan amount.

Example 7

easy

For a 4% annual rate with quarterly payments, find

i

Example 8

easy

An ordinary annuity pays at the END of each period. What is paid at the BEGINNING called?

Example 9

easy

Future value:

PMT = 200

i = 0.10

n = 3

Example 10

easy

Present value:

PMT = 100

i = 0.05

n = 1

Example 11

medium

Find the future value of an ordinary annuity:

PMT = 500

i = 0.02

n = 4

Example 12

medium

A loan of

PV = 1000

is repaid over

n=2

periods at

i = 0.10

. Find the payment

PMT

Example 13

medium

Find

FV

of an annuity due:

PMT = 100

i = 0.05

n = 2

Example 14

medium

Monthly payments of

200

for 1 year at 12% annual. Find

i

n

, and set up

FV

Example 15

medium

Find the present value of an annuity:

PMT = 300

i = 0.06

n = 3

Example 16

medium

How much more is an annuity due worth than an ordinary annuity with

FV_{ord} = 1000

and

i = 0.08

Example 17

medium

A retirement fund needs

FV = 10000

n = 5

periods at

i = 0.10

. Find the required

PMT

Example 18

medium

Identify the error: a student computes a mortgage payment using

FV = PMT\cdot\frac{(1+i)^n-1}{i}

. What should they use?

Example 19

medium

Find the present value of an annuity:

PMT = 100

i = 0.10

n = 2

Example 20

challenge

Find the future value of an ordinary annuity with

PMT=1000

, annual rate 8% compounded quarterly, for 2 years.

Example 21

challenge

A loan is amortized with

PMT = 400

i = 0.01

monthly,

n = 24

. Find the original loan amount.

Example 22

challenge

An annuity due pays

PMT = 250

for

n = 3

i = 0.04

. Find its present value.

Example 23

easy

Find the future value of an ordinary annuity with

PMT = \$300

i = 0.02

n = 2

Example 24

easy

Find the present value of an ordinary annuity with

PMT = \$100

i = 0.05

n = 2

Example 25

medium

What present value supports monthly withdrawals of \$800 for 25 years if the account earns 6% annual interest compounded monthly?

Example 26

medium

How long (in years) must \$200 monthly deposits last to accumulate \$30,000 at 3.6% annual interest compounded monthly?

Example 27

medium

Find the monthly payment on a \$250,000 mortgage at 5.4% annual interest compounded monthly over 30 years.

Example 28

medium

If monthly deposits of $250 grow to $50,000 at

i = 0.005

per month, how many months are required (round up)?

Example 29

medium

Find the present value of an ordinary annuity paying \$1,200 quarterly for 8 years at 5.2% annual interest compounded quarterly.

Example 30

medium

You take a \$15,000 car loan at 4.2% annual interest compounded monthly for 5 years. Find the monthly payment.

Example 31

hard

A 30-year mortgage of \$320,000 at 6% annual interest compounded monthly. How much total interest is paid?

Example 32

hard

An annuity due makes 10 annual payments of \$2,000 at 8% interest. Find its present value.

Example 33

hard

If \$10,000 is invested today and \$200/month is added at the end of each month for 20 years at 6% annual interest compounded monthly, find the future value.

Example 34

hard

A retirement fund will pay \$3,000 per month for 25 years starting in 20 years. If the account earns 5.4% annual interest compounded monthly, what lump sum must be invested today?

Example 35

hard

Compare an ordinary annuity and an annuity due, both with \$250 monthly payments over 8 years at 3.6% annual interest compounded monthly. How much more does the annuity due accumulate?

Example 36

challenge

A 20-year mortgage with monthly payments of \$1,150 was originally taken at 4.8% annual interest compounded monthly. Find the original loan amount.

← Back to Annuities Formula Explained Practice Mode

Related Concepts

Compound Interest Present and Future Value

Background Knowledge

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

compound interest