$100 is deposited at the end of each month into an account paying 3%… Step by Step

Step by step

The question

Question: $100 is deposited at the end of each month into an account paying 3% compounded monthly.

How much is in the account after 2 years?

1 Read the numbers

PMT is the money put in each time: $100.

r is the yearly rate: 3% is 0.03.

PMT = $100 each time, r = 3% = 0.03

n is how many times a year: 12.

t is the time: 2 years.

n = 12 a year, t = 2 years

2 Per payment

Interest is added each month.

The rate for one month is i: r over n.

0.03 divided by 12 is 0.0025.

i = rn = 0.0312 = 0.0025

i = the rate each month

And N is how many deposits: 12 times 2 is 24.

N = n × t = 12 × 2 = 24

3 Use the formula

The future value FV is what the account holds at the end: PMT times, (1 + i) to the power N, minus 1, over i.

FV = PMT × (1 + i)^N − 1i

FV = future value: the money at the end

We put the numbers in: 1 plus 0.0025 is 1.0025.

FV = 100 × 1.0025²⁴ − 10.0025

4 Work it out

With a calculator, 1.0025 to the power 24 is about 1.06176.

Minus 1, divided by 0.0025: about 24.7028.

1.0025²⁴ ≈ 1.06176

(1.06176 − 1) ÷ 0.0025 ≈ 24.7028

So FV is 100 times 24.7028.

FV ≈ 100 × 24.7028

Money is rounded to cents: about 2470 dollars and 28 cents.

FV ≈ $2,470.28

Careful!

That is only the money put in.

It leaves out all the interest.

Careful: not $100 × 24 ✗

5 Check

Let's check.

The deposits alone are 100 times 24: $2400.

The account has more, because of the interest.

deposits: $100 × 24 = $2,400 < $2,470.28

The interest earned is 70 dollars and 28 cents.

So it makes sense!

interest: $2,470.28 − $2,400 = $70.28

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Last: check.

Answer