$75 is deposited at the end of each month into an account paying… Step by Step

Step by step

The question

Question: $75 is deposited at the end of each month into an account paying 4.8% compounded monthly.

How much is in the account after 20 years?

1 Read the numbers

PMT is the money put in each time: $75.

r is the yearly rate: 4.8% is 0.048.

PMT = $75 each time, r = 4.8% = 0.048

n is how many times a year: 12.

t is the time: 20 years.

n = 12 a year, t = 20 years

2 Per payment

Interest is added each month.

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

0.048 divided by 12 is 0.004.

i = rn = 0.04812 = 0.004

i = the rate each month

And N is how many deposits: 12 times 20 is 240.

N = n × t = 12 × 20 = 240

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 0.004 is 1.004.

FV = 75 × 1.004²⁴⁰ − 10.004

4 Work it out

With a calculator, 1.004 to the power 240 is about 2.6067.

Minus 1, divided by 0.004: about 401.675.

1.004²⁴⁰ ≈ 2.6067

(2.6067 − 1) ÷ 0.004 ≈ 401.675

So FV is 75 times 401.675.

FV ≈ 75 × 401.675

Money is rounded to cents: about 30125 dollars and 63 cents.

FV ≈ $30,125.63

Careful!

That is only the money put in.

It leaves out all the interest.

Careful: not $75 × 240 ✗

5 Check

Let's check.

The deposits alone are 75 times 240: $18000.

The account has more, because of the interest.

deposits: $75 × 240 = $18,000 < $30,125.63

The interest earned is 12125 dollars and 63 cents.

So it makes sense!

interest: $30,125.63 − $18,000 = $12,125.63

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Last: check.

Answer