$120 is deposited at the end of each month into an account paying 9%… Step by Step

Step by step

The question

Question: $120 is deposited at the end of each month into an account paying 9% compounded monthly.

How much is in the account after 30 years?

1 Read the numbers

PMT is the money put in each time: $120.

r is the yearly rate: 9% is 0.09.

PMT = $120 each time, r = 9% = 0.09

n is how many times a year: 12.

t is the time: 30 years.

n = 12 a year, t = 30 years

2 Per payment

Interest is added each month.

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

0.09 divided by 12 is 0.0075.

i = rn = 0.0912 = 0.0075

i = the rate each month

And N is how many deposits: 12 times 30 is 360.

N = n × t = 12 × 30 = 360

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 0.0075 is 1.0075.

FV = 120 × 1.0075³⁶⁰ − 10.0075

4 Work it out

With a calculator, 1.0075 to the power 360 is about 14.7306.

Minus 1, divided by 0.0075: about 1830.74.

1.0075³⁶⁰ ≈ 14.7306

(14.7306 − 1) ÷ 0.0075 ≈ 1830.74

So FV is 120 times 1830.74.

FV ≈ 120 × 1830.74

Money is rounded to cents: about 219689 dollars and 22 cents.

FV ≈ $219,689.22

Careful!

That is only the money put in.

It leaves out all the interest.

Careful: not $120 × 360 ✗

5 Check

Let's check.

The deposits alone are 120 times 360: $43200.

The account has more, because of the interest.

deposits: $120 × 360 = $43,200 < $219,689.22

The interest earned is 176489 dollars and 22 cents.

So it makes sense!

interest: $219,689.22 − $43,200 = $176,489.22

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Last: check.

Answer