$10000 is invested at 5% interest compounded monthly. How much is in… Step by Step

Step by step

The question

Question: $10000 is invested at 5% interest compounded monthly.

How much is in the account after 18 years?

1 Read the numbers

P is the money put in: $10000.

P = $10000: the money put in

r is the rate.

5% is 5/100: 0.05.

r = 5% = 0.05

n is how many times a year the interest is added: monthly, so n = 12.

n = 12: monthly: 12 times a year

t is the time in years: 18.

t = 18 years

The amount after t years is A = P times (1 + r/n) to the power nt.

Each time, the money is multiplied by 1 + r over n.

A = P(1 + rn)^nt

Careful!

The 5% is for a whole year.

Each time we add only r/n: 0.05 divided by 12.

Careful: not 10000 × 1.05²¹⁶ ✗

2 Put them in

We put the numbers in.

A = 10000(1 + 0.0512)^{12 × 18}

3 Work it out

Inside the bracket first: 0.05 divided by 12 is about 0.00416667.

The power: 12 times 18 is 216.

0.05 ÷ 12 ≈ 0.00416667

12 × 18 = 216

So A is 10000 times the bracket to the power 216.

A = 10000 × (1 + 0.0512)²¹⁶

With a calculator, that power is about 2.45501.

(1 + 0.0512)²¹⁶ ≈ 2.45501

Now times 10000.

A ≈ 10000 × 2.45501

24550.0842...

Money is rounded to cents: about $24550.08.

A ≈ $24550.08

4 Check

Let's check by going back: 24550.08 divided by 2.45501 is about $10000, the money put in.

$24550.08 ÷ 2.45501 ≈ $10000

And it makes sense.

Simple interest would give $19000.

Interest on interest gives a bit more.

So the answer is right!

simple interest: $10000 + $9000 = $19000 < $24550.08

Let's look back.

First: read the numbers.

Then: put them in.

Then: work it out.

Last: check.

Answer