$1000 is invested at 5% interest compounded continuously. How much… Step by Step

Step by step

The question

Question: $1000 is invested at 5% interest compounded continuously.

How much is in the account after 4 years?

1 Read the numbers

P is the money put in: $1000.

P = $1000: the money put in

r is the rate.

5% is 5/100: 0.05.

r = 5% = 0.05

t is the time in years: 4.

t = 4 years

With continuous interest, the amount is A = P times e to the power rt.

e is the special number from the ln lesson, about 2.718.

A = Pe^rt

e ≈ 2.718

Careful!

The rate goes in as a decimal: 0.05, not 5.

Careful: not r = 5 ✗

2 Put them in

We put the numbers in.

A = 1000 × e^{0.05 × 4}

3 Work it out

The power first: 0.05 times 4 is 0.2.

0.05 × 4 = 0.2

So A is 1000 times e to the power 0.2.

A = 1000 × e^0.2

With a calculator, e to the power 0.2 is about 1.2214.

e^0.2 ≈ 1.2214

Now times 1000.

A ≈ 1000 × 1.2214

1221.4028...

Money is rounded to cents: about $1221.40.

A ≈ $1221.40

4 Check

Let's check by going back: 1221.40 divided by 1.2214 is about $1000, the money put in.

$1221.40 ÷ 1.2214 ≈ $1000

And it makes sense.

Simple interest would give $1200.

Interest on interest gives a bit more.

So the answer is right!

simple interest: $1000 + $200 = $1200 < $1221.40

Let's look back.

First: read the numbers.

Then: put them in.

Then: work it out.

Last: check.

Answer