$2,000 is invested at 5% interest compounded yearly. How much is in… Step by Step

Step by step

The question

Question: $2000 is invested at 5% interest compounded yearly.

How much is in the account after 2 years?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Read the numbers

P is the money put in: $2000.

P = $2000: 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: once a year, so n = 1.

n = 1: once a year

t is the time in years: 2.

t = 2 years

2 Find the rule

Let's see one year first.

The interest is 5% of the money: 0.05 times 2000.

The money plus its interest is 2000 lots of 1, plus 2000 lots of 0.05: 2000 times 1 + 0.05, which is times 1.05.

1 year: 2000 + 0.05 × 2000 = 2000 × (1 + 0.05) = 2000 × 1.05

Next year, the interest is on the new amount.

So again times 1.05.

every year: × 1.05 again

2 years: times 1.05, 2 times.

That is 1.05 to the power 2.

2 years: 2000 × 1.05 × 1.05 = 2000 × 1.05²

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

n times t counts how many times the interest is added.

Here n = 1: once a year, so r/n is just r.

A = P(1 + rn)^nt

Careful!

The rate goes in as a decimal: 1 + 0.05 is 1.05.

Not 1 + 5 = 6.

Careful: not 2000 × 6² ✗

3 Put them in

We put the numbers in.

A = 2000(1 + 0.051)^{1 × 2}

4 Work it out

Inside the bracket first: 0.05 divided by 1 is 0.05.

The power: 1 times 2 is 2.

1 × 2 = 2

1 + 0.05 is 1.05.

So A is 2000 times 1.05 to the power 2.

A = 2000 × 1.05²

With a calculator, that power is about 1.1025.

1.05² ≈ 1.1025

Now times 2000.

A ≈ 2000 × 1.1025

That is 2205.

Money is rounded to cents: about $2205.00.

A ≈ $2205.00

5 Check

Let's check by going back.

Dividing undoes the times: 2205.00 divided by 1.1025 is about $2000, the money put in.

$2205.00 ÷ 1.1025 ≈ $2000

And it makes sense.

Simple interest is only on the money put in: 2000 times 0.05 times 2 is $200.

simple interest: $2000 × 0.05 × 2 = $200

That gives $2200.

Interest on interest gives a bit more.

So the answer makes sense!

$2000 + $200 = $2200 < $2205.00

6 Pick the choice

Our answer is ≈ $2205.00.

That is choice C.

C) $2205

7 Why not the others

A is $2100.

Wrong: that is after 1 year.

A) $2100 ✗ that is after 1 year

B is $2200.

Wrong: simple interest: the second year also earns on the first year's interest.

B) $2200 ✗ simple interest: the second year also earns on the first year's interest

D is $4500.

Wrong: used 1.5: 5% is 0.05, so we multiply by 1.05.

D) $4500 ✗ used 1.5: 5% is 0.05, so we multiply by 1.05

Let's look back.

First: read the numbers.

Then: find the rule.

Then: put them in.

Then: work it out.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer