Step by step
The question
Question: $1500 is invested at 2% interest compounded yearly.
How much is in the account after 3 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: $1500.
r is the rate.
2% is 2/100: 0.02.
n is how many times a year the interest is added: once a year, so n = 1.
t is the time in years: 3.
2 Find the rule
Let's see one year first.
The interest is 2% of the money: 0.02 times 1500.
The money plus its interest is 1500 lots of 1, plus 1500 lots of 0.02: 1500 times 1 + 0.02, which is times 1.02.
Next year, the interest is on the new amount.
So again times 1.02.
3 years: times 1.02, 3 times.
That is 1.02 to the power 3.
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.
Careful!
The rate goes in as a decimal: 1 + 0.02 is 1.02.
Not 1 + 2 = 3.
3 Put them in
We put the numbers in.
4 Work it out
Inside the bracket first: 0.02 divided by 1 is 0.02.
The power: 1 times 3 is 3.
1 + 0.02 is 1.02.
So A is 1500 times 1.02 to the power 3.
With a calculator, that power is about 1.061208.
Now times 1500.
That is 1591.812.
Money is rounded to cents: about $1591.81.
5 Check
Let's check by going back.
Dividing undoes the times: 1591.81 divided by 1.061208 is about $1500, the money put in.
And it makes sense.
Simple interest is only on the money put in: 1500 times 0.02 times 3 is $90.
That gives $1590.
Interest on interest gives a bit more.
So the answer makes sense!
6 Pick the choice
Our answer is ≈ $1591.81.
That is choice C.
7 Why not the others
A is $1530.00.
Wrong: that is after 1 year.
B is $1590.00.
Wrong: simple interest: each year also earns on the interest before.
D is $2592.00.
Wrong: used 1.2: 2% is 0.02, so we multiply by 1.02.
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.