Worked examples
The question
Question: $1000 is invested at 5% interest compounded yearly.
How much is in the account after 3 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
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: 3.
t = 3 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 rate goes in as a decimal: 0.05, not 5.
Careful: not 1000 × 1.053 ✗
2 Put them in
We put the numbers in.
A = 1000(1 + 0.051)1 × 3
3 Work it out
Inside the bracket first.
The power: 1 times 3 is 3.
1 × 3 = 3
1 + 0.05 is 1.05.
So A is 1000 times 1.05 to the power 3.
A = 1000 × 1.053
With a calculator, that power is about 1.15763.
1.053 ≈ 1.15763
Now times 1000.
A ≈ 1000 × 1.15763
1157.625...
Money is rounded to cents: about $1157.63.
A ≈ $1157.63
4 Check
Let's check by going back: 1157.63 divided by 1.15763 is about $1000, the money put in.
$1157.63 ÷ 1.15763 ≈ $1000
And it makes sense.
Simple interest would give $1150.
Interest on interest gives a bit more.
So the answer is right!
simple interest: $1000 + $150 = $1150 < $1157.63
Let's look back.
First: read the numbers.
Then: put them in.
Then: work it out.
Last: check.
▶ Watch it solved
The question
Question: $1000 is invested at 6% interest compounded monthly.
How much is in the account after 5 years?
1 Read the numbers
P is the money put in: $1000.
P = $1000: the money put in
r is the rate.
6% is 6/100: 0.06.
r = 6% = 0.06
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: 5.
t = 5 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 6% is for a whole year.
Each time we add only r/n: 0.06 divided by 12.
Careful: not 1000 × 1.0660 ✗
2 Put them in
We put the numbers in.
A = 1000(1 + 0.0612)12 × 5
3 Work it out
Inside the bracket first: 0.06 divided by 12 is 0.005.
The power: 12 times 5 is 60.
0.06 ÷ 12 = 0.005
12 × 5 = 60
1 + 0.005 is 1.005.
So A is 1000 times 1.005 to the power 60.
A = 1000 × 1.00560
With a calculator, that power is about 1.34885.
1.00560 ≈ 1.34885
Now times 1000.
A ≈ 1000 × 1.34885
1348.8502...
Money is rounded to cents: about $1348.85.
A ≈ $1348.85
4 Check
Let's check by going back: 1348.85 divided by 1.34885 is about $1000, the money put in.
$1348.85 ÷ 1.34885 ≈ $1000
And it makes sense.
Simple interest would give $1300.
Interest on interest gives a bit more.
So the answer is right!
simple interest: $1000 + $300 = $1300 < $1348.85
Let's look back.
First: read the numbers.
Then: put them in.
Then: work it out.
Last: check.
▶ Watch it solved
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.
▶ Watch it solved
