Worked examples
The question
Question: $200 is deposited at the end of each month into an account paying 6% compounded monthly.
How much is in the account after 5 years?
1 Read the numbers
PMT is the money put in each time: $200.
r is the yearly rate: 6% is 0.06.
PMT = $200 each time, r = 6% = 0.06
n is how many times a year: 12.
t is the time: 5 years.
n = 12 a year, t = 5 years
2 Per payment
Interest is added each month.
The rate for one month is i: r over n.
0.06 divided by 12 is 0.005.
i = rn = 0.0612 = 0.005
i = the rate each month
And N is how many deposits: 12 times 5 is 60.
N = n × t = 12 × 5 = 60
3 Use the formula
The future value FV is what the account holds at the end: PMT times, (1 + i) to the power N, minus 1, over i.
FV = PMT × (1 + i)N − 1i
FV = future value: the money at the end
We put the numbers in: 1 plus 0.005 is 1.005.
FV = 200 × 1.00560 − 10.005
4 Work it out
With a calculator, 1.005 to the power 60 is about 1.34885.
Minus 1, divided by 0.005: about 69.77.
1.00560 ≈ 1.34885
(1.34885 − 1) ÷ 0.005 ≈ 69.77
So FV is 200 times 69.77.
FV ≈ 200 × 69.77
Money is rounded to cents: about 13954 dollars and 1 cent.
FV ≈ $13,954.01
Careful!
That is only the money put in.
It leaves out all the interest.
Careful: not $200 × 60 ✗
5 Check
Let's check.
The deposits alone are 200 times 60: $12000.
The account has more, because of the interest.
deposits: $200 × 60 = $12,000 < $13,954.01
The interest earned is 1954 dollars and 1 cent.
So it makes sense!
interest: $13,954.01 − $12,000 = $1,954.01
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Last: check.
Answer Careful: not $200 × 60 ✗
▶ Watch it solved
The question
Question: $250 is deposited at the end of each month into an account paying 3.75% compounded monthly.
How much is in the account after 8 years?
1 Read the numbers
PMT is the money put in each time: $250.
r is the yearly rate: 3.75% is 0.0375.
PMT = $250 each time, r = 3.75% = 0.0375
n is how many times a year: 12.
t is the time: 8 years.
n = 12 a year, t = 8 years
2 Per payment
Interest is added each month.
The rate for one month is i: r over n.
0.0375 divided by 12 is 0.003125.
i = rn = 0.037512 = 0.003125
i = the rate each month
And N is how many deposits: 12 times 8 is 96.
N = n × t = 12 × 8 = 96
3 Use the formula
The future value FV is what the account holds at the end: PMT times, (1 + i) to the power N, minus 1, over i.
FV = PMT × (1 + i)N − 1i
FV = future value: the money at the end
We put the numbers in: 1 plus 0.003125 is 1.003125.
FV = 250 × 1.00312596 − 10.003125
4 Work it out
With a calculator, 1.003125 to the power 96 is about 1.34923.
Minus 1, divided by 0.003125: about 111.753.
1.00312596 ≈ 1.34923
(1.34923 − 1) ÷ 0.003125 ≈ 111.753
So FV is 250 times 111.753.
FV ≈ 250 × 111.753
Money is rounded to cents: about 27938 dollars and 20 cents.
FV ≈ $27,938.20
Careful!
That is only the money put in.
It leaves out all the interest.
Careful: not $250 × 96 ✗
5 Check
Let's check.
The deposits alone are 250 times 96: $24000.
The account has more, because of the interest.
deposits: $250 × 96 = $24,000 < $27,938.20
The interest earned is 3938 dollars and 20 cents.
So it makes sense!
interest: $27,938.20 − $24,000 = $3,938.20
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Last: check.
Answer Careful: not $250 × 96 ✗
▶ Watch it solved
The question
Question: How much must be deposited at the end of each month at 4% compounded monthly to have $10000 after 5 years?
1 Read the numbers
FV is the money we want at the end: $10000.
We need PMT, each deposit.
r is the yearly rate: 4% is 0.04.
FV = $10,000 goal, r = 4% = 0.04
n is how many times a year: 12.
t is the time: 5 years.
n = 12 a year, t = 5 years
2 Per payment
Interest is added each month.
The rate for one month is i: r over n.
0.04 divided by 12 is 0.003333.
i = rn = 0.0412 = 0.003333
i = the rate each month
And N is how many deposits: 12 times 5 is 60.
N = n × t = 12 × 5 = 60
3 Use the formula
The savings formula: FV = PMT times, (1 + i) to the power N, minus 1, over i.
FV = PMT × (1 + i)N − 1i
FV = future value: the money at the end
We put the numbers in.
PMT stays a letter: we want it.
10000 = PMT × 1.0033360 − 10.003333
4 Work it out
With a calculator: 1.00333 to the power 60 is about 1.221.
Minus 1, divided by 0.003333: about 66.299.
1.0033360 ≈ 1.221
(1.221 − 1) ÷ 0.003333 ≈ 66.299
So 10000 is PMT times 66.299.
10000 ≈ PMT × 66.299
5 Get PMT alone
To remove times 66.299, we divide both sides by 66.299.
PMT ≈ 1000066.299
10000 divided by 66.299 is 150.8319 and a bit.
PMT ≈ 150.8319…
We round up to the next cent, so the goal is reached: 150 dollars and 84 cents.
PMT = $150.84
Careful!
Rounding down would leave the account a little short.
Careful: round UP, or the goal is not reached
6 Check
Let's check: 150 dollars and 84 cents times 66.299 is about 10000 dollars and 54 cents.
That reaches the goal.
So the answer is right!
$150.84 × 66.299 ≈ $10,000.54 ≥ $10,000
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Then: get PMT alone.
Last: check.
Answer Careful: round UP, or the goal is not reached
▶ Watch it solved
