Worked examples
The question
Question: A car loan of $15000 at 6% for 5 years is paid monthly.
Find the monthly payment.
1 Read the numbers
P is the money borrowed: $15000.
r is the yearly rate: 6% is 0.06.
P = $15,000 loan, 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 payments: 12 times 5 is 60.
N = n × t = 12 × 5 = 60
3 Use the formula
The payment each month: PMT = P times i, over 1 minus (1 + i) to the power negative N.
PMT = P × i1 − (1 + i)−N
PMT = the payment each month
We put the numbers in: 1 plus 0.005 is 1.005.
PMT = 15000 × 0.0051 − 1.005−60
4 Work it out
The top: 75.
With a calculator, 1.005 to the power negative 60 is about 0.741372.
1 minus that is 0.258628.
15000 × 0.005 = 75
1.005−60 ≈ 0.741372
1 − 0.741372 = 0.258628
Now the top over the bottom.
PMT ≈ 750.258628
75 divided by 0.258628 is 289.9920 and a bit.
PMT ≈ 289.9920…
Lenders round up to the next cent: $290 a month.
PMT = $290.00
Careful!
That pays back only the loan.
The interest must be paid too.
Careful: not $15,000 ÷ 60 ✗
5 Check
Let's check with the first month.
The interest that month is 15000 times 0.005: $75.
month 1 interest: $15,000 × 0.005 ≈ $75.00
The rest of the payment, $215, pays the loan down.
So the loan gets smaller every month.
It makes sense!
$290.00 − $75.00 = $215.00 pays the loan down
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Last: check.
Answer Careful: not $15,000 ÷ 60 ✗
▶ Watch it solved
The question
Question: A car loan of $28500 at 3.99% for 5 years is paid monthly.
Find the monthly payment.
1 Read the numbers
P is the money borrowed: $28500.
r is the yearly rate: 3.99% is 0.0399.
P = $28,500 loan, r = 3.99% = 0.0399
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.
0.0399 divided by 12 is 0.003325.
i = rn = 0.039912 = 0.003325
i = the rate each month
And N is how many payments: 12 times 5 is 60.
N = n × t = 12 × 5 = 60
3 Use the formula
The payment each month: PMT = P times i, over 1 minus (1 + i) to the power negative N.
PMT = P × i1 − (1 + i)−N
PMT = the payment each month
We put the numbers in: 1 plus 0.003325 is 1.003325.
PMT = 28500 × 0.0033251 − 1.003325−60
4 Work it out
The top: 94.7625.
With a calculator, 1.003325 to the power negative 60 is about 0.819411.
1 minus that is 0.180589.
28500 × 0.003325 = 94.7625
1.003325−60 ≈ 0.819411
1 − 0.819411 = 0.180589
Now the top over the bottom.
PMT ≈ 94.76250.180589
94.7625 divided by 0.180589 is 524.7423 and a bit.
PMT ≈ 524.7423…
Lenders round up to the next cent: 524 dollars and 75 cents a month.
PMT = $524.75
Careful!
That pays back only the loan.
The interest must be paid too.
Careful: not $28,500 ÷ 60 ✗
5 Check
Let's check with the first month.
The interest that month is 28500 times 0.003325: 94 dollars and 76 cents.
month 1 interest: $28,500 × 0.003325 ≈ $94.76
The rest of the payment, 429 dollars and 99 cents, pays the loan down.
So the loan gets smaller every month.
It makes sense!
$524.75 − $94.76 = $429.99 pays the loan down
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Last: check.
Answer Careful: not $28,500 ÷ 60 ✗
▶ Watch it solved
The question
Question: A car loan of $15000 at 6% for 5 years is paid monthly.
Find the monthly payment and the total interest.
1 Read the numbers
P is the money borrowed: $15000.
r is the yearly rate: 6% is 0.06.
P = $15,000 loan, 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 payments: 12 times 5 is 60.
N = n × t = 12 × 5 = 60
3 Use the formula
The payment each month: PMT = P times i, over 1 minus (1 + i) to the power negative N.
PMT = P × i1 − (1 + i)−N
PMT = the payment each month
We put the numbers in: 1 plus 0.005 is 1.005.
PMT = 15000 × 0.0051 − 1.005−60
4 Work it out
The top: 75.
With a calculator, 1.005 to the power negative 60 is about 0.741372.
1 minus that is 0.258628.
15000 × 0.005 = 75
1.005−60 ≈ 0.741372
1 − 0.741372 = 0.258628
Now the top over the bottom.
PMT ≈ 750.258628
75 divided by 0.258628 is 289.9920 and a bit.
PMT ≈ 289.9920…
Lenders round up to the next cent: $290 a month.
PMT = $290.00
Careful!
That pays back only the loan.
The interest must be paid too.
Careful: not $15,000 ÷ 60 ✗
5 Total interest
60 payments of $290: $17400 in all.
total paid = $290.00 × 60 = $17,400.00
The interest is what is paid on top of the loan: $2400.
interest = $17,400.00 − $15,000 = $2,400.00
6 Check
Let's check with the first month.
The interest that month is 15000 times 0.005: $75.
month 1 interest: $15,000 × 0.005 ≈ $75.00
The rest of the payment, $215, pays the loan down.
So the loan gets smaller every month.
It makes sense!
$290.00 − $75.00 = $215.00 pays the loan down
Let's look back.
First: read the numbers.
Then: per payment.
Then: use the formula.
Then: work it out.
Then: total interest.
Last: check.
Answer Careful: not $15,000 ÷ 60 ✗
▶ Watch it solved
