A loan of $10,000 at 9% for 3 years is paid monthly. Find the… Step by Step

Step by step

The question

Question: A loan of $10000 at 9% for 3 years is paid monthly.

Find the monthly payment and the total interest.

1 Read the numbers

P is the money borrowed: $10000.

r is the yearly rate: 9% is 0.09.

P = $10,000 loan, r = 9% = 0.09

n is how many times a year: 12.

t is the time: 3 years.

n = 12 a year, t = 3 years

2 Per payment

Interest is added each month.

The rate for one month is i: r over n.

0.09 divided by 12 is 0.0075.

i = rn = 0.0912 = 0.0075

i = the rate each month

And N is how many payments: 12 times 3 is 36.

N = n × t = 12 × 3 = 36

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 0.0075 is 1.0075.

PMT = 10000 × 0.00751 − 1.0075⁻³⁶

4 Work it out

The top: 75.

With a calculator, 1.0075 to the power negative 36 is about 0.764149.

1 minus that is 0.235851.

10000 × 0.0075 = 75

1.0075⁻³⁶ ≈ 0.764149

1 − 0.764149 = 0.235851

Now the top over the bottom.

PMT ≈ 750.235851

75 divided by 0.235851 is 317.9973 and a bit.

PMT ≈ 317.9973…

Lenders round up to the next cent: $318 a month.

PMT = $318.00

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $10,000 ÷ 36 ✗

5 Total interest

36 payments of $318: $11448 in all.

total paid = $318.00 × 36 = $11,448.00

The interest is what is paid on top of the loan: $1448.

interest = $11,448.00 − $10,000 = $1,448.00

6 Check

Let's check with the first month.

The interest that month is 10000 times 0.0075: $75.

month 1 interest: $10,000 × 0.0075 ≈ $75.00

The rest of the payment, $243, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$318.00 − $75.00 = $243.00 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