A car loan of $20,000 at 3% for 5 years is paid monthly. Find the… Step by Step

Step by step

The question

Question: A car loan of $20000 at 3% for 5 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $20000.

r is the yearly rate: 3% is 0.03.

P = $20,000 loan, r = 3% = 0.03

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.

0.03 divided by 12 is 0.0025.

i = rn = 0.0312 = 0.0025

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 0.0025 is 1.0025.

PMT = 20000 × 0.00251 − 1.0025⁻⁶⁰

4 Work it out

The top: 50.

With a calculator, 1.0025 to the power negative 60 is about 0.860869.

1 minus that is 0.139131.

20000 × 0.0025 = 50

1.0025⁻⁶⁰ ≈ 0.860869

1 − 0.860869 = 0.139131

Now the top over the bottom.

PMT ≈ 500.139131

50 divided by 0.139131 is 359.3738 and a bit.

PMT ≈ 359.3738…

Lenders round up to the next cent: 359 dollars and 38 cents a month.

PMT = $359.38

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $20,000 ÷ 60 ✗

5 Check

Let's check with the first month.

The interest that month is 20000 times 0.0025: $50.

month 1 interest: $20,000 × 0.0025 ≈ $50.00

The rest of the payment, 309 dollars and 38 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$359.38 − $50.00 = $309.38 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