A car loan of $28,500 at 3.99% for 5 years is paid monthly. Find the… Step by Step

Step by step

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