A loan of $10,000 at 6% for 4 years is paid monthly. Find the… Step by Step

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The question

Question: A loan of $10000 at 6% for 4 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $10000.

r is the yearly rate: 6% is 0.06.

P = $10,000 loan, r = 6% = 0.06

n is how many times a year: 12.

t is the time: 4 years.

n = 12 a year, t = 4 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 4 is 48.

N = n × t = 12 × 4 = 48

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 = 10000 × 0.0051 − 1.005⁻⁴⁸

4 Work it out

The top: 50.

With a calculator, 1.005 to the power negative 48 is about 0.787098.

1 minus that is 0.212902.

10000 × 0.005 = 50

1.005⁻⁴⁸ ≈ 0.787098

1 − 0.787098 = 0.212902

Now the top over the bottom.

PMT ≈ 500.212902

50 divided by 0.212902 is 234.8503 and a bit.

PMT ≈ 234.8503…

Lenders round up to the next cent: 234 dollars and 86 cents a month.

PMT = $234.86

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $10,000 ÷ 48 ✗

5 Check

Let's check with the first month.

The interest that month is 10000 times 0.005: $50.

month 1 interest: $10,000 × 0.005 ≈ $50.00

The rest of the payment, 184 dollars and 86 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$234.86 − $50.00 = $184.86 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