A loan of $5,000 at 12% for 2 years is paid monthly. Find the… Step by Step

Step by step

The question

Question: A loan of $5000 at 12% for 2 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $5000.

r is the yearly rate: 12% is 0.12.

P = $5,000 loan, r = 12% = 0.12

n is how many times a year: 12.

t is the time: 2 years.

n = 12 a year, t = 2 years

2 Per payment

Interest is added each month.

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

0.12 divided by 12 is 0.01.

i = rn = 0.1212 = 0.01

i = the rate each month

And N is how many payments: 12 times 2 is 24.

N = n × t = 12 × 2 = 24

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 0.01 is 1.01.

PMT = 5000 × 0.011 − 1.01⁻²⁴

4 Work it out

The top: 50.

With a calculator, 1.01 to the power negative 24 is about 0.787566.

1 minus that is 0.212434.

5000 × 0.01 = 50

1.01⁻²⁴ ≈ 0.787566

1 − 0.787566 = 0.212434

Now the top over the bottom.

PMT ≈ 500.212434

50 divided by 0.212434 is 235.3674 and a bit.

PMT ≈ 235.3674…

Lenders round up to the next cent: 235 dollars and 37 cents a month.

PMT = $235.37

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $5,000 ÷ 24 ✗

5 Check

Let's check with the first month.

The interest that month is 5000 times 0.01: $50.

month 1 interest: $5,000 × 0.01 ≈ $50.00

The rest of the payment, 185 dollars and 37 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$235.37 − $50.00 = $185.37 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