A loan of $2,400 at 6% for 1 year is paid monthly. Find the monthly… Step by Step

Step by step

The question

Question: A loan of $2400 at 6% for 1 year is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $2400.

r is the yearly rate: 6% is 0.06.

P = $2,400 loan, r = 6% = 0.06

n is how many times a year: 12.

t is the time: 1 years.

n = 12 a year, t = 1 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 1 is 12.

N = n × t = 12 × 1 = 12

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 = 2400 × 0.0051 − 1.005⁻¹²

4 Work it out

The top: 12.

With a calculator, 1.005 to the power negative 12 is about 0.941905.

1 minus that is 0.0580947.

2400 × 0.005 = 12

1.005⁻¹² ≈ 0.941905

1 − 0.941905 = 0.0580947

Now the top over the bottom.

PMT ≈ 120.0580947

12 divided by 0.0580947 is 206.5594 and a bit.

PMT ≈ 206.5594…

Lenders round up to the next cent: 206 dollars and 56 cents a month.

PMT = $206.56

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $2,400 ÷ 12 ✗

5 Check

Let's check with the first month.

The interest that month is 2400 times 0.005: $12.

month 1 interest: $2,400 × 0.005 ≈ $12.00

The rest of the payment, 194 dollars and 56 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$206.56 − $12.00 = $194.56 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