A loan of $3,600 at 15% for 2 years is paid monthly. Find the… Step by Step

Step by step

The question

Question: A loan of $3600 at 15% for 2 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $3600.

r is the yearly rate: 15% is 0.15.

P = $3,600 loan, r = 15% = 0.15

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.

0.15 divided by 12 is 0.0125.

i = rn = 0.1512 = 0.0125

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 0.0125 is 1.0125.

PMT = 3600 × 0.01251 − 1.0125⁻²⁴

4 Work it out

The top: 45.

With a calculator, 1.0125 to the power negative 24 is about 0.742197.

1 minus that is 0.257803.

3600 × 0.0125 = 45

1.0125⁻²⁴ ≈ 0.742197

1 − 0.742197 = 0.257803

Now the top over the bottom.

PMT ≈ 450.257803

45 divided by 0.257803 is 174.5519 and a bit.

PMT ≈ 174.5519…

Lenders round up to the next cent: 174 dollars and 56 cents a month.

PMT = $174.56

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $3,600 ÷ 24 ✗

5 Check

Let's check with the first month.

The interest that month is 3600 times 0.0125: $45.

month 1 interest: $3,600 × 0.0125 ≈ $45.00

The rest of the payment, 129 dollars and 56 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$174.56 − $45.00 = $129.56 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