A student loan of $12,000 at 4.8% for 10 years is paid monthly. Find… Step by Step

Step by step

The question

Question: A student loan of $12000 at 4.8% for 10 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $12000.

r is the yearly rate: 4.8% is 0.048.

P = $12,000 loan, r = 4.8% = 0.048

n is how many times a year: 12.

t is the time: 10 years.

n = 12 a year, t = 10 years

2 Per payment

Interest is added each month.

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

0.048 divided by 12 is 0.004.

i = rn = 0.04812 = 0.004

i = the rate each month

And N is how many payments: 12 times 10 is 120.

N = n × t = 12 × 10 = 120

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 0.004 is 1.004.

PMT = 12000 × 0.0041 − 1.004⁻¹²⁰

4 Work it out

The top: 48.

With a calculator, 1.004 to the power negative 120 is about 0.619376.

1 minus that is 0.380624.

12000 × 0.004 = 48

1.004⁻¹²⁰ ≈ 0.619376

1 − 0.619376 = 0.380624

Now the top over the bottom.

PMT ≈ 480.380624

48 divided by 0.380624 is 126.1087 and a bit.

PMT ≈ 126.1087…

Lenders round up to the next cent: 126 dollars and 11 cents a month.

PMT = $126.11

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $12,000 ÷ 120 ✗

5 Check

Let's check with the first month.

The interest that month is 12000 times 0.004: $48.

month 1 interest: $12,000 × 0.004 ≈ $48.00

The rest of the payment, 78 dollars and 11 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$126.11 − $48.00 = $78.11 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