A personal loan of $6,000 at 10.2% for 3 years is paid monthly. Find… Step by Step

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The question

Question: A personal loan of $6000 at 10.2% for 3 years is paid monthly.

Find the monthly payment.

1 Read the numbers

P is the money borrowed: $6000.

r is the yearly rate: 10.2% is 0.102.

P = $6,000 loan, r = 10.2% = 0.102

n is how many times a year: 12.

t is the time: 3 years.

n = 12 a year, t = 3 years

2 Per payment

Interest is added each month.

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

0.102 divided by 12 is 0.0085.

i = rn = 0.10212 = 0.0085

i = the rate each month

And N is how many payments: 12 times 3 is 36.

N = n × t = 12 × 3 = 36

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 0.0085 is 1.0085.

PMT = 6000 × 0.00851 − 1.0085⁻³⁶

4 Work it out

The top: 51.

With a calculator, 1.0085 to the power negative 36 is about 0.73734.

1 minus that is 0.26266.

6000 × 0.0085 = 51

1.0085⁻³⁶ ≈ 0.73734

1 − 0.73734 = 0.26266

Now the top over the bottom.

PMT ≈ 510.26266

51 divided by 0.26266 is 194.1670 and a bit.

PMT ≈ 194.1670…

Lenders round up to the next cent: 194 dollars and 17 cents a month.

PMT = $194.17

Careful!

That pays back only the loan.

The interest must be paid too.

Careful: not $6,000 ÷ 36 ✗

5 Check

Let's check with the first month.

The interest that month is 6000 times 0.0085: $51.

month 1 interest: $6,000 × 0.0085 ≈ $51.00

The rest of the payment, 143 dollars and 17 cents, pays the loan down.

So the loan gets smaller every month.

It makes sense!

$194.17 − $51.00 = $143.17 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