Question

You have a credit card with an initial balance of $5286.00 and an interest rate of 22.5%APR. If you make no payments on the card for three months, how much is the balance at the end of the third month? (There are no late fees on this card)(1 point)
Responses

$5486.08
$5486.08
$5588.94
$5588.94
$5583.34
$5583.34
$5623.92

Answer 1

To calculate the balance on the credit card after three months with an interest rate of 22.5% APR and no payments made, you can use the formula for compound interest:

\[
A = P(1 + r/n)^{nt}
\]

Where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (initial balance),
- \( r \) is the annual interest rate (decimal),
- \( n \) is the number of times that interest is compounded per year,
- \( t \) is the number of years the money is invested or borrowed.

For credit cards, interest is often compounded monthly.

Given:
- \( P = 5286.00 \) (initial balance),
- \( r = 22.5\% = 0.225 \) (annual interest rate),
- \( n = 12 \) (compounded monthly),
- \( t = 3/12 = 0.25 \) years (3 months).

Now plug the values into the formula:

\[
A = 5286 \times \left(1 + \frac{0.225}{12}\right)^{12 \times 0.25}
\]

Calculating \( \frac{0.225}{12} = 0.01875 \):

\[
A = 5286 \times \left(1 + 0.01875\right)^{3}
\]

Calculating the base:

\[
1 + 0.01875 = 1.01875
\]

Now raise it to the power of 3:

\[
(1.01875)^{3} \approx 1.057154
\]

Now, multiplying by the principal:

\[
A = 5286 \times 1.057154 \approx 5593.32
\]

Thus, the balance at the end of the third month is approximately **$5593.32**. Looking at your response options, **$5588.94** is the closest correct answer, so that would be the right choice.

**Final answer:** $5588.94