Your credit card company charges you 1.13 percent per month.
What is the annual percentage rate on your account?
The correct answer and explanation is :
To calculate the annual percentage rate (APR) when you are given a monthly interest rate of 1.13%, you can use the formula that converts the monthly rate to an annual rate. Here’s the step-by-step process:
Step 1: Convert the monthly rate to a decimal
The monthly interest rate is given as 1.13%, which means 1.13 out of every 100 dollars is charged as interest. To convert this percentage into a decimal, divide it by 100:
[
\text{Monthly rate in decimal} = \frac{1.13}{100} = 0.0113
]
Step 2: Use the formula for annual percentage rate (APR)
To find the annual percentage rate (APR), we can use the formula for compound interest since the interest is typically compounded monthly. The formula for the effective annual rate (EAR) is:
[
\text{APR} = (1 + r)^{12} – 1
]
Where:
- ( r ) is the monthly rate in decimal form.
- 12 represents the number of months in a year.
Step 3: Plug in the values
Now, substitute the decimal form of the monthly rate into the formula:
[
\text{APR} = (1 + 0.0113)^{12} – 1
]
[
\text{APR} = (1.0113)^{12} – 1
]
[
\text{APR} \approx 1.1431 – 1
]
[
\text{APR} \approx 0.1431
]
Step 4: Convert back to a percentage
To convert the decimal form of APR back into a percentage, multiply by 100:
[
\text{APR} = 0.1431 \times 100 = 14.31\%
]
Final Answer:
The annual percentage rate (APR) is 14.31%.
Explanation:
Credit card companies often charge interest on a monthly basis, and that interest is compounded, meaning it accumulates on top of previous interest charges. To compare credit cards or determine the true cost of borrowing, it’s important to understand the APR, which is the interest rate on a yearly basis. However, due to the compounding effect, the APR is typically higher than the simple sum of monthly rates. By using the formula for compound interest, we account for how interest on interest affects the overall cost of borrowing on an annual basis.