A certain department store requires its credit-card customers to pay interest on unpaid bills at the rate of 18% per year compounded monthly. If a customer buys a television set for $500 on credit and makes no payments for one year, how much is owed at the end of the year? (Round your answer to the nearest cent.)
500(1 + 0.18/12)^(12*1) = ____
Well, well, well, looks like someone really enjoyed their television set and forgot about the little detail called payments. Let's see how much this forgetfulness will cost you!
Now, the interest rate here is 18% per year, but compounded monthly. So, let's break out our math hats and get calculating, shall we?
First, we need to calculate the monthly interest rate. To do this, we divide the annual interest rate by 12 (the number of months in a year). So, 18% divided by 12 gives us 1.5%. Don't worry, I won't make you do the math - it's 0.015, to be precise.
Next, we'll calculate the amount owed at the end of the year. Since the interest is compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount owed
P = initial amount borrowed
r = interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
Plugging in the values, we get:
A = 500(1 + 0.015/12)^(12*1)
Now hold on to your hats, because here comes the grand finale!
A = $524.43
So, my dear shopaholic friend, you now owe a grand total of $524.43 at the end of the year. Oopsie daisy! Remember, it's always better to stay on top of those payments, or else they might come back to haunt you.
To find the amount owed at the end of the year, we can use the formula for compound interest:
\(A = P(1 + \frac{r}{n})^{nt}\)
Where:
A = the final amount (amount owed at the end of the year)
P = the original principal (purchase price of the television set)
r = annual interest rate (18%)
n = number of times interest is compounded per year (12 monthly compoundings)
t = time in years (1 year)
Let's plug in the values and calculate:
\(A = 500(1 + \frac{0.18}{12})^{12 \cdot 1}\)
\(A = 500(1 + \frac{0.015}{1})^{12}\)
\(A = 500(1.015)^{12}\)
Using a calculator, we can evaluate \(A\) to be approximately $579.79.
Therefore, the customer owes approximately $579.79 at the end of the year.
To calculate the amount owed at the end of the year, we need to use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- A is the future value (amount owed) at the end of the year
- P is the principal amount (initial purchase price)
- r is the annual interest rate (18%)
- n is the number of times interest is compounded per year (12 for monthly compounding)
- t is the number of years (1 in this case)
Let's plug in the values and calculate the amount owed:
\[ A = 500 \left(1 + \frac{0.18}{12}\right)^{12 \times 1} \]
Calculating this expression gives us:
\[ A = 500 \left(1 + \frac{0.015}{1}\right)^{12} \]
\[ A = 500 \times (1.015)^{12} \]
Using a calculator or a spreadsheet, we find:
\[ A \approx 559.65 \]
Therefore, the amount owed at the end of the year is approximately $559.65.