5. [0/5 Points] DETAILS MY NOTES If you want to earn \( 9 \% \) annual simple interest on an investment, how much should you pay for a note that will be worth \( \$ 5,400 \) in 6 months? (Round your answer to two decimal places.) \( \$ 5,177.87 \times \)

To find out how much you should pay for a note that will be worth \$5,400 in 6 months with an annual simple interest rate of 9%, we can use the formula for simple interest: \[ A = P(1 + rt) \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the time the money is invested for in years. ### Step 1: Identify the known values - \( A = 5400 \) - \( r = 0.09 \) (9% as a decimal) - \( t = 0.5 \) (6 months is half a year) ### Step 2: Rearrange the formula to solve for \( P \) We need to isolate \( P \): \[ P = \frac{A}{1 + rt} \] ### Step 3: Substitute the known values into the formula Now we can substitute the values into the rearranged formula: \[ P = \frac{5400}{1 + (0.09 \times 0.5)} \] ### Step 4: Calculate the denominator First, calculate \( rt \): \[ rt = 0.09 \times 0.5 = 0.045 \] Now, add 1 to this value: \[ 1 + rt = 1 + 0.045 = 1.045 \] ### Step 5: Calculate \( P \) Now substitute back into the equation for \( P \): \[ P = \frac{5400}{1.045} \] ### Step 6: Perform the calculation Now, let's calculate \( P \): \[ P \approx \frac{5400}{1.045} \approx 5177.87 \] Thus, the amount you should pay for the note is approximately: \[ \boxed{5177.87} \] This confirms that the answer you provided, \( \$ 5,177.87 \), is correct.