A firm of attorneys deposits $15,000 of profit
sharing money every six months in an account at 3% compounded semiannually for 7 1/2
years. Find the amount of
interest earned.
To find the amount of interest earned, we need to calculate the total balance accumulated at the end of 7 1/2 years and subtract the initial deposit amount.
First, we need to determine the interest rate per compounding period. Since the account compounds semiannually, we have 3% interest rate per year divided by 2 (for semiannual compounding). So the interest rate per compounding period is 3%/2 = 1.5%.
Next, we need to calculate the number of compounding periods. Since the money is deposited every six months for 7 1/2 years, we have 7.5 years * 2 compounding periods per year = 15 compounding periods.
Now, we can use the compound interest formula to calculate the total balance:
A = P(1 + r/n)^(nt)
Where:
A = Total balance
P = Initial deposit
r = Interest rate per compounding period (as a decimal)
n = Number of compounding periods per year
t = Number of years
Given values:
P = $15,000
r = 1.5% (0.015 as a decimal)
n = 2 (semiannual compounding)
t = 7.5 years
Plugging in these values into the formula:
A = $15,000 * (1 + 0.015/2)^(2 * 7.5)
Now, we can calculate this using a calculator or a spreadsheet to find A. Let's do the calculation:
A ≈ $19,614.31
Finally, we can find the amount of interest earned by subtracting the initial deposit from the total balance:
Interest earned = A - P
Interest earned ≈ $19,614.31 - $15,000
Interest earned ≈ $4,614.31
Therefore, the amount of interest earned is approximately $4,614.31.