If an apartment complex will need painting in 3 1/2 years and the job will cost $45,000, what amount needs to be deposited into an account now in order to have the necessary funds? The account pays 12% interest compounded semiannually.
solve for p in
p(1+.12/2)^(2*7/2) = 45000
To calculate the amount that needs to be deposited into an account now, we need to determine the future value of the $45,000 in 3 1/2 years, taking into account the interest earned over time.
First, we need to convert the time period to semiannual periods, as the interest is compounded semiannually. Since each year has two semiannual periods, 3 1/2 years would be equivalent to 7 semiannual periods.
Next, we can use the formula for compound interest to calculate the future value of the $45,000 investment:
Future Value = Present Value * (1 + (Interest Rate / Number of Compounding Periods)) ^ (Number of Compounding Periods * Time)
In this case:
Present Value = ?
Future Value = $45,000
Interest Rate = 12% = 0.12 (as a decimal)
Number of Compounding Periods = 2 (semiannual compounding)
Time = 7 (semiannual periods)
Plugging in these values into the formula, we have:
$45,000 = Present Value * (1 + (0.12 / 2)) ^ (2 * 7)
Simplifying the equation, we get:
1.06^14 = Present Value
Using a calculator or a spreadsheet, calculate 1.06^14 which equals approximately 1.985925.
Therefore, Present Value = $45,000 / 1.985925
Calculating this value, we find that the amount that needs to be deposited into the account now is:
Present Value ≈ $22,648.80
So, approximately $22,648.80 needs to be deposited into the account now in order to have the necessary funds of $45,000 for the painting job in 3 1/2 years, considering a 12% interest rate compounded semiannually.