If you deposit $900 semiannually in a bank which pays 6% interest compounded semiannually and make no more deposits after 7 years, what is the balance in the account 8 yrs. later from the last deposit?
will assume you are making 14 deposits, the first one 6 months from now, and then leave the balance alone for another year.
Amount at the last deposit
= 900[ 1.03^14 - 1]/.03 = 15377.69
Value at the end of 8 years = 15377.69(1.03)^2 = 16314.19
To find the balance in the account 8 years after the last deposit, we can break down the problem into several steps.
Step 1: Calculate the number of compounding periods
Since the interest is compounded semiannually, we need to determine the total number of compounding periods over the 8-year period. Since there are two compounding periods in a year, the total number of compounding periods is 8 x 2 = 16.
Step 2: Calculate the interest rate per compounding period
The annual interest rate is 6%. Since interest is compounded semiannually, we need to divide the annual interest rate by 2 to get the interest rate per compounding period: 6% / 2 = 3%.
Step 3: Calculate the balance after 8 years
Using the formula for compound interest, we can calculate the balance after 8 years:
Balance = Principal x (1 + interest rate per period)^(number of periods)
In this case, the principal is $900, the interest rate per period is 3%, and the number of periods is 16. Plugging in these values, we have:
Balance = $900 x (1 + 0.03)^16
Step 4: Solve the equation
Calculating the equation gives us:
Balance = $900 x (1.03)^16
Using a calculator, we find that (1.03)^16 is approximately 1.6297.
Balance = $900 x 1.6297
Calculating the equation gives us:
Balance = $1,466.77
Therefore, the balance in the account 8 years after the last deposit would be approximately $1,466.77.
To calculate the balance in the account 8 years later from the last deposit, we can start by calculating the number of compounding periods:
Number of compounding periods = Number of years × Number of compounding periods per year
Number of compounding periods = 8 × 2 (since interest is compounded semiannually)
Number of compounding periods = 16
Next, we can calculate the value of each deposit after 7 years using the formula for compound interest:
Future Value = Present Value × (1 + Interest Rate / Number of compounding periods)^(Number of compounding periods × Number of years)
Future Value = $900 × (1 + 0.06 / 2)^(2 × 7)
Future Value = $900 × (1 + 0.03)^14
Future Value = $900 × (1.03)^14
Future Value = $900 × 1.50540
Future Value ≈ $1354.86
Finally, we can calculate the balance in the account 8 years later from the last deposit by multiplying the future value by the interest rate:
Balance = Future Value × (1 + Interest Rate / Number of compounding periods)
Balance = $1354.86 × (1 + 0.06 / 2)
Balance = $1354.86 × (1 + 0.03)
Balance = $1354.86 × 1.03
Balance ≈ $1393.78
Therefore, the balance in the account 8 years later from the last deposit would be approximately $1393.78.