Lee Holmes deposited $15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% intrest compounded semiannually. At the end of 6 years, what is the balance in Lee's account?
Lance Jackson deposited $5,000 at Basil Bank at 9% interest compounded daily. What is Lance's investment at the end of 4 years?
To solve this problem step-by-step, we'll first calculate the balance at the end of year 3 and then use it to calculate the balance at the end of year 6.
Step 1: Calculate the balance at the end of year 3.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future balance
P = the principal amount (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times the interest is compounded per year
t = number of years
Given:
P = $15,000
r = 0.09 (9% interest rate)
n = 2 (compounded semiannually)
t = 3 (end of year 3)
We can plug these values into the formula:
A1 = 15,000 * (1 + 0.09/2)^(2*3)
Simplifying the equation:
A1 = 15,000 * (1.045)^6
A1 ≈ 15,000 * 1.340097
A1 ≈ $20,101.46
So, at the end of year 3, the balance in Lee's account is approximately $20,101.46.
Step 2: Calculate the balance at the end of year 6.
Now we'll calculate the balance at the end of year 6, taking into account the additional deposit of $40,000 at the beginning of year 4.
Given:
P2 = P1 + $40,000 (balance at the end of year 3 + additional deposit)
r = 0.09 (9% interest rate)
n = 2 (compounded semiannually)
t = 3 (years from the end of year 3 to year 6)
Using the compound interest formula:
A2 = P2 * (1 + r/n)^(n*t)
Substituting the values:
A2 = 20,101.46 * (1 + 0.09/2)^(2*3)
Simplifying the equation:
A2 = 20,101.46 * (1.045)^6
A2 ≈ 20,101.46 * 1.340097
A2 ≈ $26,902.57
Therefore, at the end of 6 years, the balance in Lee's account is approximately $26,902.57.
To find the balance in Lee's account at the end of 6 years, we need to calculate the future value of both deposits separately and then add them together.
Let's break down the problem step by step:
1. Calculate the future value of the first deposit of $15,000 after 6 years with 9% interest compounded semiannually. To do this, we can use the formula for compound interest:
Future Value = Principal * (1 + (Interest Rate / Compounding Period))^ (Number of Compounding Periods)
In this case, the principal is $15,000, the interest rate is 9% or 0.09, and the compounding period is semiannually, which means 2 compounding periods per year. So the formula becomes:
Future Value1 = $15,000 * (1 + (0.09 / 2))^ (2 * 6)
Calculate this value and we get the future value of the first deposit.
2. Calculate the future value of the second deposit of $40,000 after 4 years with 9% interest compounded semiannually. Using the same formula as above, but now the principal is $40,000 and the compounding period is 2 compounding periods per year for 4 years:
Future Value2 = $40,000 * (1 + (0.09 / 2))^ (2 * 4)
Calculate this value and we get the future value of the second deposit.
3. Add the two future values calculated in steps 1 and 2 to find the total balance:
Total Balance = Future Value1 + Future Value2
Calculate this value to get the final answer, which is the balance in Lee's account at the end of 6 years.