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?
Erica invests $10,000 at 5% interest compounded annually.
How much interest will Erica earn in 3 years
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
Let's calculate the balance in Lee's account step-by-step.
Step 1: Calculate the balance after the first deposit.
P1 = $15,000
r = 9% = 0.09 (as a decimal)
n = 2 (compounded semiannually)
t1 = 4 years
A1 = P1(1 + r/n)^(nt1)
A1 = $15,000(1 + 0.09/2)^(2*4)
A1 = $15,000(1.045)^8
A1 ≈ $22,647.16
After the first 4 years, the balance in Lee's account is approximately $22,647.16.
Step 2: Calculate the balance after the second deposit.
P2 = $40,000
r = 9% = 0.09 (as a decimal)
n = 2 (compounded semiannually)
t2 = 6 years (4 + 2 years)
A2 = P2(1 + r/n)^(nt2)
A2 = $40,000(1 + 0.09/2)^(2*6)
A2 = $40,000(1.045)^12
A2 ≈ $79,488.87
After the additional deposit and another 2 years, the balance in Lee's account is approximately $79,488.87.
Step 3: Calculate the final balance.
Final balance = A1 + A2
Final balance ≈ $22,647.16 + $79,488.87
Final balance ≈ $102,136.03
At the end of 6 years, the balance in Lee's account is approximately $102,136.03.
To find the balance in Lee's account at the end of 6 years, we need to calculate the compound interest for each deposit separately and then add the results.
First, let's calculate the interest on the initial deposit of $15,000 for 6 years at a rate of 9% compounded semiannually. To do this, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the investment
P is the principal amount (initial deposit)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years
For the initial deposit:
P = $15,000
r = 9% = 0.09 (as a decimal)
n = 2 (compounded semiannually)
t = 6 years
Using the formula, we can calculate the future value of the initial deposit:
A1 = $15,000(1 + 0.09/2)^(2*6)
A1 = $15,000(1 + 0.045)^(12)
A1 = $15,000(1.045)^(12)
A1 ≈ $24,220.50
So, after 6 years, the initial deposit of $15,000 will grow to approximately $24,220.50.
Now let's calculate the interest on the additional deposit of $40,000 made at the beginning of year 4. We'll consider the remaining 3 years after the initial deposit.
For the additional deposit:
P = $40,000
r = 9% = 0.09 (as a decimal)
n = 2 (compounded semiannually)
t = 3 years
Using the same formula:
A2 = $40,000(1 + 0.09/2)^(2*3)
A2 = $40,000(1 + 0.045)^(6)
A2 ≈ $50,912.52
So, after 3 years, the additional deposit of $40,000 will grow to approximately $50,912.52.
Finally, to find the total balance in Lee's account at the end of 6 years, we add the results of the two deposits:
Total balance = A1 + A2
Total balance ≈ $24,220.50 + $50,912.52
Total balance ≈ $75,133.02
Therefore, the balance in Lee's account at the end of 6 years will be approximately $75,133.02.