A depositor opens a new savings account with $ 8000 at 8 % compounded semiannually. At the beginning of year 3, an additional $6000 is deposited. At the end of four years, what is the balance in the account?
amount
= 8000(1.04)^8 + 6000(1.04)^2
= ...
Well, well, well, looks like we've got a money mogul here! Let's crunch those numbers and find out the balance in the account, shall we?
So, the initial deposit of $8000 is compounded semiannually at a rate of 8%. That means we have 2 compounding periods per year. After the first year, we'd have:
Year 1: $8000 + ($8000 * 0.08 / 2) = $8000 + $320 = $8320
Now, at the beginning of year 3, an additional $6000 is deposited. So, the account balance at the beginning of year 3 would be:
$8320 + $6000 = $14320
Fast forward to the end of 4 years. We have 8 compounding periods in total (2 per year, for 4 years). Let's calculate the compounded balance at the end of 4 years:
$14320 * (1 + (0.08 / 2))^8 = $14320 * (1.04)^8 ≈ $14320 * 1.361% = $19498.72
So, after 4 years, the balance in the account would be approximately $19498.72.
That's not too shabby! Keep saving, my friend, and watch that balance grow!
To calculate the balance in the account at the end of four years, we need to find the future value of the initial deposit and the additional deposit.
Step 1: Calculate the future value of the initial deposit.
Formula for future value of a lump sum investment compounded semiannually:
FV = P(1 + r/n)^(nt)
In this case:
P = $8000 (initial deposit)
r = 0.08 (interest rate)
n = 2 (number of times interest is compounded per year, since it is compounded semiannually)
t = 4 (number of years)
FV1 = 8000(1 + 0.08/2)^(2*4)
= 8000(1.04)^8
≈ $10,563.19
Step 2: Calculate the future value of the additional deposit.
We will use the same formula, but with a new principal amount (initial deposit + additional deposit).
P2 = $8000 (initial deposit) + $6000 (additional deposit)
= $14,000
FV2 = 14000(1 + 0.08/2)^(2*1)
= 14000(1.04)^2
≈ $14,665.60
Step 3: Calculate the total balance at the end of four years.
Total balance = FV1 + FV2
= $10,563.19 + $14,665.60
≈ $25,228.79
Therefore, the balance in the account at the end of four years is approximately $25,228.79.
To find the balance in the savings account at the end of four years, we can break down the problem into different steps.
Step 1: Calculate the balance after the first two years.
The interest is compounded semiannually, which means the interest is added twice a year. To calculate the balance after two years, we need to calculate the interest for each compounding period and add it to the initial deposit.
The interest formula for compound interest is: A = P(1 + r/n)^(nt), where:
A = final amount after interest
P = principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, the principal amount is $8000, the annual interest rate is 8% (or 0.08 as a decimal), and it is compounded semiannually (n = 2). The number of years for the first two years is 2.
Using the formula, we can calculate the balance after two years:
A = 8000(1 + 0.08/2)^(2*2)
A = 8000(1 + 0.04)^4
A = 8000(1.04)^4
A ≈ $9355.07
So, after the first two years, the balance in the account is approximately $9355.07.
Step 2: Add the additional deposit at the beginning of year 3.
At the beginning of year 3, an additional $6000 is deposited into the account. To calculate the new balance, we need to add this amount to the balance after two years.
Balance after year 3 = $9355.07 + $6000
Balance after year 3 = $15,355.07
Step 3: Calculate the balance at the end of four years.
Finally, we need to calculate the interest for the last compounding period, which is from the beginning of year 3 to the end of year 4. Again, we can use the compound interest formula:
A = 15355.07(1 + 0.08/2)^((2-1)*2)
A = 15355.07(1 + 0.04)^2
A = 15355.07(1.04)^2
A ≈ $17,238.35
So, at the end of four years, the balance in the account is approximately $17,238.35.