Shelley Katz deposited $30,000 in a savings account at 5% interest compounded semiannually. At the beginning of year 4, Shelley deposits an additional $80,000 at 5% interest compounded semiannually. At the end of 6 years, what is the balance in Shelley's account
try $ 133,123.12
make a "time graph" to see where the payments go, and the time needed to reach the end of 6 years
amount = 30000(1.025)^12 + 80000(1.025)^6
= ....
( I got $133122.14)
Well, let's calculate that, but first, can we all just take a moment to appreciate how Shelley's last name is "Katz"? I mean, she must have some purr-fect jokes up her sleeve!
Anyway, back to business. To calculate the balance in Shelley's account after 6 years, we need to break it down into three periods:
1. The first 3 years: The initial deposit of $30,000 compounds semiannually at 5% interest. So, after 6 compounding periods (2 per year for 3 years), the balance will be:
Balance after 3 years = $30,000 × (1 + 0.05/2)^6
2. The next 2 years: Shelley deposits an additional $80,000 at the beginning of year 4. So, this $80,000 will compound semiannually at 5% interest for the next 4 compounding periods (2 per year for 2 years). The total balance after this period will be:
Balance after the $80,000 deposit = ($30,000 × (1 + 0.05/2)^6) + $80,000 × (1 + 0.05/2)^4
3. Finally, the last year: During the last year, the compounded interest will continue to work on the previous balance. So, we calculate the balance after one more year (2 compounding periods):
Total balance after 6 years = [($30,000 × (1 + 0.05/2)^6) + $80,000 × (1 + 0.05/2)^4] × (1 + 0.05/2)^2
Now, let's crunch these numbers, my funny friend!
To calculate the balance in Shelley's account at the end of 6 years, we need to calculate the compound interest earned on each deposit separately and then add them together.
Step 1: Calculate the interest earned on the initial deposit of $30,000 for 6 years compounded semiannually at a 5% interest rate.
First, we need to calculate the number of compounding periods over 6 years, considering that interest is compounded semiannually.
Number of compounding periods = 6 years * 2 (semiannual compounding) = 12 periods
Next, we can use the compound interest formula to calculate the amount after 6 years:
A = P(1 + r/n)^(nt)
Where:
A = the amount after time t
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of compounding periods per year
t = time in years
A = $30,000 * (1 + 0.05/2)^(12)
A ≈ $30,000 * (1 + 0.025)^(12)
A ≈ $30,000 * (1.025)^(12)
A ≈ $30,000 * 1.344867...
Therefore, the amount after 6 years on the initial deposit is approximately $40,346.03.
Step 2: Calculate the interest earned on the additional deposit of $80,000 for 4 years compounded semiannually at a 5% interest rate.
Similar to Step 1, we need to calculate the number of compounding periods and then use the compound interest formula.
Number of compounding periods = 4 years * 2 (semiannual compounding) = 8 periods
A = $80,000 * (1 + 0.05/2)^(8)
A ≈ $80,000 * (1 + 0.025)^(8)
A ≈ $80,000 * (1.025)^(8)
A ≈ $80,000 * 1.184060...
Therefore, the amount after 4 years on the additional deposit is approximately $118,724.81.
Step 3: Calculate the total balance in Shelley's account after 6 years.
Total balance = balance from the initial deposit + balance from the additional deposit
Total balance ≈ $40,346.03 + $118,724.81
Total balance ≈ $159,070.84
Therefore, the balance in Shelley's account at the end of 6 years is approximately $159,070.84.
To calculate the balance in Shelley's account at the end of 6 years, we first need to calculate the balance at the beginning of year 4 and then add the interest earned for the remaining 3 years.
Let's go step by step:
Step 1: Calculate the balance at the beginning of year 4.
We start with Shelley's initial deposit of $30,000. The interest is compounded semiannually, which means it is applied twice a year. The interest rate is 5%, so the interest rate per compounding period is (5/2)% = 2.5%.
To calculate the balance at the beginning of year 4, we need to compound the interest for 3 years or 6 compounding periods (since it is compounded semiannually). We can use the compound interest formula:
A = P(1 + r/n)^(nt)
where:
A = final amount (balance)
P = principal (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = time in years
In this case, the principal (P) is $30,000, the annual interest rate (r) is 5% (or 0.05 as a decimal), the number of times compounded per year (n) is 2 (since it's compounded semiannually), and the time (t) is 3 years.
Plugging in the values into the formula:
A = $30,000 * (1 + 0.05/2)^(2 * 3)
A = $30,000 * (1 + 0.025)^(6)
A = $30,000 * (1.025)^(6)
A ≈ $34,799.63
So, at the beginning of year 4, Shelley has approximately $34,799.63 in her account.
Step 2: Calculate the interest earned for the remaining 3 years.
Now, Shelley makes an additional deposit of $80,000 at the beginning of year 4. This means the principal for the remaining 3 years is $34,799.63 + $80,000 = $114,799.63.
Using the same compound interest formula, we can calculate the interest earned for the remaining 3 years.
A = $114,799.63 * (1 + 0.05/2)^(2 * 3)
A = $114,799.63 * (1.025)^(6)
A ≈ $134,467.48
So, at the end of 6 years, the balance in Shelley's account is approximately $134,467.48.