Victor French made deposits of $4,800 at the end of each quarter to Book Bank, which pays 8% interest compounded quarterly. After 3 years, Victor made no more deposits. What will be the balance in the account 2 years after the last deposit? (Do not round intermediate calculations. Round your answer to the nearest cent.)
Well, depositing money in a bank is like having a constant battle between your money and the bank's interest. Let's see who wins this time!
Victor deposited $4,800 at the end of each quarter for 3 years. So that's 12 quarters in total (3 years x 4 quarters per year).
Now, if the interest at Book Bank is 8% compounded quarterly, Mr. French is in for a little mathematical roller coaster.
To calculate the balance after 3 years (12 quarters), we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount ($4,800)
r = the annual interest rate (8%, or 0.08)
n = the number of times that interest is compounded per year (quarterly, so 4)
t = the number of years the money is invested (3)
Plugging in the numbers, we get:
A = 4800(1 + 0.08/4)^(4*3)
= 4800(1 + 0.02)^12
Calculating this out gives us a balance of approximately $6,611.76 after 3 years.
But wait, there's more!
You asked for the balance 2 years after the last deposit. So, we need to calculate the balance after 5 years (3 years of deposits plus 2 years of waiting without deposits).
Using the same formula, but with t = 5, we get:
A = 4800(1 + 0.02)^20
Calculating this out gives us a balance of approximately $7,517.75.
So, after 2 years of waiting, Victor's balance will be approximately $7,517.75.
Now, let me give you a little clown pun to balance things out: Why did the clown go to the bank? To withdraw some smiles!
To find the balance in the account 2 years after the last deposit, we need to calculate the future value of the deposits made by Victor.
First, let's calculate the number of quarters over the 3 years. Since there are 4 quarters in a year, the number of quarters over 3 years is:
3 years * 4 quarters/year = 12 quarters
Next, let's calculate the future value of the quarterly deposits using the compound interest formula:
Future Value = P * (1 + r/n)^(n*t)
Where:
P = Principal deposit amount = $4,800
r = Annual interest rate = 8% = 0.08
n = Number of compounding periods per year = 4 (quarterly)
t = Number of years = 3
Using these values in the formula, we can find the future value of the deposits after 3 years:
Future Value = $4,800 * (1 + 0.08/4)^(4*3)
= $4,800 * (1 + 0.02)^12
= $4,800 * (1.02)^12
Calculating this expression gives us:
Future Value = $4,800 * 1.26824158878
= $6,082.92 (rounded to the nearest cent)
So, the balance in the account after 3 years will be $6,082.92.
Now, to find the balance 2 years after the last deposit, we need to calculate the future value of $6,082.92 after 2 years.
Using the same formula and substituting the values:
P = $6,082.92
r = Annual interest rate = 8% = 0.08
n = Number of compounding periods per year = 4 (quarterly)
t = Number of years = 2
Future Value = $6,082.92 * (1 + 0.08/4)^(4*2)
= $6,082.92 * (1 + 0.02)^8
= $6,082.92 * (1.02)^8
Calculating this expression gives us:
Future Value = $6,082.92 * 1.17241841503
= $7,139.67 (rounded to the nearest cent)
Therefore, the balance in the account 2 years after the last deposit will be approximately $7,139.67.
To find the balance in the account 2 years after the last deposit, we need to calculate the future value of the deposits and interest earned.
The formula for the future value of an investment with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = Future Value
P = Principal Amount (deposit)
r = Annual Interest Rate
n = Number of times interest is compounded per year
t = Number of years
In this case, the principal amount (deposit) is $4,800, the annual interest rate is 8% (0.08), the interest is compounded quarterly, and we want to calculate the balance 2 years after the last deposit.
First, let's calculate the number of compounding periods:
Since the interest is compounded quarterly, there are 4 compounding periods in a year. So, over 3 years, the total number of compounding periods is 3 x 4 = 12.
Next, we calculate the future value of the deposits and interest earned over the 3 years:
FV = $4,800(1 + 0.08/4)^(4 x 3)
FV = $4,800(1 + 0.02)^12
FV = $4,800(1.02)^12
Using a calculator, calculate the value of (1.02)^12 and multiply it by $4,800 to find the future value of the deposits and interest earned.
FV = $4,800 x (1.02)^12
After finding the future value, we want to calculate the balance 2 years after the last deposit. Since no more deposits were made, we need to calculate the future value for an additional 2 years. We will use the same formula, but this time t will be 2.
FV = $4,800 x (1.02)^12 x (1.02)^2
Using a calculator, calculate the value of (1.02)^2 and multiply it by the previously calculated future value.
FV = $4,800 x (1.02)^12 x (1.02)^2
Finally, round the result to the nearest cent to find the balance in the account 2 years after the last deposit.