Zach deposited $4,000 into an account that earns 6% annual interest compounded quarterly. He did not make any deposits or withdrawals. How much money was in the account after 5 years?
The interest rate is 6% per year, so the quarterly interest rate is 6%/4 = 1.5% per quarter.
After 5 years, there are 5*4 = <<5*4=20>>20 quarters.
The interest earned for each quarter is 4000*$0.015 = $<<4000*0.015=60>>60.
The principal amount is $4000.
After 5 years, the account will have 4000+ 60*20 = $<<4000+60*20=5200>>5200. Answer: \boxed{5200}.
To calculate how much money is in the account after 5 years with 6% annual interest compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years the money is invested/loaned for
In this case:
P = $4,000
r = 6% = 0.06 (in decimal form)
n = 4 (compounded quarterly)
t = 5 years
Using this information, let's calculate the future value of the investment:
A = 4000(1 + 0.06/4)^(4*5)
Step 1: Solve for (1 + 0.06/4):
1 + 0.06/4 = 1.015
A = 4000(1.015)^(20)
Step 2: Compute (1.015)^(20):
(1.015)^(20) ≈ 1.348178659
A ≈ 4000(1.348178659)
Step 3: Calculate 4000(1.348178659):
A ≈ $5,392.71
Therefore, after 5 years, there will be approximately $5,392.71 in the account.
To find the amount of money in the account after 5 years, we can 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 amount (the initial deposit)
r = the annual interest rate (written as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, the principal investment amount (P) is $4,000, the annual interest rate (r) is 6% (0.06 as a decimal), the interest is compounded quarterly (n = 4), and the investment is made for 5 years (t = 5).
Plugging these values into the formula:
A = 4000(1 + 0.06/4)^(4*5)
First, divide the annual interest rate (0.06) by the number of times interest is compounded per year (4), and then add 1 to it. This gives us (1 + 0.06/4).
Next, multiply the number of times interest is compounded per year (4) by the number of years (5), which gives us (4*5).
Finally, raise the result from the first step to the power of the result from the second step, giving us the future value of the investment (A).
Calculating this:
A = 4000(1 + 0.015)^(20)
A = 4000(1.015)^(20)
Using a calculator or spreadsheet, calculate (1.015)^(20) and we get 1.3382251426.
Multiply this value by the initial deposit of $4,000:
A = 4000 * 1.3382251426
A ≈ $5,352.90
Therefore, there would be approximately $5,352.90 in the account after 5 years.