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?
A. $538.74
B $120,000
C $1,200.00
D $5,387.42
The formula for compound interest is given by:
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 or loan amount)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested or borrowed for
In this case, Zach deposited $4,000 (P) into an account that earns 6% (r = 0.06) annual interest compounded quarterly (n = 4). The money was invested for 5 years (t = 5).
Plugging the values into the formula, we have:
A = 4000(1 + 0.06/4)^(4*5)
Calculating the values inside the parentheses first:
1 + 0.06/4 = 1.015
Now, calculate the exponent:
4*5 = 20
Plugging the values into the formula:
A = 4000(1.015)^20
Using a calculator to calculate (1.015)^20:
A ≈ 4,871.70
Therefore, after 5 years, there will be approximately $4,871.70 in the account.
The closest answer choice to this result is D. $5,387.42.
To calculate 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 final amount of money in the account after 5 years
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal, i.e., 6% = 0.06)
n = the number of times interest is compounded per year (quarterly means 4 times a year)
t = the number of years
Given:
P = $4,000
r = 6% = 0.06
n = 4 (quarterly compounding)
t = 5 years
Let's substitute these values into the formula:
A = $4,000(1 + 0.06/4)^(4*5)
First, let's calculate the exponent:
4*5 = 20
Now, let's calculate inside the parentheses:
1 + 0.06/4 = 1.015
Now, let's calculate the value of A:
A = $4,000 * (1.015)^20
Using a calculator, the value of (1.015)^20 is approximately 1.34824122.
A = $4,000 * 1.34824122
A ≈ $5,392.97
We can round this to the nearest cent:
A ≈ $5,392.97
Therefore, the amount of money in the account after 5 years is approximately $5,392.97.
The correct answer is not among the options provided.
To calculate 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 = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, Zach deposited $4,000, the annual interest rate is 6% (or 0.06 as a decimal), and interest is compounded quarterly, so n = 4.
Plugging these values into the formula, we have:
A = 4000(1 + 0.06/4)^(4*5)
Now, let's calculate the answer.
First, calculate the value inside the parentheses:
1 + 0.06/4 = 1.015
Next, calculate the exponent:
4 * 5 = 20
Finally, substitute these values into the formula:
A = 4000 * (1.015)^(20)
Calculating this on a calculator or using a spreadsheet, we find that A ≈ 5387.42.
Therefore, the correct answer is D. $5,387.42.