Suppose $10,000 is deposited into a savings account earning 2% interest compounded quarterly. Find the balance in the account 5 years, rounded to the nearest cent.
The formula for the balance in an account with interest compounded quarterly is \[A = P \left(1 + \frac{r}{n}\right)^{nt},\] where $A$ is the final amount (the balance), $P$ is the principal (the initial amount), $r$ is the annual interest rate (expressed as a decimal), $n$ is the number of times per year interest is compounded, and $t$ is the number of years. Substituting, we have \[A = 10000 \left(1 + \frac{0.02}{4}\right)^{4 \cdot 5} = 10000 \left(1 + 0.005\right)^{20} \approx \boxed{\$11,\!048.45}.\]
To find the balance in the savings account after 5 years with quarterly compounding, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount/balance in the account
P = principal/initial amount (in this case, $10,000)
r = annual interest rate (2% or 0.02 in decimal form)
n = number of times interest is compounded per year (quarterly means 4 times a year)
t = number of years
Plugging in the given values into the formula, we have:
A = 10000(1 + 0.02/4)^(4 * 5)
Simplifying the formula:
A = 10000(1 + 0.005)^(20)
A = 10000(1.005)^(20)
Using a calculator, we can evaluate this expression:
A ≈ 10000 * 1.104622
Rounding the answer to the nearest cent, we get:
A ≈ $11,046.22
Therefore, the balance in the savings account after 5 years, rounded to the nearest cent, is $11,046.22.