Suppose $5000 is deposited in a bank account that compounds interest four times per year. The bank account contains $9900 after 13 years. What is the annual interest rate for this bank account?
i already set it up but i don't know where to go from there. Please help
To find the annual interest rate for this bank account, you can use the formula for compound interest:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
where:
- A is the future value of the account
- P is the initial deposit (principal)
- r is the annual interest rate (in decimal form)
- n is the number of times the interest is compounded per year
- t is the number of years
In this case, the initial deposit is $5000, and the future value is $9900. The interest is compounded four times per year, so n = 4. And the account was held for 13 years, so t = 13.
The equation becomes:
\[9900 = 5000\left(1 + \frac{r}{4}\right)^{4 \times 13}\]
To solve for r, you need to isolate it on one side of the equation. Here's how you can proceed:
1. Divide both sides of the equation by $5000:
\[\frac{9900}{5000} = \left(1 + \frac{r}{4}\right)^{4 \times 13}\]
2. Take the 13th root of both sides:
\[\left(\frac{9900}{5000}\right)^{\frac{1}{13}} = 1 + \frac{r}{4}\]
3. Subtract 1 from both sides:
\[\left(\frac{9900}{5000}\right)^{\frac{1}{13}} - 1 = \frac{r}{4}\]
4. Multiply both sides by 4:
\[4 \times \left(\frac{9900}{5000}\right)^{\frac{1}{13}} - 4 = r\]
Now, you can use a calculator to evaluate the expression on the right-hand side to find the annual interest rate.