Find the accumulated amount after 9 years if $19,000 is invested at 5.5%/year compounded in the following ways. (Round your answers to the nearest cent.)
(a) compounded annually
$ 



(b) compounded semiannually
$ 



(c) compounded quarterly
$ 



(d) compounded monthly
$ 


To solve for the accumulated amount in each case, we use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the accumulated amount (final amount)
- $P$ is the principal (initial investment)
- $r$ is the annual interest rate (in decimal form)
- $n$ is the number of compounding periods per year
- $t$ is the time the money is invested for (in years)

Given:
- $P = 19,000$
- $r = 5.5\% = 0.055$
- $t = 9$ years

### (a) Compounded annually:
Here, $n = 1$ (since compounding occurs once a year).

\[
A = 19,000 \left(1 + \frac{0.055}{1}\right)^{1 \times 9}
\]

### (b) Compounded semiannually:
Here, $n = 2$ (since compounding occurs twice a year).

\[
A = 19,000 \left(1 + \frac{0.055}{2}\right)^{2 \times 9}
\]

### (c) Compounded quarterly:
Here, $n = 4$ (since compounding occurs four times a year).

\[
A = 19,000 \left(1 + \frac{0.055}{4}\right)^{4 \times 9}
\]

### (d) Compounded monthly:
Here, $n = 12$ (since compounding occurs twelve times a year).

\[
A = 19,000 \left(1 + \frac{0.055}{12}\right)^{12 \times 9}
\]

Let's calculate the accumulated amounts for each case.Here are the accumulated amounts after 9 years for the different compounding frequencies:

(a) Compounded annually: **$30,762.79**

(b) Compounded semiannually: **$30,961.82**

(c) Compounded quarterly: **$31,064.53**

(d) Compounded monthly: **$31,134.24**

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. What happens to the accumulated amount if the interest rate is increased to 6%?
2. How does the investment grow if compounded daily instead of monthly?
3. How much interest would be earned in each case?
4. What if the investment period was 15 years instead of 9 years?
5. How does simple interest compare to compound interest over the same time period?

**Tip:** The more frequently the interest is compounded, the higher the accumulated amount will be.

Find the accumulated amount after 9 years if $19,000 is invested at 5.5%/year compounded in the following ways. (Round your answers to the nearest cent.) (a) compounded annually (b) compounded semiannually (c) compounded quarterly (d) compounded monthly

Learn how to calculate the accumulated amount of $19,000 invested for 9 years at an interest rate of 5.5% per year, compounded annually, semiannually, quarterly, and monthly. This problem covers financial mathematics and compound interest concepts, suitable for high school students in Grades 10-12.

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