Find the required annual interest rate to the nearest tenth of a percent for ​$1300 to grow to ​$1800 if interest is compounded quarterly for 9 yr.
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Part 1
The required annual interest rate is 
  
enter your response here​%. ​(Round to the nearest tenth as​ needed.)

To solve this problem, we can use the compound interest formula:

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

Where:
- $A$ is the amount of money accumulated after interest (future value), which is $1800.
- $P$ is the principal amount (initial investment), which is $1300.
- $r$ is the annual interest rate (which we need to find).
- $n$ is the number of times the interest is compounded per year (quarterly means $n = 4$).
- $t$ is the number of years the money is invested or borrowed for, which is 9 years.

Now, plug in the given values into the formula:

\[
1800 = 1300 \left(1 + \frac{r}{4}\right)^{4 \times 9}
\]

Simplify:

\[
1800 = 1300 \left(1 + \frac{r}{4}\right)^{36}
\]

To isolate $\left(1 + \frac{r}{4}\right)$, divide both sides by 1300:

\[
\frac{1800}{1300} = \left(1 + \frac{r}{4}\right)^{36}
\]

\[
1.3846 \approx \left(1 + \frac{r}{4}\right)^{36}
\]

Next, take the 36th root of both sides:

\[
(1.3846)^{\frac{1}{36}} \approx 1 + \frac{r}{4}
\]

Use a calculator to find $(1.3846)^{\frac{1}{36}}$:

\[
1.009 \approx 1 + \frac{r}{4}
\]

Subtract 1 from both sides:

\[
0.009 \approx \frac{r}{4}
\]

Now, multiply both sides by 4:

\[
r \approx 0.036
\]

Convert this to a percentage by multiplying by 100:

\[
r \approx 3.6\%
\]

Thus, the required annual interest rate is approximately **3.6%**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the compound interest formula when interest is compounded monthly?
2. How would the interest rate change if the investment duration was shorter?
3. What would be the future value if the interest rate was 5% instead of the calculated 3.6%?
4. How does the frequency of compounding affect the final amount of an investment?
5. How is simple interest different from compound interest?

**Tip:** The more frequently interest is compounded, the higher the accumulated amount, even if the interest rate stays the same.

Find the required annual interest rate to the nearest tenth of a percent for $1300 to grow to $1800 if interest is compounded quarterly for 9 years.

Calculate the required annual interest rate to grow $1300 to $1800 with quarterly compounding over 9 years. This step-by-step guide explains the compound interest formula and shows how to isolate the rate, resulting in an interest rate of approximately 3.6%. Suitable for high school students in Grades 10-12.

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