https://file.aiquickdraw.com/mathbot/file/1730482360237r8ftewyo.jpg

The image provided shows a compound interest problem where we need to find the annual interest rate. Here’s a breakdown of the details:

### Problem Details
- **Initial Investment (Present Value, PV)**: $16,000 (entered as -16000 in the calculator since it's an outflow).
- **Future Value (FV)**: $91,751.91.
- **Time Period**: 15 years.
- **Compounding Frequency**: Monthly (12 times per year).
- **Unknown Value**: Annual Interest Rate.

### Key Calculations
To solve for the interest rate, we can use the **Future Value formula for compound interest**:

\[
FV = PV \times \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]

Where:
- $FV$ = Future Value, $91,751.91
- $PV$ = Present Value, $16,000
- $r$ = Annual interest rate (unknown)
- $n$ = Compounding periods per year, 12
- $t$ = Time in years, 15

In this case, we solve for $r$ by rearranging the formula.

### Using the Values
1. **Set $FV$ to 91,751.91**.
2. **Set $PV$ to -16,000**.
3. **Set $N$ to 180** (15 years * 12 months).
4. **Set P/Y and C/Y to 12** (compounded monthly).
5. **Calculate the interest rate** that satisfies this equation.

### Solution
According to the image, after calculating, the answer appears to be approximately **11.7% annual interest rate**.

Would you like a detailed walkthrough of how to manually derive the annual interest rate? 

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Here are some related questions to deepen your understanding:
1. How would the interest rate change if the investment period was shorter?
2. What if the interest were compounded quarterly instead of monthly?
3. How does compounding frequency affect the growth of an investment?
4. What would be the total interest earned over the 15 years?
5. How does a change in initial investment affect the future value?

**Tip:** When using financial calculators, remember that cash outflows (like an investment) are usually entered as negative values, while inflows are positive.

You invest $16,000 in an account. The interest is compounded monthly for 15 years. The account balance will be $91,751.91 at the end of 15 years. What is the annual interest rate?

Learn how to calculate the annual interest rate required for a $16,000 investment to grow to $91,751.91 over 15 years with monthly compounding. This example uses the compound interest formula to solve for the unknown interest rate and is suitable for students in Grades 11-12 or introductory college level.

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