You currently have $5,700 (Present Value) in an account that has an interest rate of 4.5% per year compounded semi-annually (2 times per year). You want to withdraw all your money when it reaches $10,260 (Future Value). In how many years will you be able to withdraw all your money?

The number of years is 
.
Round your answer to 1 decimal place.


To determine the number of years it will take for your account balance to grow from $5,700 to $10,260 with an interest rate of 4.5% per year compounded semi-annually, we can use the formula for compound interest:

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

where:
- $ FV $ is the future value ($10,260),
- $ PV $ is the present value ($5,700),
- $ r $ is the annual interest rate (0.045),
- $ n $ is the number of times the interest is compounded per year (2),
- $ t $ is the number of years.

We need to solve for $ t $. Rearranging the formula to solve for $ t $:

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

Substitute the given values into the formula:

\[ t = \frac{\log\left(\frac{10260}{5700}\right)}{2 \log\left(1 + \frac{0.045}{2}\right)} \]

First, calculate the fraction inside the logarithm:

\[ \frac{10260}{5700} \approx 1.8 \]

Next, calculate the term inside the logarithm for the interest rate:

\[ 1 + \frac{0.045}{2} = 1 + 0.0225 = 1.0225 \]

Now, compute the logarithms:

\[ \log(1.8) \approx 0.2553 \]
\[ \log(1.0225) \approx 0.0097 \]

Finally, substitute these values back into the formula for $ t $:

\[ t = \frac{0.2553}{2 \times 0.0097} \]
\[ t \approx \frac{0.2553}{0.0194} \]
\[ t \approx 13.16 \]

So, it will take approximately 13.2 years for your account to grow to $10,260.

If you have any questions or need further details, feel free to ask.

Here are 8 related questions:
1. What is the formula for compound interest?
2. How does semi-annual compounding affect the interest calculation compared to annual compounding?
3. What is the effective annual rate when compounded semi-annually at 4.5%?
4. How would the number of years change if the interest rate were higher?
5. What is the difference between nominal and effective interest rates?
6. How can you calculate the future value with continuous compounding?
7. What is the impact of increasing the number of compounding periods per year?
8. How would you determine the present value needed to reach a specific future value in a given number of years?

**Tip:** Always double-check your calculations when using logarithms to ensure accuracy.

Learn how to calculate the number of years required for an investment to grow from $5,700 to $10,260 with a 4.5% annual interest rate compounded semi-annually. Explore the compound interest formula and logarithmic calculations. Suitable for advanced high school students.

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