Charlie invests $2000 in a bank for m years at 4% per annum compound interest. After m years he has $2631.86, find the value of m.

To find the value of $ m $ in the given problem, we can use the compound interest formula:

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

Here:
- $ A = 2631.86 $ (the future value)
- $ P = 2000 $ (the principal)
- $ r = 0.04 $ (the annual interest rate as a decimal)
- $ n = 1 $ (since the interest is compounded annually)
- $ t = m $ (the number of years, which we need to find)

Substitute the given values into the formula:

\[ 2631.86 = 2000 \left(1 + 0.04\right)^m \]

Simplify inside the parentheses:

\[ 2631.86 = 2000 \left(1.04\right)^m \]

Divide both sides by 2000 to isolate the exponential term:

\[ \frac{2631.86}{2000} = \left(1.04\right)^m \]

\[ 1.31593 = 1.04^m \]

Now, take the natural logarithm of both sides to solve for $ m $:

\[ \ln(1.31593) = \ln(1.04^m) \]

Using the property of logarithms $ \ln(a^b) = b \ln(a) $:

\[ \ln(1.31593) = m \ln(1.04) \]

Now, solve for $ m $:

\[ m = \frac{\ln(1.31593)}{\ln(1.04)} \]

Using a calculator to find the values of the natural logarithms:

\[ \ln(1.31593) \approx 0.2744 \]
\[ \ln(1.04) \approx 0.0392 \]

So,

\[ m = \frac{0.2744}{0.0392} \approx 7 \]

Therefore, the value of $ m $ is approximately 7 years.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How would the value of $ m $ change if the interest rate was 5% instead of 4%?
2. What is the formula for calculating the number of years required for an investment to double?
3. How does continuous compounding affect the calculation of $ m $?
4. What would the future value be if $ m $ was 10 years?
5. How do you calculate the annual interest rate if you know the principal, future value, and time period?
6. What is the difference between nominal and effective interest rates?
7. How do you determine the time period for an investment if the interest is compounded semiannually?
8. What impact does increasing the principal amount have on the time period needed to reach a certain future value?

**Tip:** When solving for time in compound interest problems, using logarithms can simplify the process significantly. Always check your logarithm calculations for accuracy.

Learn how to calculate the number of years required for $2000 invested at 4% compound interest to grow to $2631.86. This problem involves using the compound interest formula and logarithms. Suitable for high school students.

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