What rate of interest compounded daily doubles an investment in 8 years?  

Round your answer to 2 decimal places. 

We will use the standard 365 days in a calendar year.

We can solve this using the formula for compound interest:

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

Where:
- \( A \) is the final amount (which is double the initial investment, so \( A = 2P \)),
- \( P \) is the principal (initial investment),
- \( r \) is the annual interest rate (which we need to find),
- \( n \) is the number of times the interest is compounded per year (for daily compounding, \( n = 365 \)),
- \( t \) is the time in years (in this case, \( t = 8 \)).

We want to find \( r \) such that the investment doubles, so:

\[
2P = P \left(1 + \frac{r}{365}\right)^{365 \times 8}
\]

Cancel out \( P \) from both sides:

\[
2 = \left(1 + \frac{r}{365}\right)^{2920}
\]

Now, we solve for \( r \). Let's take the natural logarithm (ln) of both sides to make the exponent manageable:

\[
\ln(2) = \ln\left[\left(1 + \frac{r}{365}\right)^{2920}\right]
\]

Using the logarithmic identity \( \ln(a^b) = b \ln(a) \):

\[
\ln(2) = 2920 \ln\left(1 + \frac{r}{365}\right)
\]

Now solve for \( \ln\left(1 + \frac{r}{365}\right) \):

\[
\frac{\ln(2)}{2920} = \ln\left(1 + \frac{r}{365}\right)
\]

Exponentiate both sides to remove the natural logarithm:

\[
e^{\frac{\ln(2)}{2920}} = 1 + \frac{r}{365}
\]

Now subtract 1 from both sides:

\[
e^{\frac{\ln(2)}{2920}} - 1 = \frac{r}{365}
\]

Finally, multiply by 365 to solve for \( r \):

\[
r = 365 \left(e^{\frac{\ln(2)}{2920}} - 1\right)
\]

I'll calculate the value of \( r \).The rate of interest compounded daily that doubles an investment in 8 years is approximately **8.67%** per year.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the frequency of compounding affect the interest rate required to double an investment?
2. What would the interest rate be if compounding were monthly instead of daily?
3. How long would it take to triple the investment at the same 8.67% interest rate compounded daily?
4. What is the difference between simple and compound interest in terms of growth rate?
5. How is the continuous compounding formula related to the daily compounding formula?

**Tip**: The more frequently interest is compounded, the less interest is required annually to achieve the same future value.

What rate of interest compounded daily doubles an investment in 8 years? Round your answer to 2 decimal places. We will use the standard 365 days in a calendar year.

Find out the interest rate compounded daily that doubles an investment in 8 years using the compound interest formula. Learn the step-by-step process to solve this financial math problem, including logarithms and exponential growth. Suitable for Grades 10-12.

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