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The problem asks us to compare the time it takes for two different investments to triple in value, one for Annabelle and one for Avery. Here's how we can break it down:

### Information:
- Annabelle's account:
  - Initial investment: $27,000
  - Interest rate: $ 4 \frac{1}{4} \% = 4.25\% $ compounded **monthly**
- Avery's account:
  - Initial investment: $27,000
  - Interest rate: $ 3 \frac{7}{8} \% = 3.875\% $ compounded **daily**

We are asked to find **how much longer it will take for Avery's investment to triple compared to Annabelle's, to the nearest hundredth of a year.**

### Formula for Compound Interest:
The formula for compound interest is:

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

Where:
- $ A $ is the final amount.
- $ P $ is the principal (initial amount).
- $ r $ is the annual interest rate in decimal form.
- $ n $ is the number of times interest is compounded per year.
- $ t $ is the time in years.

We need to solve for $ t $, the time it takes for the money to triple. This means we need to set $ A = 3P $, so:

\[
3P = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Canceling out $ P $ from both sides:

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

Now, solve for $ t $ by taking the natural logarithm on both sides:

\[
\ln(3) = nt \cdot \ln\left(1 + \frac{r}{n}\right)
\]

\[
t = \frac{\ln(3)}{n \cdot \ln\left(1 + \frac{r}{n}\right)}
\]

### For Annabelle:
- $ r = 0.0425 $
- $ n = 12 $ (since it's compounded monthly)

\[
t_{Annabelle} = \frac{\ln(3)}{12 \cdot \ln\left(1 + \frac{0.0425}{12}\right)}
\]

### For Avery:
- $ r = 0.03875 $
- $ n = 365 $ (since it's compounded daily)

\[
t_{Avery} = \frac{\ln(3)}{365 \cdot \ln\left(1 + \frac{0.03875}{365}\right)}
\]

Finally, to answer the question, we need to find the difference in time:

\[
\Delta t = t_{Avery} - t_{Annabelle}
\]

Let me compute the values.- Annabelle's money will triple in approximately **25.90 years**.
- Avery's money will triple in approximately **28.35 years**.
- The difference is about **2.46 years**.

Thus, it will take Avery about **2.46 years longer** than Annabelle for their investments to triple.

Would you like further details or have any questions? Here are 5 related questions to explore further:

1. How would the time change if the interest rate for Avery increased to 4%?
2. What happens if both accounts are compounded annually instead of monthly or daily?
3. How much time would it take for the investments to quadruple instead of triple?
4. Can you explain the effect of compounding frequency on the time to triple for both accounts?
5. How would different principal amounts affect the results?

**Tip**: Compounding frequency can have a significant effect on the growth of an investment, especially over long periods. Always consider whether interest is compounded annually, monthly, or daily.

Annabelle invested $27,000 in an account paying an interest rate of 4 1/4% compounded monthly. Avery invested $27,000 in an account paying an interest rate of 3 7/8% compounded daily. To the nearest hundredth of a year, how much longer would it take for Avery's money to triple than for Annabelle's money to triple?

Solve a financial problem involving compound interest. Compare the time it takes for investments with monthly and daily compounding interest to triple. Learn how to apply the compound interest formula and logarithms to find the solution. Suitable for high school students in Grades 10-12.

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