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The puzzle in the image is about Julia adding $500 to an account with a 5% interest rate compounded daily. We are tasked with answering several questions based on this.

Let’s break down the questions one by one using the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $A$ is the amount of money in the account after time $t$,
- $P$ is the principal amount (initial deposit),
- $r$ is the annual interest rate (in decimal),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the time in years.

Here:
- $P = 500$,
- $r = 0.05$,
- $n = 365$ (compounded daily),
- We will vary $t$ based on the specific question.

### 1. **How much interest will accrue in her account after 5 years?**
We need to find the value of the account after 5 years and subtract the initial principal.

\[
A = 500 \left(1 + \frac{0.05}{365}\right)^{365 \times 5}
\]

I will calculate this.After 5 years, the account will have $642.00, and the interest accrued will be approximately **$142.00**.

### 2. **Will her account's value double by year 10?**
For this, we check if the account value at year 10 is at least $1000.

\[
A = 500 \left(1 + \frac{0.05}{365}\right)^{365 \times 10}
\]

Let me calculate this now.After 10 years, the account will be worth approximately **$824.33**, which is less than double the original $500. So, the answer is **NO**, it will not double by year 10.

### 3. **What will her account be worth in 20 years?**
Let’s calculate the value after 20 years.

\[
A = 500 \left(1 + \frac{0.05}{365}\right)^{365 \times 20}
\] 

I will calculate this now.After 20 years, the account will be worth approximately **$1359.05**.

### 4. **Will her account's value quadruple by year 30?**
For this, we check if the account value at year 30 reaches at least $2000.

\[
A = 500 \left(1 + \frac{0.05}{365}\right)^{365 \times 30}
\]

I will calculate this now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the account's value at 30 years using the same compound interest formula. If you have any more questions or need further clarification, feel free to ask!

### Related questions:
1. How would the value change if the interest was compounded monthly instead of daily?
2. What if Julia deposited $1000 instead of $500 with the same interest rate?
3. How much time would it take for the account to double at this rate?
4. What would be the final amount if the interest rate was 6% instead of 5%?
5. How does compound interest differ from simple interest in this context?

**Tip**: Compounding more frequently (like daily) increases the total amount slightly more than less frequent compounding.

Julia added $500 to an account paying a 5% interest rate, compounded daily. How much interest will accrue in her account after 5 years? Will her account double by year 10? What will her account be worth in 20 years? Will her account quadruple by year 30?

Solve a compound interest problem where Julia adds $500 to an account with a 5% interest rate compounded daily. Find out how much interest accrues in 5 years, whether the account doubles in 10 years, and more. This problem demonstrates how compound interest works over time using daily compounding.

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