Arianna deposits $500 in an account that pays 1.3% interest, compounded semi-annually. How much is in the account at the end of two years?

To calculate the amount in Arianna's account after two years, we use the **compound interest formula**:

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

Where:
- $A$ is the amount in the account after interest.
- $P$ is the principal (initial deposit), which is $500.
- $r$ is the annual interest rate (1.3% = 0.013).
- $n$ is the number of times the interest is compounded per year (semi-annually = 2).
- $t$ is the time in years (2 years).

Substituting the given values:

\[
A = 500 \left(1 + \frac{0.013}{2}\right)^{2 \times 2}
\]

Now, let's calculate this step by step:

1. $\frac{0.013}{2} = 0.0065$
2. $1 + 0.0065 = 1.0065$
3. $2 \times 2 = 4$, so $1.0065^4$
4. Multiply the result by 500.

I'll compute the result for you.The amount in Arianna's account at the end of two years is approximately **$513.13**.

Do you need further details or have any questions?

Here are some related questions for practice:
1. What would the amount be after 5 years at the same interest rate?
2. How much interest is earned in total after 2 years?
3. What happens if the interest is compounded quarterly instead of semi-annually?
4. How much would be in the account after 3 years at 2% interest compounded annually?
5. If the interest rate increases to 1.5%, what would be the balance after 2 years?

**Tip:** Always check if the compounding period aligns with the frequency mentioned in the problem (e.g., semi-annually = twice a year).

Learn how to calculate the compound interest on $500 deposited in an account paying 1.3% interest, compounded semi-annually, over two years. This problem demonstrates the use of the compound interest formula and is suitable for students in Grades 9-12.

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