In 2021, a bank account is started with an initial deposit of
$7,500.00. The interest rate on this account was 2.25%. Find
the amount in this account after 10 years if interest is
compounded:
a. yearly
b. monthly
c. continuously

To calculate the amount in the account after 10 years, we will use different formulas depending on how the interest is compounded. Here are the formulas for each case:

### a) **Yearly Compounding**:
For yearly compounding, the formula is:

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

Where:
- $ A $ is the amount after time $ t $.
- $ P $ is the principal (initial deposit) = 7,500.
- $ r $ is the annual interest rate = 2.25% = 0.0225.
- $ n $ is the number of times interest is compounded per year. Since it is yearly compounding, $ n = 1 $.
- $ t $ is the time in years = 10.

\[
A = 7500 \left(1 + \frac{0.0225}{1}\right)^{1 \times 10}
\]
\[
A = 7500 \left(1.0225\right)^{10}
\]
\[
A \approx 7500 \times 1.2516 = 9,387.23
\]

So, after 10 years with yearly compounding, the amount will be **$9,387.23**.

### b) **Monthly Compounding**:
For monthly compounding, we use the same formula, but now $ n = 12 $ (since the interest is compounded monthly):

\[
A = 7500 \left(1 + \frac{0.0225}{12}\right)^{12 \times 10}
\]
\[
A = 7500 \left(1 + 0.001875\right)^{120}
\]
\[
A = 7500 \left(1.001875\right)^{120}
\]
\[
A \approx 7500 \times 1.2531 = 9,398.38
\]

So, after 10 years with monthly compounding, the amount will be **$9,398.38**.

### c) **Continuous Compounding**:
For continuous compounding, we use the formula:

\[
A = Pe^{rt}
\]

Where:
- $ e $ is Euler’s number (approximately 2.71828),
- $ r = 0.0225 $,
- $ t = 10 $.

\[
A = 7500 \times e^{0.0225 \times 10}
\]
\[
A = 7500 \times e^{0.225} \approx 7500 \times 1.2528 = 9,396.00
\]

So, after 10 years with continuous compounding, the amount will be **$9,396.00**.

### Summary of Results:
- **Yearly Compounding**: $9,387.23
- **Monthly Compounding**: $9,398.38
- **Continuous Compounding**: $9,396.00

---

Would you like more details on any specific calculation? Here are some related questions to deepen your understanding:

1. What if the interest rate was higher, say 3%? How would the amounts change?
2. How would the results differ if the account was only held for 5 years?
3. What is the difference between continuous and periodic compounding in terms of long-term growth?
4. What would the balance be with daily compounding instead of monthly or yearly?
5. How does compound interest differ from simple interest, and how do the calculations compare?

**Tip**: Continuous compounding generally yields slightly higher returns over time because interest is added at every instant!

In 2021, a bank account is started with an initial deposit of $7,500.00. The interest rate on this account was 2.25%. Find the amount in this account after 10 years if interest is compounded: a. yearly, b. monthly, c. continuously.

Find out how to calculate the amount in a bank account after 10 years with an initial deposit of $7,500 and a 2.25% interest rate, compounded yearly, monthly, and continuously. This guide uses essential compound interest formulas and explores exponential growth. Suitable for students in Grades 10-12.

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