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The images you uploaded discuss a financial investment problem involving compound interest. The goal is to compute the future value (A) for different interest rates and compounding frequencies. The formula provided is:

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

Where:
- \(A\) = future value
- \(P\) = principal (initial amount invested) = 10,000 PHP
- \(r\) = interest rate (as a decimal)
- \(t\) = time in years (15 years)
- \(k\) = number of compounding periods per year

You are given three scenarios:
1. **1.5% compounded annually** (\(r = 0.015\), \(k = 1\))
2. **1.35% compounded semi-annually** (\(r = 0.0135\), \(k = 2\))
3. **1.4% compounded quarterly** (\(r = 0.014\), \(k = 4\))

### Let's calculate the future value for each scenario.

1. **1.5% Compounded Annually:**
\[
A = 10,000 \left(1 + \frac{0.015}{1}\right)^{1 \times 15}
\]
\[
A = 10,000 (1.015)^{15} = 10,000 (1.246716) \approx 12,467.16 \, \text{PHP}
\]

2. **1.35% Compounded Semi-Annually:**
\[
A = 10,000 \left(1 + \frac{0.0135}{2}\right)^{2 \times 15}
\]
\[
A = 10,000 \left(1 + 0.00675\right)^{30} = 10,000 (1.00675)^{30} \approx 11,605.89 \, \text{PHP}
\]

3. **1.4% Compounded Quarterly:**
\[
A = 10,000 \left(1 + \frac{0.014}{4}\right)^{4 \times 15}
\]
\[
A = 10,000 \left(1 + 0.0035\right)^{60} = 10,000 (1.0035)^{60} \approx 12,249.26 \, \text{PHP}
\]

### Summary:
- **1.5% compounded annually** = **12,467.16 PHP**
- **1.35% compounded semi-annually** = **11,605.89 PHP**
- **1.4% compounded quarterly** = **12,249.26 PHP**

#### Recommendations:
The best option for your client based on these future values is **1.5% compounded annually**, as it provides the highest return after 15 years (12,467.16 PHP).

Would you like further details or have any questions about these calculations?

Here are 5 related questions to expand on this:

1. What would happen if the investment period was 10 years instead of 15?
2. How does increasing the compounding frequency impact the future value?
3. What is the difference in returns if the principal was 50,000 PHP instead?
4. Can you explain the advantage of compound interest compared to simple interest over long periods?
5. How do inflation and interest rates affect investment growth?

**Tip:** Compounding more frequently increases the future value, but the impact diminishes as the compounding periods increase.

Several banks in the Philippines offer different investment options involving compound interest. A client is looking to invest ₱10,000 for 15 years at different rates and compounding frequencies (1.5% annually, 1.35% semi-annually, 1.4% quarterly). The task is to compute the future value for each scenario using the formula A = P(1 + r/k)^(kt).

Learn how to calculate the future value of a ₱10,000 investment over 15 years using compound interest with different rates and compounding frequencies. This problem applies the compound interest formula and covers investment concepts. Suitable for high school students.

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