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To solve this problem using the annuity formulas provided in the image, we will calculate both the future value (F) and the present value (P) for each scenario in the table.

### The formulas for future value (F) and present value (P) of an annuity are:
\[
F = R \cdot \frac{(1 + j)^n - 1}{j}
\]
\[
P = R \cdot \frac{1 - (1 + j)^{-n}}{j}
\]

Where:
- \(R\) is the periodic payment amount.
- \(j\) is the interest rate per period (rate divided by the number of compounding periods per year).
- \(n\) is the total number of periods (\(n = \text{{number of years}} \times \text{{number of compounding periods per year}}\)).

### Step-by-step for each case:

#### **a. Php 2,500, 2%, Monthly, 3 years:**
- \(R = 2,500\)
- Rate = 2% annually, so \(j = \frac{2\%}{12} = \frac{0.02}{12} \approx 0.00167\)
- Term = 3 years, compounded monthly, so \(n = 3 \times 12 = 36\)

#### **b. Php 3,000, 3%, Quarterly, 4 years:**
- \(R = 3,000\)
- Rate = 3% annually, so \(j = \frac{3\%}{4} = \frac{0.03}{4} = 0.0075\)
- Term = 4 years, compounded quarterly, so \(n = 4 \times 4 = 16\)

#### **c. Php 5,000, 4%, Quarterly, 5 years:**
- \(R = 5,000\)
- Rate = 4% annually, so \(j = \frac{4\%}{4} = \frac{0.04}{4} = 0.01\)
- Term = 5 years, compounded quarterly, so \(n = 5 \times 4 = 20\)

#### **d. Php 7,500, 5%, Semi-annually, 10 years:**
- \(R = 7,500\)
- Rate = 5% annually, so \(j = \frac{5\%}{2} = \frac{0.05}{2} = 0.025\)
- Term = 10 years, compounded semi-annually, so \(n = 10 \times 2 = 20\)

#### **e. Php 12,000, 10%, Annually, 15 years:**
- \(R = 12,000\)
- Rate = 10% annually, so \(j = \frac{10\%}{1} = 0.10\)
- Term = 15 years, compounded annually, so \(n = 15 \times 1 = 15\)

I will now calculate the future value and present value for each of these scenarios.Here are the calculated future value (F) and present value (P) for each case:

#### **a. Php 2,500, 2%, Monthly, 3 years:**
- **Future Value (F):** Php 92,675.27
- **Present Value (P):** Php 87,282.64

#### **b. Php 3,000, 3%, Quarterly, 4 years:**
- **Future Value (F):** Php 50,796.85
- **Present Value (P):** Php 45,072.94

#### **c. Php 5,000, 4%, Quarterly, 5 years:**
- **Future Value (F):** Php 110,095.02
- **Present Value (P):** Php 90,227.76

#### **d. Php 7,500, 5%, Semi-annually, 10 years:**
- **Future Value (F):** Php 191,584.93
- **Present Value (P):** Php 116,918.72

#### **e. Php 12,000, 10%, Annually, 15 years:**
- **Future Value (F):** Php 381,269.78
- **Present Value (P):** Php 91,272.95

Would you like further details on any calculation or explanation? Here are some questions to extend your understanding:

1. How does the frequency of compounding affect the future and present values?
2. What would happen if the payment frequency changed from monthly to annually in case (a)?
3. How would an increase in the interest rate affect the future value for case (b)?
4. How does extending the payment term by 5 years influence the future value in case (d)?
5. What are the practical applications of future value and present value in financial planning?

**Tip:** A higher number of compounding periods leads to greater future values due to the effect of compound interest.

Learn how to solve annuities with compound interest using future value (F) and present value (P) formulas. This solution includes step-by-step calculations for different cases, including monthly, quarterly, semi-annual, and annual payments. Key financial formulas are explored with examples for various interest rates and terms.

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