You have provided an image that contains details about annuities and formulas for future value (F) and present value (P) of annuities. To help solve for the future and present values, let's break down the information provided:

Given Data:

1. Periodic Payments (R): Amounts for each scenario (a, b, c, d, e) are given.
2. Interest Rates (j): Rates are provided (2%, 3%, 4%, 5%, 10%).
3. Compounding Frequency: Depends on the payment mode (monthly, quarterly, semi-annually, annually).
4. Term (n): The number of years is provided for each annuity.

Key Formulas:

1. Future Value (F): $F = R \times \frac{(1 + j)^n - 1}{j}$
2. Present Value (P): $P = R \times \frac{1 - (1 + j)^{-n}}{j}$

Interpretation of Variables:

• R: The periodic payment.
• j: The interest rate per period. For instance, if the rate is annual but the payment is monthly, you need to divide the annual rate by the number of periods in a year.
• n: The total number of periods. This is the number of payments over the term of the annuity. For example, for a monthly payment over 3 years, $n = 3 \times 12 = 36$ months.

Applying to Each Scenario:

We can now use the formulas for each case. Here's the breakdown for each of the given options (a-e):

a. Php 2,500, 2%, Monthly, 3 years

• R = 2500 (payment)
• Annual Rate = 2%, so monthly rate = 2% / 12 = 0.0016667
• n = 3 years → 36 months
• Use the formula to find $F$ and $P$.

b. Php 3,000, 3%, Quarterly, 4 years

• R = 3000 (payment)
• Annual Rate = 3%, so quarterly rate = 3% / 4 = 0.0075
• n = 4 years → 16 quarters
• Use the formula to find $F$ and $P$.

c. Php 5,000, 4%, Quarterly, 5 years

• R = 5000 (payment)
• Annual Rate = 4%, so quarterly rate = 4% / 4 = 0.01
• n = 5 years → 20 quarters
• Use the formula to find $F$ and $P$.

d. Php 7,500, 5%, Semi-annually, 10 years

• R = 7500 (payment)
• Annual Rate = 5%, so semi-annual rate = 5% / 2 = 0.025
• n = 10 years → 20 semi-annual periods
• Use the formula to find $F$ and $P$.

e. Php 12,000, 10%, Annually, 15 years

• R = 12000 (payment)
• Annual Rate = 10%, so annual rate = 10% = 0.10
• n = 15 years → 15 periods
• Use the formula to find $F$ and $P$.

Do you want me to go ahead and calculate the future and present values for each of these scenarios?