To solve this problem using the formulas for present value (P) and future value (F) of annuities, let's break it down:

Given:

• Annual payment $R = 1,000$ pesos.
• Interest rate $i = 6\%$ annually, but compounded quarterly, so the quarterly interest rate is: $j = \frac{6\%}{4} = 1.5\% = 0.015$
• Payment duration is 8 years, and since it's compounded quarterly, the number of periods is: $n = 8 \times 4 = 32 \text{ quarters}.$

Now, the formulas from the image are:

1. Present Value of an Annuity (P):

$P = R \times \frac{1 - (1 + j)^{-n}}{j}$ Substitute the known values: $P = 1,000 \times \frac{1 - (1 + 0.015)^{-32}}{0.015}$

First, calculate $(1 + 0.015)^{-32}$: $(1 + 0.015)^{-32} = 1.015^{-32} \approx 0.64107$

Now, subtract this from 1: $1 - 0.64107 = 0.35893$

Finally, calculate the present value $P$: $P = 1,000 \times \frac{0.35893}{0.015} \approx 1,000 \times 23.9287 = 23,928.7 \text{ pesos}.$

2. Future Value of an Annuity (F):

$F = R \times \frac{(1 + j)^n - 1}{j}$ Substitute the known values: $F = 1,000 \times \frac{(1 + 0.015)^{32} - 1}{0.015}$

First, calculate $(1 + 0.015)^{32}$: $(1.015)^{32} \approx 1.5597$

Now, subtract 1: $1.5597 - 1 = 0.5597$

Finally, calculate the future value $F$: $F = 1,000 \times \frac{0.5597}{0.015} \approx 1,000 \times 37.3133 = 37,313.3 \text{ pesos}.$

Conclusion:

• Present Value (P): 23,928.70 pesos
• Future Value (F): 37,313.30 pesos

Would you like further clarifications or a detailed explanation on a specific step?

Here are 5 questions related to annuities:

1. How does the compounding frequency affect the present and future value of an annuity?
2. What is the difference between an ordinary annuity and an annuity due?
3. How does the interest rate impact the future value of an annuity?
4. Can annuities be applied to calculate loan payments or savings plans?
5. How would the formula change if the interest rate were compounded monthly instead of quarterly?

Tip: When calculating annuities, always ensure that the interest rate matches the compounding period for accurate results.