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solve this using annuities,show the solutions 

Let's solve this problem using the annuities formula provided in the image. We are given the following information:

- Annual payments (\(R\)) = ₱1,000.00
- Term = 8 years
- Interest rate = 6% compounded quarterly
- We need to find both the **Present Value (P)** and **Future Value (F)**.

### Step 1: Break down the interest rate and number of periods
Since the interest is compounded quarterly, we need to adjust both the interest rate and the number of compounding periods:

- **Quarterly interest rate**: \(j = \frac{6\%}{4} = \frac{0.06}{4} = 0.015\)
- **Number of periods**: \(n = 8 \times 4 = 32\) quarters

### Step 2: Use the provided formulas
The formulas in the image are as follows:

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

### Step 3: Calculate the Present Value
Let's substitute the values into the present value formula:
\[
P = 1,000 \times \frac{1 - (1 + 0.015)^{-32}}{0.015}
\]
First, calculate \(1 + 0.015 = 1.015\), then raise it to the power of \(-32\):
\[
1.015^{-32} = 0.63196
\]
Now substitute back:
\[
P = 1,000 \times \frac{1 - 0.63196}{0.015}
= 1,000 \times \frac{0.36804}{0.015}
= 1,000 \times 24.536
= 24,536.05
\]
So, the **Present Value (P)** is ₱24,536.05.

### Step 4: Calculate the Future Value
Substitute the values into the future value formula:
\[
F = 1,000 \times \frac{(1 + 0.015)^{32} - 1}{0.015}
\]
First, calculate \(1.015^{32} = 1.61449\), then subtract 1:
\[
1.61449 - 1 = 0.61449
\]
Now substitute back:
\[
F = 1,000 \times \frac{0.61449}{0.015}
= 1,000 \times 40.966
= 40,966.14
\]
So, the **Future Value (F)** is ₱40,966.14.

### Summary of Results:
- **Present Value (P)**: ₱24,536.05
- **Future Value (F)**: ₱40,966.14

Would you like further explanation on any step?

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Here are five questions for you to explore further:

1. How would the values change if the interest rate was compounded monthly instead of quarterly?
2. What is the effect of increasing the number of years on the future value?
3. Can you derive the formulas for annuities based on the definition of compounding interest?
4. How would the results change if the payment was made at the beginning of each period (annuity due)?
5. What is the significance of using the present value formula in financial planning?

**Tip:** When solving annuity problems, always ensure that the interest rate and the number of periods are consistent in terms of compounding frequency (monthly, quarterly, etc.).

Annual payments of ₱1,000.00 at the end of each term for 8 years with interest rate of 6% compounded quarterly. Find the present and the future value.

Solve the problem of finding the present and future value of annuities with annual payments of ₱1,000.00 for 8 years at an interest rate of 6% compounded quarterly. Learn how to use the annuity formulas to compute the values, breaking down each step for easy understanding. Suitable for students in Grades 10-12.

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