mr.akira buys a tv set with a quarterly payment of ₱2,000 at the end of each term for 5 years with an interest rate of 8% compounded annually. how much is the tv set after 5 years?? find its annuities and solve it

To find the cost of the TV set using the given information, we will calculate the present value of the annuities. The problem involves an annuity where regular payments are made (₱2,000 per quarter for 5 years), and the interest is compounded annually at 8%.

### Key Information:
- **Quarterly payment**: ₱2,000
- **Number of years**: 5
- **Interest rate**: 8% compounded annually
- **Number of payments per year**: 4 (since payments are made quarterly)
- **Total number of payments**: \( 5 \times 4 = 20 \) payments
- **Interest per period (quarterly rate)**: Since the interest is compounded annually but payments are quarterly, we need to convert the annual interest rate to a quarterly rate.

  \[
  \text{Quarterly interest rate} = \frac{8\%}{4} = 2\% \text{ per quarter or } 0.02 \text{ in decimal form}.
  \]

### Present Value of Annuity Formula:
The present value \( PV \) of an ordinary annuity (payments made at the end of each period) is given by the formula:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- \( P \) is the regular payment (₱2,000),
- \( r \) is the interest rate per period (0.02),
- \( n \) is the total number of periods (20).

### Substituting the values:

\[
PV = 2000 \times \frac{1 - (1 + 0.02)^{-20}}{0.02}
\]

### Step-by-step calculation:
1. Calculate \( (1 + 0.02)^{-20} \):
   \[
   (1 + 0.02)^{-20} = 1.02^{-20} \approx 0.6730
   \]

2. Subtract this from 1:
   \[
   1 - 0.6730 = 0.3270
   \]

3. Divide by the interest rate per period (0.02):
   \[
   \frac{0.3270}{0.02} = 16.35
   \]

4. Multiply by the payment amount (₱2,000):
   \[
   PV = 2000 \times 16.35 = 32,700
   \]

### Final Answer:
The present value (cost of the TV set) is **₱32,700**.

This is the total cost of the TV set after 5 years, considering the payments and interest.

---

Would you like more details or further explanation?

Here are 5 related questions to expand your understanding:
1. How does changing the interest rate affect the present value of an annuity?
2. What would happen if the payments were made monthly instead of quarterly?
3. How is the formula for present value derived from compound interest principles?
4. Can you use the future value formula for this problem instead of present value? Why or why not?
5. How would the total cost change if the interest were compounded quarterly instead of annually?

**Tip**: Always ensure you adjust the interest rate to match the payment frequency (e.g., quarterly, monthly) to avoid incorrect calculations.

Mr. Akira buys a TV set with a quarterly payment of ₱2,000 at the end of each term for 5 years with an interest rate of 8% compounded annually. How much is the TV set after 5 years? Find its annuities and solve it.

Learn how to determine the cost of a TV set purchased with quarterly payments of ₱2,000 for 5 years at an 8% annual interest rate. This solution explores annuities, the present value formula, and the time value of money. Suitable for students in Grades 10-12.

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