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To solve this problem, we need to calculate the **present value (PV)** of each offer using the formula for compound interest and annuity, considering a 3% annual interest rate compounded annually. Let's break down each offer:

### Offer from Buyer X:
- Buyer X offers ₱80,000 now, and ₱900,000 in a lump-sum payment after 4 years.
- We need to compute the present value of the ₱900,000 lump-sum payment.

The **Present Value (PV)** formula for a lump sum is:

\[
PV = \frac{FV}{(1 + r)^t}
\]

Where:
- \(FV = ₱900,000\) (future value),
- \(r = 0.03\) (3% annual interest rate),
- \(t = 4\) (years).

The present value of ₱900,000 after 4 years is:

\[
PV_X = \frac{900,000}{(1 + 0.03)^4} = \frac{900,000}{1.1255} \approx ₱799,200
\]

Adding the immediate ₱80,000, the total present value of Buyer X's offer is:

\[
PV_{\text{total}_X} = 80,000 + 799,200 = ₱879,200
\]

### Offer from Buyer Y:
- Buyer Y offers ₱90,000 now and ₱35,000 every quarter for 4 years.
  
To find the total present value of Buyer Y’s offer, we need to calculate:
1. The present value of the ₱90,000 immediate payment.
2. The present value of the quarterly payments of ₱35,000 over 4 years.

#### Immediate Payment of ₱90,000:
This is already in present value, so:

\[
PV_{\text{90,000}} = ₱90,000
\]

#### Quarterly Payments:
Since there are quarterly payments over 4 years, there will be \(4 \times 4 = 16\) payments of ₱35,000. The interest rate per quarter is \(\frac{3\%}{4} = 0.0075\).

We use the **Present Value of an Annuity** formula:

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

Where:
- \(P = ₱35,000\),
- \(r = 0.0075\) (quarterly interest rate),
- \(n = 16\) (number of quarters).

\[
PV_{\text{quarterly}} = 35,000 \times \frac{1 - (1 + 0.0075)^{-16}}{0.0075} 
\]

\[
PV_{\text{quarterly}} = 35,000 \times \frac{1 - 0.88849}{0.0075} 
\]

\[
PV_{\text{quarterly}} = 35,000 \times \frac{0.11151}{0.0075} \approx ₱520,380
\]

#### Total Present Value of Buyer Y’s Offer:
Adding the present value of the immediate ₱90,000 and the quarterly payments:

\[
PV_{\text{total}_Y} = 90,000 + 520,380 = ₱610,380
\]

### Comparison:
- Present value of Buyer X's offer: ₱879,200
- Present value of Buyer Y's offer: ₱610,380

Thus, **Buyer X’s offer has a higher market value**.

---

Would you like further details on any step? Here are 5 related questions to consider:

1. How does the interest rate affect the present value in financial decisions?
2. What is the difference between the present value of a lump sum and an annuity?
3. How would a higher or lower interest rate change the outcome of the comparison?
4. What are the uses of the present value formula in everyday financial decisions?
5. How does compounding frequency (e.g., annually vs quarterly) impact future and present values?

**Tip:** When comparing offers that include future payments, always calculate their present value to make an accurate comparison.

Your father received two offers on a lot that he wants to sell. Buyer X has offered ₱80,000 and a ₱900,000 lump-sum payment 4 years from now. Buyer Y has offered ₱90,000 plus ₱35,000 every quarter for 4 years. Compare the fair market values of the two offers if money can earn 3% compounded annually. Which offer has a higher market value?

Solve a problem involving two different real estate offers, one involving a lump sum and the other involving quarterly payments. Use present value and annuity formulas with a 3% interest rate compounded annually to determine which offer has a higher market value.

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