A debt of P200,000 whose interest is 15% compounded quarterly is to be paid a series of 12 quarterly payments. The first 6 payments will be P30,000 each while the remaining 6 payments will be equal to liquidate the debt. What will be the annuity for the last 6 payments?

To solve this problem, we will calculate the size of the equal annuity payments for the last 6 payments. Here's the step-by-step process:

### Step 1: Break down the problem
- **Principal (debt)**: P200,000
- **Interest rate**: 15% compounded quarterly, which gives us a quarterly interest rate of:
  \[
  i = \frac{15\%}{4} = 3.75\% = 0.0375
  \]
- **First 6 payments**: P30,000 each.
- **Remaining debt**: After 6 payments, the remaining debt will be paid with 6 equal annuity payments.

### Step 2: Determine the balance after the first 6 payments
We will use the **present value of an annuity** formula to calculate how much of the debt is paid off by the first 6 payments and how much remains.

The formula for the present value of an annuity is:
\[
PV = R \times \frac{1 - (1 + i)^{-n}}{i}
\]
Where:
- \( PV \) is the present value of the annuity,
- \( R \) is the periodic payment (P30,000),
- \( i \) is the interest rate per period (0.0375),
- \( n \) is the number of periods (6).

Substituting the values:
\[
PV_{first6} = 30,000 \times \frac{1 - (1 + 0.0375)^{-6}}{0.0375}
\]

Calculating this gives the total value of the first 6 payments in today's terms (present value).

### Step 3: Calculate the remaining balance
The remaining balance after the first 6 payments is the original debt minus the present value of the first 6 payments.

\[
Remaining\,Balance = 200,000 - PV_{first6}
\]

### Step 4: Determine the annuity for the last 6 payments
Now that we know the remaining balance, we can calculate the annuity for the last 6 payments. Again, we use the formula for the present value of an annuity, but this time we solve for \( R \) (the annuity payment) for the remaining balance over 6 periods.

The formula becomes:
\[
Remaining\,Balance = R \times \frac{1 - (1 + i)^{-6}}{i}
\]

Solving for \( R \):
\[
R = \frac{Remaining\,Balance}{\frac{1 - (1 + i)^{-6}}{i}}
\]

Substitute the values of \( Remaining\,Balance \) and \( i \) (0.0375) into the equation to calculate \( R \), which is the amount of each of the final 6 payments.

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Would you like me to calculate each step explicitly and provide the exact values?

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### Relative questions:
1. How would the payments change if the interest rate were compounded monthly instead of quarterly?
2. What if the initial debt was P250,000 with the same structure of payments?
3. How would the payment plan change if the first 6 payments were smaller (e.g., P25,000)?
4. What happens to the annuity if the interest rate is variable over the payment periods?
5. How does the number of payments (e.g., 8 instead of 6 for the final payments) impact the annuity?

### Tip:
When dealing with compound interest problems, always be careful about the compounding period and ensure the interest rate matches the payment period.

Learn how to calculate the annuity for the last 6 payments of a P200,000 debt with 15% interest compounded quarterly. The first 6 payments are P30,000 each, and this problem involves key concepts such as compound interest and present value of annuities. Step-by-step solution provided.

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