A loan for a machine shop costs P60,000 if paid in cash. On the installment plan, a purchaser should pay P20,000 down payment and 10 quarterly installments, the first due at the end of the first quarter after purchase. If money is worth r% compounded quarterly same as the rate of interest 8.24% compounded annually. Find the quarterly installment and construct an amortization schedule.

To solve this problem, we need to determine the **quarterly installment** amount and construct an **amortization schedule**. The key steps are:

1. **Translate the interest rate**: Since the annual interest rate is 8.24%, compounded quarterly, we need to convert this into a quarterly rate.
   - **Annual rate \( r = 8.24\% \)**.
   - Convert this to a **quarterly rate** \( i \). The formula for the quarterly interest rate is:
     \[
     i = \frac{r}{4} = \frac{8.24\%}{4} = 2.06\% = 0.0206
     \]

2. **Determine the loan amount that needs to be paid in installments**: 
   The total price of the machine is **P60,000**, but the purchaser makes a **P20,000 down payment**. So, the amount to be financed through installments is:
   \[
   \text{Loan principal} = P60,000 - P20,000 = P40,000
   \]

3. **Quarterly installment computation**: The quarterly payment can be computed using the **annuity formula** for installments:
   \[
   A = \frac{P \cdot i}{1 - (1 + i)^{-n}}
   \]
   where:
   - \( A \) is the quarterly installment,
   - \( P \) is the loan amount \( P = 40,000 \),
   - \( i \) is the quarterly interest rate \( 0.0206 \),
   - \( n \) is the number of payments \( 10 \).

Let's calculate the quarterly installment using this formula.The **quarterly installment** is approximately **P4,467.05**.

### Amortization Schedule Construction:
To build an amortization schedule, we need to track how much of each payment goes toward the interest and how much reduces the principal.

The steps are:
1. Start with the **loan balance** \( P40,000 \).
2. For each quarter:
   - Calculate the **interest** for the period, which is the loan balance multiplied by the quarterly interest rate.
   - Subtract the interest from the quarterly installment to determine the **principal repayment**.
   - Subtract the principal repayment from the loan balance to update it for the next period.

Let's calculate the full amortization schedule for the 10 installments.Here is the **amortization schedule** for the 10 quarterly installments:

| Payment | Quarterly Installment | Interest | Principal | Remaining Balance |
|---------|-----------------------|----------|-----------|-------------------|
| 1       | P4,467.05              | P824.00  | P3,643.05 | P36,356.95        |
| 2       | P4,467.05              | P748.95  | P3,718.10 | P32,638.85        |
| 3       | P4,467.05              | P672.36  | P3,794.69 | P28,844.16        |
| 4       | P4,467.05              | P594.19  | P3,872.86 | P24,971.30        |
| 5       | P4,467.05              | P514.41  | P3,952.64 | P21,018.65        |
| 6       | P4,467.05              | P432.98  | P4,034.07 | P16,984.58        |
| 7       | P4,467.05              | P349.88  | P4,117.17 | P12,867.42        |
| 8       | P4,467.05              | P265.07  | P4,201.98 | P8,665.43         |
| 9       | P4,467.05              | P178.51  | P4,288.54 | P4,376.89         |
| 10      | P4,467.05              | P90.16   | P4,376.89 | P0.00             |

Each payment is **P4,467.05**, with varying amounts of interest and principal repaid each quarter.

Do you want further details or explanations? Here are five related questions to explore:

1. How does the interest portion of each payment change over time?
2. Can we calculate the total interest paid over the entire loan period?
3. How would the amortization change if the interest rate were different?
4. What would happen if more or fewer installments were chosen?
5. How does this method compare with continuous compounding?

**Tip:** To reduce the total interest paid on a loan, you can opt to make extra payments towards the principal.

A loan for a machine shop costs P60,000 if paid in cash. On the installment plan, a purchaser should pay P20,000 down payment and 10 quarterly installments, the first due at the end of the first quarter after purchase. If money is worth r% compounded quarterly same as the rate of interest 8.24% compounded annually, find the quarterly installment and construct an amortization schedule.

Learn how to calculate the quarterly installment for a P60,000 loan with a P20,000 down payment and 10 quarterly installments at 8.24% annual interest compounded quarterly. Includes a detailed amortization schedule. Suitable for finance students and advanced high school math learners.

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