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To solve this problem using the concept of annuities, we need to find the present value (or cash price) of the car. Mrs. Bartolome is making monthly payments of ₱10,000 for 5 years at an interest rate of 8% compounded monthly.

### Step-by-step solution:

#### Given Data:
- Monthly payment (\(PMT\)) = ₱10,000
- Number of years = 5 years
- Interest rate per year (\(r\)) = 8% = 0.08
- Payments are compounded monthly, so:
  - Number of months (\(n\)) = 5 years \(\times\) 12 months/year = 60 months
  - Monthly interest rate = \(\frac{8\%}{12} = 0.08/12 = 0.00667\) (rounded to 4 decimal places)

#### Formula for Present Value of an Ordinary Annuity:
\[
PV = PMT \times \left( \frac{1 - (1 + i)^{-n}}{i} \right)
\]
Where:
- \(PMT\) is the monthly payment
- \(i\) is the monthly interest rate
- \(n\) is the total number of payments (months)

#### Substituting the given values:
\[
PV = 10,000 \times \left( \frac{1 - (1 + 0.00667)^{-60}}{0.00667} \right)
\]

First, calculate \((1 + 0.00667)^{-60}\):
\[
(1 + 0.00667) = 1.00667
\]
\[
1.00667^{-60} = 0.69897 \quad \text{(approximately)}
\]

Now substitute this back into the equation:
\[
PV = 10,000 \times \left( \frac{1 - 0.69897}{0.00667} \right)
\]
\[
PV = 10,000 \times \left( \frac{0.30103}{0.00667} \right)
\]
\[
PV = 10,000 \times 45.136
\]
\[
PV = 451,360
\]

### The cash price of the car is approximately **₱451,360**.

Would you like more details or clarifications? 

Here are 5 related questions to further explore this topic:
1. How would the result change if the interest rate was compounded quarterly instead of monthly?
2. What is the total amount paid by Mrs. Bartolome over 5 years, including interest?
3. Can you explain the difference between the present value and future value of an annuity?
4. How would you solve this problem if the payments were made at the beginning of each month (annuity due)?
5. What would be the monthly payment if the cash price of the car was known instead?

### Tip:
When solving annuity problems, remember that the interest rate and number of periods must always match the payment frequency (monthly, quarterly, etc.).

Mrs. Bartolome buys a car and pays ₱10,000 monthly for 5 years with an interest rate of 8% compounded monthly. What is the cash price of the car?

Learn how to calculate the cash price of a car based on monthly payments of ₱10,000 over 5 years with an 8% interest rate compounded monthly. This problem uses the Present Value formula for an ordinary annuity. Ideal for senior high school students.

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