A car was bought at PHP 400 000. The value of the car depreciates at the rate of 10% per annum. What will be the value of the car on the 6th year? If the car will be sold on the 10th year, how much is its market value?

To solve this problem, we will use the formula for exponential depreciation, which is:

\[
V = P(1 - r)^t
\]

Where:
- \( V \) is the value of the car after \( t \) years.
- \( P \) is the original value of the car.
- \( r \) is the rate of depreciation (expressed as a decimal).
- \( t \) is the number of years.

### Step 1: Given values
- \( P = 400,000 \, \text{PHP} \)
- \( r = 0.10 \) (since 10% is the depreciation rate)
- For the 6th year, \( t = 6 \)
- For the 10th year, \( t = 10 \)

### Step 2: Calculate the value on the 6th year
\[
V_6 = 400,000(1 - 0.10)^6 = 400,000(0.90)^6
\]

### Step 3: Calculate the value on the 10th year
\[
V_{10} = 400,000(1 - 0.10)^{10} = 400,000(0.90)^{10}
\]

Let's calculate both values.### Results:
- The value of the car after 6 years will be approximately **PHP 212,576.40**.
- The value of the car after 10 years will be approximately **PHP 139,471.38**.

Would you like more details on how these calculations were done, or do you have any other questions?

Here are five related questions to further your understanding:
1. How does the value of the car change if the depreciation rate is higher or lower?
2. What if the car is sold after 8 years instead of 10? How would the value change?
3. How would the depreciation formula change if the rate of depreciation varied each year?
4. What if the depreciation rate is compounded monthly instead of annually?
5. How could you estimate the car’s value using other depreciation methods like straight-line depreciation?

**Tip:** In real-life scenarios, factors like market conditions and car model popularity can also affect depreciation rates.

Learn how to calculate the depreciated value of a car bought for PHP 400,000 with a 10% annual depreciation rate. This problem demonstrates exponential decay and includes solutions for the 6th and 10th years. Ideal for students in Grades 10-12.

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