A car depreciates **$1680.00** the first year, and **$475** each year thereafter.
How many years will it take a **$22000.00** car to depreciate such that its price becomes first time **less than a half of the original price**?

To find out how many years it will take for a $22000.00 car to depreciate to less than half its original price, we need to calculate the depreciation each year and track the car's value over time.

Given:
- Original price of the car $ P_0 = \$22000.00 $
- Depreciation in the first year $ D_1 = \$1680.00 $
- Depreciation each subsequent year $ D = \$475.00 $

Half of the original price is:
\[ \frac{P_0}{2} = \frac{22000.00}{2} = 11000.00 \]

We will determine the number of years $ n $ it takes for the car's value to drop below $11000.00.

### Year-by-Year Depreciation Calculation:
1. **First Year:**
   \[ P_1 = P_0 - D_1 = 22000.00 - 1680.00 = 20320.00 \]

2. **Subsequent Years:**
   For each following year $ n \geq 2 $:
   \[ P_n = P_{n-1} - D \]

Let's calculate this iteratively until the car's value is less than $11000.00:

- **Year 2:**
  \[ P_2 = 20320.00 - 475.00 = 19845.00 \]

- **Year 3:**
  \[ P_3 = 19845.00 - 475.00 = 19370.00 \]

- **Year 4:**
  \[ P_4 = 19370.00 - 475.00 = 18895.00 \]

- **Year 5:**
  \[ P_5 = 18895.00 - 475.00 = 18420.00 \]

- **Year 6:**
  \[ P_6 = 18420.00 - 475.00 = 17945.00 \]

- **Year 7:**
  \[ P_7 = 17945.00 - 475.00 = 17470.00 \]

- **Year 8:**
  \[ P_8 = 17470.00 - 475.00 = 16995.00 \]

- **Year 9:**
  \[ P_9 = 16995.00 - 475.00 = 16520.00 \]

- **Year 10:**
  \[ P_{10} = 16520.00 - 475.00 = 16045.00 \]

- **Year 11:**
  \[ P_{11} = 16045.00 - 475.00 = 15570.00 \]

- **Year 12:**
  \[ P_{12} = 15570.00 - 475.00 = 15095.00 \]

- **Year 13:**
  \[ P_{13} = 15095.00 - 475.00 = 14620.00 \]

- **Year 14:**
  \[ P_{14} = 14620.00 - 475.00 = 14145.00 \]

- **Year 15:**
  \[ P_{15} = 14145.00 - 475.00 = 13670.00 \]

- **Year 16:**
  \[ P_{16} = 13670.00 - 475.00 = 13195.00 \]

- **Year 17:**
  \[ P_{17} = 13195.00 - 475.00 = 12720.00 \]

- **Year 18:**
  \[ P_{18} = 12720.00 - 475.00 = 12245.00 \]

- **Year 19:**
  \[ P_{19} = 12245.00 - 475.00 = 11770.00 \]

- **Year 20:**
  \[ P_{20} = 11770.00 - 475.00 = 11295.00 \]

- **Year 21:**
  \[ P_{21} = 11295.00 - 475.00 = 10820.00 \]

After 21 years, the value of the car drops to $10820.00, which is less than half of the original price. Therefore, it will take **21 years** for the car's value to become less than half of its original price.

Do you want more details or have any questions?

Here are some related questions:
1. How would the depreciation change if the first year's depreciation was $2000.00?
2. What is the value of the car after 10 years?
3. How does compound depreciation differ from linear depreciation?
4. How would the total depreciation after 21 years be calculated?
5. If the car's value depreciated at a constant rate of 5% per year, how many years would it take to reach half the original value?
6. What is the mathematical formula to calculate the depreciation for each year?
7. How does the concept of depreciation apply to real estate properties?
8. Can you calculate the future value of the car using an exponential decay model?

**Tip:** Understanding linear and exponential depreciation is crucial for accurately assessing the long-term value of assets.

Learn how to calculate car depreciation over time using arithmetic principles. Find out how many years it will take for a $22000.00 car to depreciate to less than half its original price of $22000.00. Suitable for Grades 7-9.

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