Problem 22 - Algebra 1: Concepts and Skills

Question

You buy a used truck for 20,000 dollar. The truck depreciates 7% per year. Find the value of the truck after the given number of years. $$3 years$$

Step-by-Step Solution

Answer

The value of the truck after 3 years is approximately $16087.14.

Step 1: Understand the depreciation rate

Here, we are given a depreciation rate of 7%. In decimal format, this is 0.07.

Step 2: Use the compound depreciation formula

Value after depreciation = Original price * (1 - Depreciation rate)^Number of years. Substituting the given values: Value after 3 years = 20000 * (1 - 0.07)^3.

Step 3: Calculate the value

Perform the calculation. Start by subtracting the rate from 1 to get: 1 - 0.07 = 0.93. Then, calculate 0.93^3 = 0.804357. Lastly multiply this result with the original price: 20000 * 0.804357 = $16087.14.

Key Concepts

Depreciation

Depreciation is the process by which an asset loses value over time. For instance, when you buy a car, its worth typically decreases every year. Understanding why and how depreciation happens is crucial. The rate of depreciation is often expressed as a percentage. In our example, the truck has a depreciation rate of 7% per year.
To measure depreciation, we use a formula:

• New Value = Original Value \( \times (1 - \text{Depreciation Rate})^{n} \) where \( n \) is the number of years.

The above formula helps us calculate how much an asset is worth after a certain number of years. In our case, after 3 years, the truck's value will be adjusted down because of its depreciation.
It's essential to grasp this concept since it assists in understanding asset management, especially when considering buying or selling property.

Compound Interest

Unlike depreciation, compound interest is more about earning rather than losing value. It's a process that allows interest to be added to the original principal, and then each successive period's interest is calculated based on the new principal. This is why compound interest can grow an investment more quickly than simple interest.
When thinking about compound interest, it relies heavily on percentages like depreciation. Here, too, percentages define what amount is added over each period of time. Although solving problems about depreciation of cars and compound interest might seem different, they both use a similar formula structure:

• Determine initial value (principal for interest calculations)
• Apply a rate expressed as a percentage
• Calculate over a set number of periods

This shared use of exponential formulas underpins why thorough comprehension of both concepts is beneficial in personal finance management methods.

Percentages

Percentages are a mathematical way to express proportions. They are crucial in situations involving depreciation and compound interest. A percentage is just a number or rate expressed as a fraction of 100. For our truck example, a 7% depreciation means the truck loses 7% of its value each year.
Working with percentages involves simple yet important steps:

• Convert the percentage to a decimal for calculations by dividing by 100 (e.g., 7% becomes 0.07).
• Use this decimal in formulas to find the effect of depreciation or interest over time.

Percentages may also help us understand more complex financial situations, making them a powerful tool in various decision-making processes. Mastery of percentages can simplify both personal and professional financial analysis.