Problem 42 - Algebra 2

Question

COMPUTERS Suppose a computer that costs \(\$ 3000\) new is only worth \(\$ 600\) after 3 years. What is the average annual rate of depreciation?

Step-by-Step Solution

Answer

The average annual rate of depreciation is approximately 26.67%.

Step 1: Identify Initial and Final Values

Start by identifying the initial cost of the computer, which is given as $3000. The final value after depreciation over 3 years is $600. We will use these values to find the average annual rate of depreciation.

Step 2: Calculate Total Depreciation Amount

Subtract the final value from the initial value to find the total depreciation amount over the 3-year period.\[\text{Total Depreciation} = 3000 - 600 = 2400\]

Step 3: Compute Annual Depreciation

Divide the total depreciation amount by the number of years to find the annual depreciation in dollar terms.\[\text{Annual Depreciation} = \frac{2400}{3} = 800\]

Step 4: Calculate Average Annual Rate of Depreciation

Divide the annual depreciation amount by the initial value to determine the average annual rate of depreciation, then convert it to a percentage.\[\text{Average Annual Rate of Depreciation} = \frac{800}{3000} \approx 0.2667 \text{ or } 26.67\%\]

Key Concepts

Initial Value

The initial value is the original cost or price of an asset at the time of purchase. In this example, the initial value of the computer is $3,000. Understanding the initial value is crucial because it serves as the starting point for calculating depreciation. Depreciation helps determine how much value the asset loses over time.

This initial cost is important in various calculations like depreciation, resale value, and accounting purposes. When acquiring any asset, documenting this initial cost is essential for future financial assessments and reports.

Final Value

The final value refers to the worth of an asset after a certain period, considering factors like depreciation. In the given problem, the final value of the computer after 3 years is $600. This is significantly less than the initial value, highlighting the effect of depreciation over time.

Determining the final value requires you to consider several factors, such as the asset's condition and market demand. In depreciation calculations, the final value is used to measure how much worth has been lost and helps in understanding the asset's declining value pattern.

Depreciation Calculation

Depreciation calculation is the process of determining how much an asset’s value has decreased over time. The first step in calculating depreciation is to find the total depreciation amount. This is done by subtracting the final value from the initial value. In the exercise, this looks like:

• Total Depreciation = Initial Value - Final Value
• For the computer: 3000 - 600 = 2400

Next, you must calculate the annual depreciation. This is the total depreciation divided by the number of years. For a 3-year period, the annual depreciation is:

• Annual Depreciation = \(\frac{2400}{3} = 800\)

This method ensures you understand how the asset depreciated each year and helps in financial planning and tax calculations.

Annual Depreciation Rate

The annual depreciation rate is the percentage by which an asset’s value decreases each year. This rate is useful for understanding the speed at which an asset is losing value. To find this rate, divide the annual depreciation by the initial value and convert it into a percentage. Here, for the computer:

• Annual Depreciation Rate = \(\frac{800}{3000} \approx 0.2667 \)
• Convert to percentage: \(0.2667 \times 100\% = 26.67\%\)

This means that each year, the computer loses about 26.67% of its original value. Knowledge of the annual depreciation rate is pivotal for budgeting, asset management, and financial projections.