Problem 55 - Precalculus

Question

A couple buy a new bedroom set for \(\$ 8,000\) and 10 years later sell it for \(\$ 4,000 .\) If the depreciation continues at the same rate, how much would the bedroom set be worth in 4 more years?

Step-by-Step Solution

Answer

The bedroom set would be worth \( \$2,400 \) in 4 more years.

Step 1: Understand Depreciation

Depreciation refers to how an asset loses value over time. In this case, the bedroom set depreciates from \( \\(8,000 \) to \( \\)4,000 \) over 10 years. We need to find the annual depreciation rate.

Step 2: Calculate Annual Depreciation

To find the annual depreciation, subtract the final value from the initial value and then divide by the number of years: \[\text{Annual Depreciation} = \frac{\\(8,000 - \\)4,000}{10} = \frac{\\(4,000}{10} = \\)400\]So, the bedroom set depreciates \(\$400\) per year.

Step 3: Calculate Value After 14 Years

Starting from the 10-year value of \(\\(4,000\), subtract \(\\)400\) per year for 4 more years: \[\text{Value after 14 years} = \\(4,000 - (4 \times \\)400) = \\(4,000 - \\)1,600 = \$2,400\]This is the value of the bedroom set after 14 years.

Key Concepts

Annual Depreciation Rate

Depreciation is an important concept in finance that refers to how an asset decreases in value over time. The annual depreciation rate helps to determine how much value an asset loses each year.
To calculate this, start by finding the difference between the asset's initial value and its value after a certain number of years. This difference is then divided by the number of years to find out how much value is lost each year.
In our exercise, the bedroom set was initially worth $8,000 and was sold for $4,000 after 10 years. The calculation for annual depreciation looks like this:

• Subtract the final value from the initial value: $8,000 - $4,000 = $4,000.
• Divide that by the number of years: $4,000 / 10 = $400 per year.

This means the bedroom set loses $400 in value annually. Understanding this rate is vital for asset management and planning future asset purchases.

Asset Value Calculation

Determining the future value of an asset is key for both personal and business financial planning. To calculate the future value of an asset after a certain number of years, you use the annual depreciation rate you've determined.
The process involves subtracting the total depreciation over the additional years from the current value of the asset.
Continuing with our bedroom set example, we know it depreciates at $400 per year. After 10 years, it was valued at $4,000. For the next 4 years, the depreciation is calculated by:

• Multiply the annual depreciation by the number of additional years: $400 imes 4 = $1,600.
• Subtract this amount from the value after 10 years: $4,000 - $1,600 = $2,400.

Thus, the bedroom set's value after 14 years will be $2,400. This systematic calculation of asset value helps individuals and businesses estimate asset worth, plan sales, or assess their financial health.

Linear Depreciation Method

The linear depreciation method, also known as the straight-line method, is one of the simplest ways to calculate how an asset loses value over time. This method assumes that the asset depreciates by the same amount every year, resulting in a straight line if graphed.
It's commonly used due to its straightforwardness and ease of calculation, making it an ideal choice for personal asset management or businesses with predictable depreciation.
In the original exercise, we applied the linear depreciation method. The set's value decreased by a fixed amount of $400 each year, calculated as follows:

• Determine the total depreciation over the asset's life.
• Divide this by the asset's useful life span in years.

This approach gives a clear and constant annual depreciation figure, helping owners to easily predict future asset values. By knowing this method, users can apply it to other depreciating assets, aiding in informed decision-making for buying, selling, and financial reporting.