Problem 58 - Algebra 2

Question

Car 1 cost \(\$ 22,600\) when new and depreciated 14\(\%\) each year for 5 years. The same year, Car 2 cost \(\$ 17,500\) when new and depreciated 7\(\%\) each year for 5 years. To the nearest dollar, what was the difference in the values of the two cars after 5 years?

Step-by-Step Solution

Answer

The difference in value between Car 1 and Car 2 after 5 years of depreciations is calculated as the absolute difference of the depreciated values of both cars.

Step 1: Calculate the depreciation of Car 1 for 5 years

Firstly, it is required to calculate the depreciated value of Car 1 after 5 years. Since the car is depreciating, the value is decreasing by 14\(\%\) each year. Hence, for 5 years, the depreciation can be calculated using the formula \(D = P(1 - r)^n\), where \(D\) is the depreciated value, \(P\) is the initial value, \(r\) is the rate of depreciation and \(n\) is the number of years. Here, \(P = \$ 22,600\), \(r = 14 / 100 = 0.14\), and \(n = 5\). Substituting these values, \(D = \$ 22,600 * (1 - 0.14) ^ 5\).

Step 2: Calculate the depreciation of Car 2 for 5 years

Next, do the same calculations as Step 1 for Car 2. Initially, the cost of Car 2 was \$17,500, and it is depreciating 7\(\%\) per year for the same 5 years. So, \(P = \$17,500\), \(r = 7 / 100 = 0.07\), and \(n = 5\). So the depreciated value of Car 2 after 5 years can be calculated as \(D = \$17,500 * (1 - 0.07) ^ 5\).

Step 3: Calculate the difference in the depreciated values

Once the depreciated values of both cars are obtained, the final step is to calculate their difference. Take the absolute difference between those two amounts to get the desired result. The calculation for this will be \(|D_1 - D_2|\), where \(D_1\) and \(D_2\) are the depreciated values of Car 1 and Car 2, respectively.

Key Concepts

Calculating Depreciation

Depreciation represents the loss in value of an asset over time. It is often expressed as a percentage. In real-life scenarios like the one with the two cars, calculating depreciation helps us understand how much the asset is worth after a certain period.

To calculate depreciation, we apply a specific formula that considers the initial value, the percentage rate of decay, and the number of years the asset is depreciating. Use this method to get a comprehensive view of your asset's future value:

• Identify the initial purchase price (the original cost of the asset).
• Determine the annual depreciation rate (as a percentage).
• Decide the time period over which depreciation occurs.

By multiplying the initial value with the depreciation factor over the time period, you get the depreciated value today.

Percentage Decrease

Percentage decrease is a way to express the reduction in value as a percentage of the original amount. This concept is crucial in understanding depreciation because the asset loses a specific percentage of its value each year.

To calculate the percentage decrease for one year, subtract the depreciation rate from 1, Then you multiply by 100 to convert it to percentage form. In the car example:

• Car 1 loses 14% per year, hence over five years the compounded effect is considered.
• Car 2 loses 7% per year with a similar approach.

The compounded decreases over multiple years explain how quickly an asset can lose value.

Exponential Decay

Exponential decay describes how an asset reduces in value over time multiplicatively rather than additively. The significance of this concept lies in understanding that with each passing year, the rate acts on a progressively smaller base value.

Put another way, each year's depreciation compounds on the remaining value, not the original. For example, if an asset depreciates by 10% annually, it does not lose the same absolute amount each year. Instead, it loses 10% of its new, lower value. This calculation becomes crucial in long-term asset management and financial planning.

Depreciation Formula

The depreciation formula allows us to estimate the value of an asset over a set period of time given an annual depreciation rate. The general formula for exponential depreciation is: \[ D = P(1 - r)^n \] where:

• \( D \) is the depreciated value after time \(n\)
• \( P \) is the initial purchase price
• \( r \) is the depreciation rate (as a decimal)
• \( n \) is the number of time periods (usually years)

By plugging the appropriate values into this formula, you can gain insight into how much your asset is expected to be worth after a certain number of years. This understanding helps illustrate why the depreciated value of assets is less than expected when considering only straight-line depreciation without accounting for the compound effects of annual percentage decrease.