Problem 120 - Elementary and Intermediate Algebra

Question

In business, equipment is often depreciated using the double declining-balance method. In this method, a piece of equipment with a life expectancy of \(N\) years, costing \(\mathrm{SC},\) will depreciate to a value of \(\mathrm{S} V\) in \(n\) years, where \(n\) is given by the formula $$n=\frac{\log V-\log C}{\log \left(1-\frac{2}{N}\right)}$$ A computer that cost \(\$ 37,000\) has a life expectancy of 5 years. If it has depreciated to a value of \(\$ 8,000,\) how old is it?

Step-by-Step Solution

Answer

The computer is approximately 3 years old.

Step 1: Identify the Given Values

We need to determine the age of the computer denoted by variable \( n \) given its parameters. The cost of the computer \( C \) is \(37,000, the depreciated value \( V \) is \)8,000, and the life expectancy \( N \) is 5 years.

Step 2: Understand the Formula

The age \( n \) of the equipment can be calculated with the formula: \[ n = \frac{\log V - \log C}{\log \left(1 - \frac{2}{N}\right)} \] Here, \( V \) is the current value, \( C \) is the original cost, and \( N \) is the lifespan in years.

Step 3: Calculate the Logarithms

Plug the values into the formula: \[ \log V = \log 8000 \] and \[ \log C = \log 37000 \] Calculate these logarithms using a calculator to get values approximately \( \log 8000 = 3.903 \) and \( \log 37000 = 4.568 \).

Step 4: Compute the Growth Rate

Calculate the decline rate:\( 1 - \frac{2}{N} = 1 - \frac{2}{5} = 0.6 \).Now find the logarithm: \( \log(0.6) \approx -0.222 \).

Step 5: Substitute and Solve for n

Substitute the solved values into the formula:\[ n = \frac{3.903 - 4.568}{-0.222} \]Calculate the numerator and the denominator:Numerator: \( 3.903 - 4.568 = -0.665 \)Denominator: \( -0.222 \)Thus, \[ n = \frac{-0.665}{-0.222} \approx 2.996 \] which rounds to 3.

Key Concepts

Double Declining-Balance Method

The double declining-balance method is a form of accelerated depreciation. This method is used to calculate and allocate the depreciation expense of an asset more quickly in the earlier years of the asset's life. The formula focuses on twice the straight-line depreciation rate.

• It emphasizes front-loading depreciation amounts, meaning you'll take larger depreciation deductions at the beginning of the asset's life.
• After the initial larger deductions, the amounts gradually decrease over time.

This method is beneficial for equipment expected to lose more value in the early years of use, like technology and vehicles. It offers tax advantages by providing greater write-offs, helping businesses to better match expenses with revenue generation early on. Thus, when a company buys a new computer or machine, using the double declining-balance method allows the entity to report substantial depreciation expenses in the early years of owning the asset.
In essence, this approach helps ensure the business reflects the reality of an asset's more rapid depreciation in practical use.

Logarithms in Financial Calculations

Logarithms are incredibly useful in financial calculations for their ability to linearize exponential relationships. In the context of depreciation, logarithms help solve equations related to the exponential decay of the equipment's value.

• They transform multiplication or division processes into simpler addition and subtraction tasks.
• When working with depreciation, logarithms allow us to calculate the period over which value diminishes with precision.

In the solution of our exercise, we involved the use of logarithms to deduce how many years it took for the computer's value to drop from $37,000 to $8,000.
By calculating the logarithms of the starting and ending values, we could effectively manage data that exponentially shrinks. Logarithms, thus, underpin financial calculations that deal with compounding rates, growth, decay, and proportional scales, proving indispensable for accountants and financial analysts alike.

Life Expectancy of Equipment

Estimating the life expectancy of equipment is fundamental in assessing its depreciation schedule. Life expectancy refers to the estimated number of years a particular piece of equipment is expected to remain useful for its intended purpose.

• This figure is essential in calculating depreciation accurately and matching expense recognition to revenue production.
• Typically, life expectancy is determined by considering the asset's manufacturer specifications, warranty, and industry norms.

For example, in our exercise, the computer's lifespan is set to 5 years. This number impacts the rate of depreciation, directly affecting financial statements and tax filings.
An accurate life expectancy ensures that a business doesn’t overestimate or underestimate an asset’s duration of use. This accuracy allows companies to balance financial reporting, maintain efficient capital allocation, and make informed decisions about new investments or disposals.