Problem 95 - Intermediate Algebra

Question

Accounting. As a piece of equipment gets older, its value usually lessens. One way to calculate depreciation is to use the formula $$V=C-\left(\frac{C-S}{L}\right) N$$ where \(V\) denotes the value of the equipment at the end of year \(N, L\) is its useful lifetime (in years), \(C\) is its cost new, and \(S\) is its salvage value at the end of its useful life. a. Solve the formula for \(L\) b. Determine what an accountant considered the useful lifetime of a forklift that cost \(\$ 25,000\) new, was worth \(\$ 13,000\) after 4 years, and has a salvage value of \(\$ 1,000\).

Step-by-Step Solution

Answer

a. Solve for \( L \) as \( L = \frac{N(C-S)}{C-V} \). b. The forklift's useful lifetime is 8 years.

Step 1: Rearrange the Formula to Solve for L

Start with the formula for depreciation: \[ V = C - \left(\frac{C-S}{L}\right) N \]Rearrange it to isolate \( L \) on one side of the equation. First, subtract \( V \) from \( C \) on both sides:\[ C - V = \left(\frac{C-S}{L}\right) N \]Next, multiply both sides by \( L \) to eliminate the fraction:\[ L(C - V) = N(C-S) \]Finally, divide both sides by \( (C - V) \) to solve for \( L \):\[ L = \frac{N(C-S)}{C-V} \]

Step 2: Identify Given Values for the Forklift Problem

From the problem, we identify the following values:- Cost new, \( C = \\(25,000\)- Salvage value, \( S = \\)1,000\)- Forklift value after 4 years, \( V = \$13,000\)- Time, \( N = 4 \) years.

Step 3: Plug Values into the Rearranged Equation

Using the formula derived in Step 1:\[ L = \frac{N(C-S)}{C-V} \]Substitute the known values into the equation:\[ L = \frac{4(25,000-1,000)}{25,000-13,000} \]

Step 4: Calculate the Useful Lifetime, L

Simplify the equation:\[ L = \frac{4 \times 24,000}{12,000} \]Calculate \( L \):\[ L = \frac{96,000}{12,000} = 8 \]Therefore, the useful lifetime of the forklift is 8 years.

Key Concepts

Depreciation Formula

Depreciation helps us understand how the value of an asset, like equipment, decreases over time. It’s crucial in accounting to predict how much an item will be worth in the future. The basic formula for depreciation allows us to calculate the current value of an item:
\[ V = C - \left(\frac{C-S}{L}\right) N \]
Here’s a simple way to look at these parts:

• \(V\): Current value of the equipment.
• \(C\): Cost of the equipment when it was new.
• \(S\): Salvage value, or what you’d get if you sold it at its end of life.
• \(L\): Useful lifetime in years.
• \(N\): The number of years since purchasing.

The formula essentially tells us how much value is lost each year due to usage and passage of time. Understanding this can help in making decisions about when to replace old equipment and budget for future purchases.

Rearranging Equations

In algebra, rearranging equations is a common technique to find unknown variables. With our depreciation formula, we needed to find \(L\), the useful lifetime:
First, by subtracting the remaining value \(V\) from the original cost \(C\), we express how much of the initial value has been used up:\[ C - V = \left(\frac{C-S}{L}\right) N \]
Next, multiply through by \(L\) to eliminate the fraction. This isolates \(L\) on one side of the equation and makes it easier to solve:\[ L(C - V) = N(C-S) \]Finally, divide both sides by \(C - V\) to solve for \(L\):\[ L = \frac{N(C-S)}{C-V} \]These steps might seem complicated at first, but getting comfortable with rearranging formulas will make solving many algebra problems easier in the long run. The trick is to take it step by step and ensure each action keeps the equation balanced.

Useful Lifetime Calculation

The useful lifetime of an asset is the total time it can effectively perform its intended function before it’s not worthwhile keeping it. By determining this period using our formula for \(L\), accountants can schedule replacements and budget appropriately.
Given values for the forklift were:

• Initial cost, \(C = \\(25,000\).
• Salvage value, \(S = \\)1,000\).
• Value after 4 years, \(V = \$13,000\).
• Time, \(N = 4\) years.

Plug into the rearranged depreciation formula:\[ L = \frac{4(25,000-1,000)}{25,000-13,000} \]
Simplify:\[ L = \frac{96,000}{12,000} \]Calculating gives:\[ L = 8 \]So, the forklift has a useful lifetime of 8 years. Understanding this helps businesses plan for future expenses and ensures equipment is utilized efficiently before it becomes impractical to maintain.