Problem 44 - Prealgebra and Introductory Algebra

Question

Business Break-even analysis is a method used to determine the sales volume required for a company to "break even," or experience neither a profit nor a loss on the sale of its product. The break-even point represents the number of units that must be made and sold for income from sales to equal the cost of producing the product. The break-even point can be calculated using the formula \(B=\frac{F}{S-V},\) where \(F\) is the fixed costs, \(S\) is the selling price per unit, and \(V\) is the variable costs per unit. a. Solve the formula \(B=\frac{F}{S-V}\) for \(S\) b. Use your answer to part (a) to find the selling price per button pinhole video spycam required for a company to break even. The fixed costs are \(\$ 15,000,\) the variable costs per spycam are \(\$ 60,\) and the company plans to make and sell 200 spycams. c. Use your answer to part (a) to find the selling price per spy camera video lighter required for a company to break even. The fixed costs are \(\$ 18,000,\) the variable costs per lighter are \(\$ 65,\) and the company plans to make and sell 600 lighters.

Step-by-Step Solution

Answer

a) The rearranged formula is \(S=\frac{F}{B} + V\). b) The selling price per button pinhole video spycam required to break even is $135. c) The selling price per spy camera video lighter required to break even is $95.

Step 1: Rearrange the break-even formula

The formula \(B=\frac{F}{S-V}\) needs to be rearranged for \(S\). Multiply both sides by \(S-V\), so it becomes \(B(S-V)=F\). Then, distribute \(B\) on the left side of the equation, which results in \(BS-BV=F\). Re-arrange for \(S\), we end up with \(S=\frac{F}{B} + V\).

Step 2: Calculate selling price for button pinhole video spycam

Use the found formula to calculate \(S\) for the button pinhole video spycam. The fixed costs \(F = $ 15,000\), variable costs \(V = $ 60\) and the planned units \(B = 200\). Substituting these values into the formula, \(S=\frac{$15,000}{200} + $60 =$ 135.\)

Step 3: Calculate selling price for spy camera video lighter

Now we calculate \(S\) for the spy camera video lighter. Fixed costs \(F = $ 18,000\), variable costs \(V = $ 65\) and the planned units \(B = 600\). Substituting these values into the revised formula, \(S=\frac{$18,000}{600} + $65 =$ 95.\)

Key Concepts

Business Mathematics

Break-even analysis is a core concept in business mathematics. It helps businesses determine the point where they make no profit or loss. In essence, it identifies the selling price needed for a company to cover all its costs given specific sales targets and costs structures.

Here are key steps in break-even analysis:

• Identify all fixed costs, which remain unchanged regardless of the production volume.
• Identify variable costs per unit, which fluctuate depending on the number of units produced.
• Apply the formula to determine the sales volume or price required to achieve a break-even point.

Business mathematics involves using precise formulas to make strategic decisions, like determining prices or costs. It's an invaluable tool for ensuring business activities are financially viable.

Cost Accounting

Cost accounting focuses on the detailed tracking and analysis of all costs incurred by a business. It plays an essential role in break-even analysis, as it helps to classify costs into fixed and variable categories.

- Fixed costs include expenses like rent, salaries, and insurance, which do not change with production levels.
- Variable costs are expenses that vary directly with production volume, such as raw materials and direct labor.

By understanding these costs, businesses can make better financial decisions. Cost accounting provides the necessary data to perform the calculations required for break-even analysis, ensuring that all relevant costs are accounted for in determining the break-even point.

Mathematical Modeling

Mathematical modeling in break-even analysis involves using mathematical expressions to represent business scenarios. By developing an equation for break-even, businesses can simulate various pricing and cost scenarios to forecast financial outcomes.

The break-even formula \(B=\frac{F}{S-V}\) is a model that translates complex financial relationships into a simpler mathematical form.

This model allows companies to:

• Alter variables like costs or sales volume to see potential outcomes.
• Determine potential profits or losses at different production scales.
• Strategically plan pricing or production volume initiatives.

Mathematical modeling is crucial as it provides a clearer picture of how different business variables interact and affect overall financial performance.

Algebraic Manipulation

Algebraic manipulation entails rearranging equations to isolate a particular variable. This is a valuable skill for solving problems in break-even analysis. In the original exercise, the goal was to solve the break-even formula \(B=\frac{F}{S-V}\) for \(S\).

To achieve this, one must:

• Multiply both sides by \(S-V\) to eliminate the denominator.
• Distribute \(B\) across \(S-V\).
• Rearrange terms into the form \(S = \frac{F}{B} + V\).

Algebraic manipulation allows businesses to adapt existing mathematical models to derive new relationships. Mastering this skill ensures you can efficiently solve equations to inform decision-making in financial planning and analysis.