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Chapter 7: Problem 59

Break-Even Analysis In Exercises 57 and 58 , find the sales necessary to break even \((R=C)\) for the total cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Break-Even Analysis A small software company invests \(\$ 16,000\) to produce a software package that will sell for \(\$ 55.95 .\) Each unit costs \(\$ 9.45\) to produce. (a) How many units must the company sell to break even? (b) How many units must the company sell to make a profit of \(\$ 100,000 ?\)

Short Answer

Expert verified
To break even, the company must sell 344 units. To make a profit of $100,000, the company must sell 2494 units.

Step by step solution

01

Identify the relevant parameters

The fixed costs equal $16,000. The variable cost per unit is $9.45. The selling price per unit is $55.95. The revenue obtained from selling \(x\) units at $55.95 per unit is \(55.95x\). The total cost of producing \(x\) units is \(16,000+9.45x\).
02

Set up the break-even equation for part (a)

The break-even point is where total revenue equals total cost. So, set up the equation like this: \(55.95x = 16,000+9.45x\).
03

Solve the break-even equation for part (a)

In order to solve the equation, subtract \(9.45x\) from both sides to obtain \(46.5x = 16,000\). Roughly, solving for \(x\) gives \(x = 344\). Rounding this figure to the nearest whole number gives 344 units. Therefore, the company must sell 344 packages to break even.
04

Set up profit equation for part (b)

The company wants to make a profit of $100,000. So, the revenue minus the total cost should equal $100,000: \(55.95x - (16,000+9.45x) = 100,000\).
05

Solve the profit equation for part (b)

With the same approach, subtract \(9.45x\) and $16,000 from both sides to simplify the equation, resulting in \(46.5x = 116,000\). Solving this equation gives \(x=2494\). Rounding to the nearest whole unit, the company needs to sell 2494 units to achieve a profit of $100,000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Costs
Fixed costs are expenses that remain constant, regardless of how many units you produce. These costs occur no matter what, even if you don鈥檛 sell a single unit. In our example, the fixed costs are set at $16,000.
This expense is probably related to necessities like equipment, software development fees, or initial setup costs.
Understanding these costs is crucial because they impact pricing strategy and budget planning. Unlike variable costs, fixed costs don鈥檛 fluctuate with production levels.
This stability helps companies in financial forecasting and budgeting. Ensuring you account for all fixed costs when pricing a product helps maintain financial health.
Variable Costs
Variable costs change with the number of units produced. Every additional unit brings extra cost. In the exercise, for every software package produced, the variable cost is $9.45.
These costs might include raw materials or labor costs, things which directly correlate with production.
As your production quantity increases, so does your total variable cost. This is important for projections; as sales increase, so do expenses.
When calculating the break-even point, it's crucial to consider these costs, as they can significantly influence overall profitability. Understanding and managing variable costs helps companies optimize their profit margins.
Revenue Calculation
Revenue is the income generated from selling products or services. In the given example, each software package is sold for $55.95. If you sell \(x\) packages, the revenue is \(R = 55.95x\).
Revenue calculations are fundamental in determining the break-even point. By setting revenue equal to total costs, you can find out how many units must be sold to cover all expenses.
For the break-even point, the revenue must equal total costs (fixed plus variable). Solving the equation \(55.95x = 16,000 + 9.45x\) determines the minimum sales required to avoid losses.
This calculation helps businesses plan and set sales targets needed to achieve profitability.

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