Break-Even Analysis Revision Notes for Grade 11 NSC Matric Mathematical Literacy

Break-Even Analysis

What is break-even?

Break-even point represents the exact moment when a business covers all its expenses without making a profit or loss. At this critical point, the total income from sales equals the total costs of running the business.

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Understanding break-even is essential for any business because it provides crucial financial insights that determine business viability and planning strategies.

Understanding break-even is essential for any business because it shows:

How many products must be sold to cover all costs
The minimum income needed to avoid making a loss
When the business will start generating profit

Beyond business contexts, break-even can also refer to situations where two options become equally valuable, such as comparing different cell phone contracts or electricity usage systems.

Break-even has two values

Break-even analysis always produces two related values that work together to show the complete break-even picture.

Looking at this doughnut business example, we can see how break-even works in practice:

The two break-even values are:

Horizontal axis value: Number of doughnuts (approximately 67 doughnuts)
Vertical axis value: Amount in Rand (approximately R340)

This means the business breaks even when:

67 doughnuts are sold, AND
Both income and costs equal R340

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Key insight: These two values always work as a pair. You cannot have a break-even point with just one value - you need both the quantity and the monetary amount.

Methods for determining break-even values

There are two main methods for finding break-even values:

Method 1: Graphical method

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This involves:

Drawing a graph with costs and income lines
Finding where the two lines intersect
Reading both values from the intersection point

The graph clearly shows the break-even point where the red cost line meets the blue income line.

Method 2: Trial and improvement

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This mathematical approach uses equations and intelligent guessing with the following given information for the doughnut business:

Income per doughnut = R5.00
Fixed cost (stall rental) = R250.00
Variable cost per doughnut = R1.30

Formulas:

Income = R5.00 × number of doughnuts
Total cost = R250.00 + (R1.30 × number of doughnuts)

Worked examples using trial and improvement

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Worked Example 1: Testing 100 doughnuts

Income = R5.00 × 100 = R500.00
Cost = R250.00 + (R1.30 × 100) = R250.00 + R130.00 = R380.00
Result: Income > Cost (profit situation)

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Worked Example 2: Testing 50 doughnuts

Income = R5.00 × 50 = R250.00
Cost = R250.00 + (R1.30 × 50) = R250.00 + R65.00 = R315.00
Result: Income < Cost (loss situation)

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Worked Example 3: Refining the estimate

Since 50 doughnuts gives a loss and 100 doughnuts gives a profit, the break-even point lies between these values. Through continued testing, we find that approximately 68 doughnuts will give us the break-even point.

Verification:

Income = R5.00 × 68 = R340.00
Cost = R250.00 + (R1.30 × 68) = R250.00 + R88.40 = R338.40
These values are approximately equal, confirming our break-even point.

Key formulas to remember

For any break-even analysis:

$\text{Income} = \text{Selling price per unit} \times \text{Number of units}$

$\text{Total cost} = \text{Fixed costs} + (\text{Variable cost per unit} \times \text{Number of units})$

$\text{At break-even: Income} = \text{Total cost}$

Exam tips

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Critical exam strategies:

Always express break-even as two values (quantity and amount)
Show all working when using trial and improvement
Read graph coordinates carefully for both axes
Check your answer makes sense in the business context
Remember that break-even means no profit and no loss

Summary

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Key Points to Remember:

Break-even point occurs when total income equals total costs - the business makes neither profit nor loss
Break-even always has two values - one from the horizontal axis (quantity) and one from the vertical axis (monetary amount)
Two methods exist: graphical intersection method and trial and improvement using equations
Trial and improvement requires testing different quantities until income and costs are approximately equal
Break-even analysis helps businesses determine minimum sales targets needed to avoid losses

Break-Even Analysis (Grade 11 NSC Matric Mathematical Literacy): Revision Notes