Exercises: Revenue and cost forecasting: the break-even point

Explain in your own words what the break-even point is, and what selling above or below it means.

The break-even formula is $q^* = \frac{CF}{p - CV_u}$. Identify what each term represents (CF, p, CV_u) and explain the meaning of the denominator, known as the unit contribution margin.

The figure represents total revenue (IT) and total costs (CT) of a business as a function of units sold, with fixed costs (CF) as a benchmark. Explain what the point marked with '?' represents and what occurs in the loss and profit regions.

A florist's has fixed costs of $1.500\ \text{€}$ a month, sells each bouquet at $15\ \text{€}$ and it costs her $9\ \text{€}$ to prepare it. a) Calculate the monthly break-even point with the formula $q^* = \frac{CF}{p - CV_u}$. b) Explain what happens with the result if she sells exactly that number of bouquets.

A company has a break-even point of 800 units a month. Explain, without doing calculations, what result it will obtain if it sells 650 units, if it sells 800 and if it sells 1,000, reasoning from the meaning of the break-even point.

A burger food truck has these monthly figures: space rental of $800\ \text{€}$, the cook's salary of $1.400\ \text{€}$, ingredients at $4\ \text{€}$ per burger, packaging at $1\ \text{€}$ per burger, a selling price of $11\ \text{€}$ and insurance which, spread out, comes to $200\ \text{€}$ a month. a) Calculate the monthly fixed costs and the variable cost per unit. b) Calculate the break-even point.

Define the unit contribution margin and calculate it for a product that sells at $25\ \text{€}$ and costs $16\ \text{€}$ to manufacture. Explain what that margin "contributes" to within the company and what happens with it once the fixed costs are covered.

The figure shows a company's total revenue (IT) and total costs (CT) according to units sold, with fixed costs (CF) as a reference and the loss and profit zones shaded. a) State what the point marked with "?" represents. b) Estimate, by reading the axes of the graph, how many units the company must sell to neither gain nor lose. c) Reason, looking at the graph, whether the company makes or loses money when it sells 60 units.

A company has fixed costs of $6.000\ \text{€}$ per month, sells its product for $30\ \text{€}$ per unit, and each unit costs $18\ \text{€}$ to produce. a) Calculate the monthly break-even point using the formula $q^* = \frac{CF}{p - CV_u}$. b) Explain what this number means for the company.

The break-even point for a shop is 500 T-shirts per month. In June it sold 400 and in July 620. Indicate in which month it made a loss and in which a profit, reason why it is not necessary to know the exact amounts to know this, and explain what each T-shirt sold above 500 represents.

A bakery has a break-even point of 2 000 cakes per year. Analyse three levers to lower it — raising the price, reducing fixed costs, and reducing unit variable cost — explaining how each one acts on the formula and what risk using it entails.

A backpack factory has fixed costs of $18.000\ \text{€}$ a month, sells each backpack at $45\ \text{€}$ and the variable cost per unit is $25\ \text{€}$. a) Calculate the monthly break-even point. b) Calculate the month's profit if it sells 1 200 backpacks: $B = (p - CV_u) \cdot q - CF$. c) Explain what each backpack sold above the break-even point contributes to the result.

A publishing house has fixed costs of $60.000\ \text{€}$ a year, sells each book at $24\ \text{€}$ and the variable cost per unit is $14\ \text{€}$. a) Calculate the break-even point in books. b) Calculate the break-even point in euros, that is, the sales figure at which revenue and costs are equal: $IT^* = p \cdot q^*$. c) Comment on what information each of the two measures provides.

An academy has fixed costs of $9.000\ \text{€}$ a month, charges $120\ \text{€}$ per course and bears a variable cost of $60\ \text{€}$ per student. a) Calculate its break-even point. b) After a rent rise, the fixed costs become $12.000\ \text{€}$: calculate the new break-even point and comment on the change. c) Reason what other two decisions could offset the rise in fixed costs and how each would act on the formula.

A company needs 8 000 components a year and is undecided between manufacturing them, with fixed costs of $30.000\ \text{€}$ and a variable cost per unit of $2\ \text{€}$, or buying them from a supplier at $6\ \text{€}$ per unit. a) Calculate the annual cost of each option for that volume. b) Calculate at what annual volume both options would cost the same. c) Reason how the company should use that figure to decide between making and buying.

An artisan ice cream parlour opens 300 days a year. Its annual fixed costs are $54.000\ \text{€}$, it sells an ice cream at $4\ \text{€}$ and each one costs it $2,5\ \text{€}$. a) Calculate the annual break-even point in ice creams. b) Translate that break-even point into ice creams per opening day. c) Comment on whether the daily figure seems achievable and what data about the premises and the area you would like to know to assess it.

A coffee shop is open 25 days a month with 12 four-seater tables and offers a single lunch shift. Its monthly fixed costs are $4.500\ \text{€}$, the price per set menu is $12\ \text{€}$, and the variable cost per menu is $7\ \text{€}$. a) Calculate the break-even point in set menus per month and the average table occupancy rate needed to avoid incurring losses. b) If the demand forecast for the next month is $60\ \%$ of its maximum capacity, the business would incur losses. To prevent this, two alternatives are considered: 1) Offer a premium set menu at $15\ \text{€}$ with a unit variable cost of $8,5\ \text{€}$, maintaining fixed costs. 2) Reduce fixed costs by $20\ \%$ by renegotiating the rent, keeping the initial price and variable cost. Calculate the new break-even point for each alternative and justify which one allows the business to be profitable with the forecasted demand.

An entrepreneur plans to sell T-shirts with fixed costs of $5.000\ \text{€}$ per year. He intends to set a selling price of $10\ \text{€}$, even though his manufacturing variable cost is $11\ \text{€}$ per T-shirt. In his business plan, he states: “Even though the unit margin is negative, if we reach a large enough sales volume, total revenues will exceed total costs and we will cover fixed costs.” a) Refute the entrepreneur's claim both quantitatively and conceptually: demonstrate using the profit function why it is mathematically impossible to reach the break-even point, and calculate the losses if he were to sell 5.000 T-shirts. b) To make the project viable, a mentor demands that the break-even point be set at a maximum of 1.000 T-shirts. If market constraints prevent the price from exceeding $14\ \text{€}$, determine analytically: 1) The maximum allowable unit variable cost while maintaining fixed costs at $5.000\ \text{€}$. 2) The percentage reduction in fixed costs required if the unit variable cost could only be lowered to $10,5\ \text{€}$.

A food stall chain is hesitating between a city centre location (fixed costs of $6.000\ \text{€}$ per month, price $9\ \text{€}$, unit variable cost $4\ \text{€}$, and 1 400 expected units) or a neighbourhood location (fixed costs of $2.000\ \text{€}$, same price and variable cost, and 500 expected units). a) Calculate the break-even point and the expected profit of each option. b) Recommend one, also assessing the risk if actual sales fall 20% below expectations.

A comic book publisher has annual fixed costs of $80.000\ \text{€}$, sells each comic for $20\ \text{€}$, and incurs a unit variable cost of $12\ \text{€}$. Its strategic plan sets a target profit of $40.000\ \text{€}$, but a market report concludes that maximum market demand will not exceed 12.500 copies. a) Calculate the break-even point and demonstrate quantitatively why the company will not reach its target profit under current conditions despite operating above the break-even point. b) To achieve the $40.000\ \text{€}$ profit selling exactly the 12.500 comics forecasted, management analyzes two independent proposals: 1) Invest in a print-on-demand platform, which would reduce the unit variable cost while keeping fixed costs unchanged. 2) Outsource distribution to reduce fixed costs, keeping the price and unit variable cost unchanged. Calculate the required unit variable cost in proposal 1 and the maximum level of fixed costs in proposal 2. c) Justify which of the two proposals places the company in a position of lower operating risk in the event of an unexpected contraction in demand.

A bakery can produce a maximum of 9 000 loaves a month due to the capacity of its ovens. Its fixed costs are $15.000\ \text{€}$ a month, the selling price is $3,5\ \text{€}$ per loaf and the variable cost per unit is $2\ \text{€}$. a) Calculate the break-even point and compare it with the maximum capacity. b) Reason whether the business is viable as it is set up. c) Propose and evaluate two solutions (raising the price, reducing the fixed costs or expanding the capacity).

A company can produce in a craft way (fixed costs of $10.000\ \text{€}$ and a variable cost per unit of $8\ \text{€}$) or in an automated way (fixed costs of $34.000\ \text{€}$ and a variable cost per unit of $4\ \text{€}$), selling in both cases at $10\ \text{€}$ per unit. a) Calculate the break-even point of each technology. b) Calculate the volume above which the automated technology is worthwhile (indifference point between both total costs). c) Recommend a technology if sales of 4 000 units are expected and reason the risk of each option.

A juice factory has fixed costs of $21.000\ \text{€}$ a month, sells a litre at $3\ \text{€}$ and the variable cost per unit is $1,8\ \text{€}$. a) Calculate the current break-even point. b) The raw material becomes $10\ \%$ more expensive and the company decides to raise the price by $5\ \%$: calculate the new variable cost per unit, the new price and the new break-even point (round to whole litres). c) Comment on whether the price rise offsets the raw material becoming more expensive.

A natural cosmetics company has annual fixed costs of $96.000\ \text{€}$, sells each cream at $30\ \text{€}$ and bears a variable cost per unit of $18\ \text{€}$. a) Calculate the break-even point and interpret it. b) Calculate the annual profit if it sells 10 000 creams. c) A perfumery chain offers to buy 3 000 extra creams at $22\ \text{€}$ per unit, without this involving new fixed costs: reason whether it is worth accepting the order by comparing the offered price with the contribution margin.

Unit challenge. Is it viable to launch a healthy snack service for break times at your high school? Evaluate this business model by completing the following stages:

1. SWOT of the environment: Define your proposal and prepare a SWOT matrix with two factors per quadrant, consulting price data in local supermarkets or youth spending statistics from the INE.

2. Cost and price structure: Determine the selling price ($P$) by surveying five classmates about their willingness to pay. Estimate monthly fixed costs ($CF$, e.g., reduced self-employment fee or municipal permits: $80,0$ €) and the unit variable cost ($CV_u$, e.g., fruit, nuts, and packaging: $1,2$ €).

3. Break-even point: Calculate the break-even point using the formula $Q^* = \frac{CF}{P - CV_u}$. Plot the total revenue and total cost graph by hand, indicating the break-even point and the loss and profit areas.

4. Final verdict: Prepare a brief executive report (one side of a page) where you conclude whether the required sales volume is realistic and which SWOT strategy you would prioritise to mitigate risks.

Assessment criteria:

• Accuracy in the algebraic calculation and graphical representation of the break-even point ($Q^*$).

• Realism and consistency in obtaining costs and prices with verified data.

• Analytical depth of the SWOT matrix linked to the school context.

• Critical ability to argue the final viability of the project.