The point where a stall starts earning
Two linear functions of one variable, and the difference between where they cross and what crossing means.
Scenario
A market stall pays £180 for its pitch, whatever it sells. Each item it sells costs £2.40 to make, and sells for £6.00.
Let n be the number of items sold in a day. Assume every item made is sold, and that no other costs arise.
Answers in order, explanations withheld, and a report on where the chain broke.
- 21.1Linear functionsFoundation
Which pair of functions gives the day's total cost and total revenue?
- AC(n) = 180 + 2.40n and R(n) = 6.00n
- BC(n) = 2.40n and R(n) = 180 + 6.00n
- CC(n) = 180n + 2.40 and R(n) = 6.00n
- DC(n) = 180 + 6.00n and R(n) = 2.40n
Show the answer and reasoningHide the reasoning
C(n) = 180 + 2.40n and R(n) = 6.00n
The pitch is paid once, so it is the constant; the £2.40 is paid per item, so it multiplies n. Revenue has no fixed part at all — selling nothing earns nothing.
- 21.2Systems of equationsStandard
Builds on: The two functions from question 1. With the pitch fee attached to the wrong function, this equation has no sensible solution.
How many items must be sold for the day to break even?
- A30
- B50
- C75
- D108
Show the answer and reasoningHide the reasoning
50
Break-even is where revenue equals cost: 6.00n = 180 + 2.40n, so 3.60n = 180 and n = 50. Both sides then come to £300. Dividing the £180 by the £6.00 selling price gives 30, which ignores what each item costs to make.
- 21.3Interpreting linear modelsStandard
Builds on: Question 2 divided 180 by 3.60 without naming it. This asks what the number that did the dividing actually was, which is the step that makes the method transferable.
What does the figure £3.60 represent in this situation?
- AThe profit on each item sold, before the pitch fee is covered.
- BThe cost of making each item.
- CThe profit on the fiftieth item.
- DThe average profit per item across the day.
Show the answer and reasoningHide the reasoning
The profit on each item sold, before the pitch fee is covered.
£6.00 - £2.40 = £3.60 is what each sale adds, and it is the same for every item. It is not the average profit per item, which depends on how many were sold because the £180 is spread across them.
- 21.4Interpreting linear modelsChallenge
Builds on: Question 3 established that £3.60 is constant per item. That is what makes the second choice wrong, and it is not something the arithmetic in question 2 alone would settle.
The stall sells 80 items. What is the day's profit, and what would the fortieth item have been worth?
- A£108 profit; the fortieth item still added £3.60.
- B£108 profit; the fortieth item added nothing, because the stall was still at a loss.
- C£480 profit; the fortieth item added £6.00.
- D£288 profit; the fortieth item added £3.60.
Show the answer and reasoningHide the reasoning
£108 profit; the fortieth item still added £3.60.
Revenue is £480 and cost is 180 + 192 = £372, so profit is £108. Every item adds £3.60 whether or not the day is yet in profit — at 40 items the stall is £36 down, and the fortieth sale still improved that by £3.60. Being at a loss is a fact about the total, not about the next sale.