Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Solve 5x + 40 = 9x.
Answer
x = 10, because 4x = 40. - Solve y = x + 3 and x + y = 9 by substitution.
Answer
x = 3, y = 6: x + x + 3 = 9, so 2x = 6. - How many solutions have y = 4x + 1 and y = 4x − 3?
Answer
None: the lines are parallel.
Words used in this lesson
- break-even
- the point where cost equals income. For example: 12 notebooks.
- fixed cost
- a cost that does not change with the number sold. For example: hiring equipment for 48.
- income
- money coming in from sales. For example: 7 × 12 = 84.
- profit
- income minus cost. For example: 210 − 138 = 72.
- model
- a mathematical description of a situation. For example: C = 48 + 3x.
- social enterprise
- a business that uses its profit for a good cause. For example: notebooks to raise money for a library.
A stall that has to cover its costs
A school group in Accra makes recycled-paper notebooks to raise money for a library. Hiring the binding equipment costs 48 cedis, however many notebooks they make. Each notebook costs 3 cedis to make and sells for 7 cedis.
Predict first: how many notebooks must they sell before the money coming in matches the money going out: 6, 12 or 20? Write the cost as C = 48 + 3x and the income as I = 7x, where x is the number sold. Then explore with the figure, with its own numbers.
C = 30 + 2x and I = 5x meet at 10 items, where both are 50. At 6 items: a loss of 12.
Show me the steps
- C = 30 + 2x and I = 5x: each item adds 5 − 2 = 3 towards the 30, so 30 ÷ 3 = 10 items. Both are 50.
- Raise the price: C = 30 + 2x and I = 8x. Each item adds 6, so 30 ÷ 6 = 5 items. Both are 40.
- Raise the fixed cost: C = 60 + 2x and I = 5x. Each item adds 3, so 60 ÷ 3 = 20 items. Both are 100.
- C = 30 + 2x and I = 2x: each item adds nothing, so the 30 is never covered. No break-even.
Break-even is the number of items at which cost and income are equal. Cost is a fixed amount plus an amount for each item; income is the price times the number of items. Setting them equal is solving a system of two linear equations, and the solution is the intersection of the two lines.
Each item sold adds (price − cost of each item) towards the fixed cost, so break-even items = fixed cost ÷ (price − cost of each item). That only exists if the price is more than the cost of each item.
A common mistake: thinking break-even is the profit. At the break-even point profit is zero: the business has only covered its costs. Profit starts with the next item sold. And if the price is no more than the cost of each item, a business with a fixed cost never breaks even, however many it sells.
Worked examples
The notebooks: C = 48 + 3x and I = 7x. How many to break even, and what happens at 30?
- Break-even: 48 + 3x = 7x, so 48 = 4x and x = 12.
- At 12 notebooks cost and income are both 7 × 12 = 84 cedis.
- At 30 notebooks: income 7 × 30 = 210 and cost 48 + 90 = 138, so the profit is 210 − 138 = 72 cedis for the library.
Choosing a method: two print shops. Shop P charges 40 setup plus 2 a copy. Shop Q charges 10 setup plus 5 a copy. When are they equal, and which is cheaper for 20 copies?
- Both are already y = …, so set them equal (substitution): 40 + 2x = 10 + 5x.
- 30 = 3x, so x = 10 copies, where both cost 60.
- At 20 copies P costs 40 + 40 = 80 and Q costs 10 + 100 = 110. P is cheaper.
Which method?
• An equation already says y = … or x = …: substitution.
• Both equations have the same (or opposite) number in front of one unknown: elimination, straight
away.
• Neither: scale one or both equations and use elimination, or rearrange and substitute. Pick the
way with the smaller numbers.
• You want a picture or a check of an estimate: graph.
Your go. The first step is given. The group tries a price of 6 cedis. Now C = 48 + 3x and I = 6x. How many notebooks to break even, and how much money is that?
- Each notebook adds 6 − 3 = 3 towards the 48.
- Find the number of notebooks and the money yourself, then check.
Check the answer
48 ÷ 3 = 16 notebooks, and 6 × 16 = 96 cedis. Check: 48 + 3 × 16 = 96.
Check before you go on
1. Which is the quickest way to solve 3x + 4y = 20 and 3x − 2y = 8?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A stall has a fixed cost of 90 shillings. Each item costs 3 shillings to make and sells for 8. How many items must it sell to break even?
3. A stall has a fixed cost of 100 naira. Each item costs 4 naira to make and sells for 9 naira. How much money does it take in at the break-even point, in naira?
Level 2 · Building
4. A cooperative's cost is y = 90 + 5x cedis and its income is y = 12x cedis, where x is the number of whole items sold. What is the smallest number of items that gives a profit?
5. x + y = 14 and 3x + y = 26. Find x.
Level 3 · Stretch
6. A craft co-op sells sandals at 25 pesos and baskets at 40 pesos. One week it sold 12 items and took in 360 pesos. How many sandals did it sell?
7. Plan A costs 120 + 6s dirhams and plan B costs 20 + 11s dirhams, where s is the number of sessions. What is the smallest whole number of sessions for which plan A costs less than plan B?
Something to think about
What would the graph look like if the fixed cost is 0 and the price of each item equals the cost of each item? What does it say about the business?
One way to think about it
Cost and income are the same line, y = vx, so they meet at every number of items: the system has infinitely many solutions. The business never loses and never gains on each sale: it breaks even on every item, which is fine for a charity that only wants to cover its costs but cannot raise money.
Exit ticket
8. A stall has a fixed cost of 20 rupees. Each item costs 6 rupees to make and sells for 5 rupees. Which statement is true?