Interpreting intersections and break-even points | Year 12

01 / LEARN

The idea, step by step

Equal values for the same input

An intersection of cost models means equal costs at the same input. For revenue and cost, it means zero profit. Give the input and the common output with units. Compare the models on either side rather than assuming the option with the smaller initial fee remains cheaper forever.

Break-even is not maximum profit

Profit is revenue minus all costs, including fixed costs. If ticket counts must be integers, solve the continuous threshold first and then choose the first permitted integer giving the required profit. Break-even gives zero profit, not the greatest possible profit; capacity or demand restrictions are needed to locate a practical maximum.

P = R - C

A common trap

Break-even is the most profitable attendance.

Instead: Break-even is zero profit. A maximum requires the objective and all practical limits.

02 / WORKED EXAMPLES

See the thinking

Example 1

Choosing a plan

Plan A costs AUD 25 plus AUD 3 per use. Plan B costs AUD 61 plus AUD 1 per use. Compare them.

Solve $25 + 3 n = 61 + n$ , giving $n = 18$ .
Both cost AUD 79.
At 10 uses A is 55 and B is 71; at 25 uses A is 100 and B is 86.

A is cheaper below 18 uses, B above 18; equal at 18.

Example 2

An event

Tickets sell for AUD 15. Costs are AUD 120 plus AUD 7 per attendee. Find break-even and first profitable attendance.

$R = 15 n$ , $C = 120 + 7 n$ .
Set $15 n = 120 + 7 n$ : $n = 15$ .
Profit is $8 n - 120$ , positive for $n > 15$ .

Break-even at 15; first profit at 16 attendees.

Check your understanding

Revenue is $9 n$ and cost is $52 + 5 n$ . Find break-even.

Show answer

13 items; both are AUD 117.

$9 n = 52 + 5 n$ gives $4 n = 52$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find equal cost for $A = 40 + 2 n$ and $B = 16 + 5 n$ .
2 marks
For the same plans, which is cheaper at 12 uses?
2 marks
A product sells for AUD 28 and costs AUD 210 fixed plus AUD 13 per unit. Find break-even.
2 marks
Find profit on 20 products in Question 3.
2 marks
Profit is $P = 12 n - 95$ . Find the smallest profitable whole-number attendance.
2 marks
Why do $A = 30 + 4 n$ and $B = 80 + 4 n$ never have equal cost?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find equal cost for $A = 40 + 2 n$ and $B = 16 + 5 n$ .
2 marks
$40 + 2 n = 16 + 5 n$ gives $n = 8$ .
8 uses, AUD 56.
For the same plans, which is cheaper at 12 uses?
2 marks
$A = 64$ and $B = 76$ .
A, by AUD 12.
A product sells for AUD 28 and costs AUD 210 fixed plus AUD 13 per unit. Find break-even.
2 marks
$28 n = 210 + 13 n$ gives $15 n = 210$ .
14 products.
Find profit on 20 products in Question 3.
2 marks
$P = 15 n - 210$ ; at 20 this is $300 - 210 = 90$ .
AUD 90.
Profit is $P = 12 n - 95$ . Find the smallest profitable whole-number attendance.
2 marks
$n > 95 / 12$ , about 7.92; the first permitted integer is 8.
8 people.
Why do $A = 30 + 4 n$ and $B = 80 + 4 n$ never have equal cost?
2 marks
The gradients match and $B - A = 50$ for all $n$ .
B always costs AUD 50 more.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.

Official curriculum source ↗

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