Understand the idea.
Separate the fixed and changing parts
A linear cost model separates a fixed fee from a charge per unit used. In , the fee is dollars and each hour adds dollars. The intercept and gradient encode these different roles.
Define the input and how charging works. A formula allowing fractional hours differs from a service rounding each hire up to a whole hour. Use the conditions supplied and include the fixed fee once per hire.
An intersection is a comparison point
Two cost graphs intersect where both rules give the same output for the same input. Set the expressions equal, solve for that input, and substitute it into either rule to find the common cost.
An equal-cost point helps compare options, but does not make one universally cheaper. Test feasible inputs on either side or examine the cost difference. Report time and money separately, with units and any domain restrictions.
See the method in action.
Worked example
Hire A costs 18 dollars plus 4 dollars per hour. Find a rule and the cost for 5 hours.
Worked solution
- 1
Define the input.
The variable represents the quantity that changes the charge.
- 2
Combine fixed and hourly charges.
The fixed 18-dollar fee is added once; the hourly charge is multiplied by h.
- 3
Substitute five hours.
Five hourly charges plus the fixed fee give AUD 38.00.
; AUD 38.00
Worked example
Hire A costs dollars and B costs dollars. When are they equal?
Worked solution
- 1
Set the cost rules equal.
Equal-cost hire means both providers charge the same total for the same time.
- 2
Collect the fixed and hourly parts.
Subtract 6 and then 4h from both sides, before dividing by 3.
- 3
Calculate the common cost.
The intersection input must be substituted to find its output.
- 4
Check the other provider.
Both costs agree at four hours, confirming AUD 34.00.
4 hours, at AUD 34.00
Try it yourself.
Check your understanding
A service costs 12 dollars plus 3 dollars per visit. What is its cost for 6 visits?
Show answer & reasoning
AUD 30.00
Six visits contribute dollars, then the fixed fee adds . The total is dollars, or AUD 30.00.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- A class has a 20-dollar booking fee plus 8 dollars per student. Write a rule for students.2 marks
- Use to find the cost for 7 students.2 marks
- A costs dollars and B costs dollars. Find the equal-cost count.3 marks
- For A: and B: , which is cheaper for 3 items?2 marks
- A costs dollars and B costs dollars. Which is cheaper for 10 items, and by how much?2 marks
- A model gives dollars for hours. Explain why using it at needs care.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A class has a 20-dollar booking fee plus 8 dollars per student. Write a rule for students.2 marks
The fixed part is and students contribute .
dollars - Use to find the cost for 7 students.2 marks
.
AUD 76.00 - A costs dollars and B costs dollars. Find the equal-cost count.3 marks
gives .
At , both give .
, cost AUD 40.00 - For A: and B: , which is cheaper for 3 items?2 marks
A costs and B costs .
A, by AUD 6.00 - A costs dollars and B costs dollars. Which is cheaper for 10 items, and by how much?2 marks
A costs and B costs .
B, by AUD 15.00 - A model gives dollars for hours. Explain why using it at needs care.3 marks
The pricing rule is only supplied for up to hours.
It may change beyond that interval, so the formula alone does not justify a 20-hour quote.
20 hours is outside the stated domain.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Year 9 financial modelling within AC9M9A05 and AC9M9M05; supplied hypothetical rates and costs, not financial advice.
Official curriculum source ↗