Comparing linear growth/cost models | Year 9

01 / LEARN

The idea, step by step

Separate the fixed and changing parts

A hire charge of 18 dollars plus 4 dollars per hour has rule $C = 18 + 4 h$ . The 18 dollars is paid even for zero hours in this model; each extra hour adds 4 dollars. The graph is a straight line with intercept 18 and gradient 4.

Define $h$ and its allowed values. A model may allow half-hours, or it may charge only in whole-hour blocks. Those are different rules. Use the conditions supplied rather than assuming that every straight-line formula applies to every input.

An intersection is a comparison point

To compare two providers, set their cost rules equal. This gives the input at which they quote the same amount. Check a value below and above the intersection to decide which is cheaper; a lower fixed fee does not guarantee the lower total at all times.

Use a table or graph as a second representation. Report a feasible input with its units and the corresponding cost. If the algebraic intersection is outside the allowed domain, it is not a usable purchase option.

A common trap

Choose the plan with the smaller fixed fee without considering use.

Instead: Compare the whole cost at the required input, including both fixed and variable parts.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Hire A costs 18 dollars plus 4 dollars per hour. Find a rule and the cost for 5 hours.

The fixed part is $18$ and the hourly part is $4 h$ .
$C (5) = 18 + 4 (5) = 38$ .

C = 18 + 4 h

; AUD 38.00

Example 2

Worked example

Hire A costs $C_{A} = 18 + 4 h$ dollars and B costs $C_{B} = 6 + 7 h$ dollars. When are they equal?

$18 + 4 h = 6 + 7 h$ .
$12 = 3 h$ , so $h = 4$ .
Both rules give $34$ dollars.

4 hours, at AUD 34.00

Check your understanding

A service costs 12 dollars plus 3 dollars per visit. What is its cost for 6 visits?

Show answer

AUD 30.00

$12 + 3 (6) = 30$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A class has a 20-dollar booking fee plus 8 dollars per student. Write a rule for n students.
2 marks
Use $C = 20 + 8 n$ to find the cost for 7 students.
2 marks
A costs $15 + 5 n$ dollars and B costs $30 + 2 n$ dollars. Find the equal-cost count.
3 marks
For A: $15 + 5 n$ and B: $30 + 2 n$ , which is cheaper for 3 items?
2 marks
A costs $15 + 5 n$ dollars and B costs $30 + 2 n$ dollars. Which is cheaper for 10 items, and by how much?
2 marks
A model gives $C = 25 + 4 h$ dollars for $0 \leq h \leq 8$ hours. Explain why using it at h=20 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 n students.
2 marks
The fixed part is $20$ and $n$ students contribute $8 n$ .
$C = 20 + 8 n$ dollars
Use $C = 20 + 8 n$ to find the cost for 7 students.
2 marks
$20 + 8 (7) = 76$ .
AUD 76.00
A costs $15 + 5 n$ dollars and B costs $30 + 2 n$ dollars. Find the equal-cost count.
3 marks
$15 + 5 n = 30 + 2 n$ gives $3 n = 15$ .
At $n = 5$ , both give $40$ .
$n = 5$ , cost AUD 40.00
For A: $15 + 5 n$ and B: $30 + 2 n$ , which is cheaper for 3 items?
2 marks
A costs $30$ and B costs $36$ .
A, by AUD 6.00
A costs $15 + 5 n$ dollars and B costs $30 + 2 n$ dollars. Which is cheaper for 10 items, and by how much?
2 marks
A costs $65$ and B costs $50$ .
B, by AUD 15.00
A model gives $C = 25 + 4 h$ dollars for $0 \leq h \leq 8$ hours. Explain why using it at h=20 needs care.
3 marks
The pricing rule is only supplied for up to $8$ 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.

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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 ↗

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