The idea, step by step
Represent the important costs
A model is a simplified mathematical description of a situation. A fixed cost is paid once, regardless of the number of people or items. A per-item cost is multiplied by how many items there are.
For a trip with 30 dollars of transport and 6 dollars entry per student, add the fixed 30 once. A table makes the changing and unchanged parts visible. Label every amount so the model can be checked.
| Students | Transport in dollars | Entry in dollars | Total in dollars |
|---|---|---|---|
| 5 | 30 | 30 | 60 |
| 10 | 30 | 60 | 90 |
| 15 | 30 | 90 | 120 |
Check the model against the situation
A correct calculation can still give a poor prediction if the assumptions are wrong. A model might assume every ticket costs the same, transport has enough seats, and there are no extra fees. State those assumptions before using the result to make a decision.
Test a small case and ask whether the result is reasonable. Compare the prediction with a new quote or actual cost, then revise the model if a fee or discount changes. Model choice and interpretation are part of the mathematics.
A common trap
Multiply every cost by the number of people.
Instead: A fixed fee is paid once. Multiply only the per-person or per-item cost by the count.
See the thinking
Find an affordable group size
Transport costs 30 dollars and entry costs 6 dollars per student. The budget is 150 dollars. Assuming transport has enough seats, what is the largest affordable number of students?
- Remove the fixed transport cost: 150 - 30 = 120 dollars for tickets.
- 120 divided by 6 gives 20 tickets.
- Check: 30 + 6 times 20 = 150 dollars. One more student would cost 156 dollars.
Find the fundraiser surplus
A stall sells 30 items for 2.50 dollars each. Materials cost 1.50 dollars per item and setup costs 20 dollars once. Find the money left after costs.
- Income = 30 times 2.50 = 75 dollars.
- Materials = 30 times 1.50 = 45 dollars.
- Total cost = 45 + 20 = 65 dollars. Surplus = 75 - 65 = 10 dollars.
Check your understanding
A venue charges 10 dollars once plus 4 dollars per person. Find the total for 6 people.
Show answer
34 dollars
Total = 10 + 4 times 6 = 34 dollars.
Your turn
6 questions · 17 marks · Work at your own pace.
Read the questions online
- An excursion has a 20-dollar booking fee and costs 5 dollars per person. Find the cost for 8 people.2 marks
- A game venue costs 10 dollars once plus 4 dollars per person. Find the total for 0, 1 and 2 people.2 marks
- A stall sells 10 items for 3 dollars each. Materials cost 1 dollar per item, plus a 15-dollar setup fee. Find the surplus.3 marks
- A ticket model uses 7 dollars per person. Give two things that should be checked before using it to predict the cost of a school trip.3 marks
- Company A charges 40 dollars once plus 2 dollars per person. Company B charges 6 dollars per person with no fixed fee. Both can carry 12 people. Which is cheaper for 12 people?3 marks
- You have 80 dollars for kits. Delivery costs 18 dollars once and each kit costs 7.50 dollars. Find the largest number of whole kits you can buy and the money left.4 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- An excursion has a 20-dollar booking fee and costs 5 dollars per person. Find the cost for 8 people.2 marks
20 + 5 times 8 = 60 dollars.
60 dollars - A game venue costs 10 dollars once plus 4 dollars per person. Find the total for 0, 1 and 2 people.2 marks
The fixed fee remains 10 dollars. Add 0, 4 or 8 dollars.
10, 14 and 18 dollars. - A stall sells 10 items for 3 dollars each. Materials cost 1 dollar per item, plus a 15-dollar setup fee. Find the surplus.3 marks
Income = 30 dollars. Costs = 10 + 15 = 25 dollars.
Surplus = 30 - 25 = 5 dollars.
5 dollars - A ticket model uses 7 dollars per person. Give two things that should be checked before using it to predict the cost of a school trip.3 marks
The model needs an appropriate per-person price and all fixed fees. Other reasonable cost or capacity assumptions are acceptable.
For example: whether all students pay 7 dollars; whether a booking or transport fee must be added. - Company A charges 40 dollars once plus 2 dollars per person. Company B charges 6 dollars per person with no fixed fee. Both can carry 12 people. Which is cheaper for 12 people?3 marks
A costs 40 + 2 times 12 = 64 dollars.
B costs 6 times 12 = 72 dollars.
A is cheaper by 8 dollars. - You have 80 dollars for kits. Delivery costs 18 dollars once and each kit costs 7.50 dollars. Find the largest number of whole kits you can buy and the money left.4 marks
After delivery, 62 dollars remains. Eight kits cost 60 dollars; nine cost 67.50 dollars.
Total for eight = 18 + 60 = 78 dollars; 80 - 78 = 2 dollars.
8 kits; 2 dollars left.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Fractions–decimals–percentages; percentages of quantities; expressing one amount as a percentage of another; discounts and sale prices; practical spending/budget comparisons; choosing and explaining a financial model.
Official curriculum source ↗