Choose a plan for four items
- A = 2(4) + 12 = $20.
- B = 4(4) = $16.
- Plan B is $4 cheaper at this input.
Comparing plans & interpreting models
Use a table, a graph and reasoning to decide which plan fits a situation.
Lesson 09 · Start where you need
Your learning goal
Compare two linear cost models, locate a break-even point and state the limits of the comparison.
Choose the same input for a fair comparison
Plan A costs $12 to start plus $2 per item: A = 2n + 12. Plan B has no start fee and costs $4 per item: B = 4n. Compare the plans at the same number of items. A smaller starting fee does not guarantee a lower cost for every order size.
Words worth knowing
- Ordered pair
- An input and its output, written (x, y).
- Rate of change
- Change in output divided by change in input.
- Starting value
- The output when the input is zero.
- Domain
- The set of permitted inputs for the situation.
1
Make the idea visible.
Predict a change, adjust one value, then explain what stayed the same.
Plan A = 2n + 12; Plan B = 4n. Costs are dollars. This simplified model counts complete items and omits taxes or other charges.
Read both the start and the rate
Plan B starts at $0 but increases more quickly. Plan A starts higher and increases more slowly. At n = 3, A = $18 while B = $12. At n = 10, A = $32 while B = $40. The cheaper plan depends on n.
Find where the costs agree
Set the expressions equal: 2n + 12 = 4n. Subtract 2n from both sides to get 12 = 2n, then divide by 2: n = 6. Both costs are $24 at six items. On a graph, the intersection is (6, 24), which has both an input and an output.
Check what the situation leaves out
These plans assume fixed prices with no extra fees or discounts. Whole items restrict n to non-negative integers. If a mathematical intersection occurs at a fractional item count, compare the neighbouring whole-number orders rather than pretending a fraction of a complete item is available. A percentage raise compounded each year is not a constant-dollar linear increase; that distinction matters in the Salary Project.
Worked examples
Follow the reasoning.
Choose a plan for eight items
- A = 2(8) + 12 = $28.
- B = 4(8) = $32.
- Plan A is $4 cheaper. The comparison changed after the break-even point at six items.
Try it with support
Show your thinking.
Explain the point (6, 24) in the two-plan comparison.
Give me a hint
Name both coordinates and units.
Compare a worked response
At six items, both plans cost $24.
The x-coordinate is item count; the y-coordinate is cost, not a second item count.
Does a 5% raise each year add the same number of dollars forever?
Give me a hint
Compare 5% of $100 with 5% of $105.
Compare a worked response
No.
The percentage is applied to a changing base. The dollar increases are $5 and $5.25, so a compounded percentage raise is not linear in years.
Independent practice
Check your understanding.
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Check the concepts, too.
Choose a response, then use the explanation to check your reasoning. This is practice, not a school mark.
Go further
Keep the connections close.
Khan Academy: graphing lines & slope → · All related learning collections →