Comparing Linear Models and Break-Even Points
Compare options using linear equations, find where two models are equal, and explain which option is better under different conditions. A lower starting cost is not always best — the rate tells the real story.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Plan A costs $20 plus $5 per gigabyte. Plan B costs $50 plus $2 per gigabyte. Which plan is cheaper?
Without calculating — write what extra information you need before deciding. Which plan would you lean towards and why?
When comparing two options, set the two equations equal and solve for the input where both outputs match. This is the break-even point — the moment when neither option is cheaper than the other.
Fixed cost is the amount paid regardless of usage. Rate is how much each extra unit costs. A low fixed cost can be misleading if the rate is high — over time, the rate dominates.
Key facts
- A break-even point is where two models have the same output.
- The intersection point can represent equal cost, equal distance or equal savings.
- The cheaper option can change depending on the input value.
Concepts
- A lower starting cost is not always the best long-term option.
- A lower rate becomes more important as the input increases.
- A decision should be justified for a specific range or condition.
Skills
- Write two linear equations for competing options.
- Find and interpret a break-even point.
- Justify which model is better before and after the break-even point.
A linear model with a low starting value can become expensive if its rate is high. When comparing options, identify the fixed cost and the repeated rate for each option. Then compare at the input value that matters.
Do not choose the cheaper starting cost without considering the rate. For small input values, the model with the lower fixed cost tends to be cheaper. For large input values, the model with the lower rate tends to be cheaper.
Break-even: where Revenue = Cost. Below: loss zone (red). Above: profit zone (green)
What to write in your book
- A linear model takes the form $C = a + bx$ where $a$ is the fixed cost and $b$ is the rate per unit.
- To find the break-even point: set Model A = Model B and solve for $x$.
- A lower fixed cost does NOT mean cheaper overall — the rate matters more for large input values.
- Always state which option is cheaper both below and above the break-even point.
Did you get this? True or false: when comparing two linear models, the option with the lower fixed cost will always be cheaper in the long run.
After finding a break-even point algebraically, a table of values can confirm the result by showing both models producing the same output at the break-even input.
| Gigabytes | 5 | 10 | 15 |
|---|---|---|---|
| Plan A: $20 + 5g$ | $45 | $70 | $95 |
| Plan B: $50 + 2g$ | $60 | $70 | $80 |
The table confirms the break-even point at 10 GB because both costs are $70. At 5 GB, Plan A ($45) is cheaper. At 15 GB, Plan B ($80) is cheaper.
What to write in your book
- Build a table substituting values either side of the break-even point to verify your algebra.
- The break-even input is confirmed when both model outputs are equal in the table.
- Below the break-even point, the model with the lower fixed cost is usually cheaper.
- Above the break-even point, the model with the lower rate is always cheaper.
Fill the gap: Using the phone plan table above — at 5 GB Plan A costs $ and Plan B costs $60, so Plan is cheaper below the break-even point.
Worked examples · 3 in a row, reveal as you go
Plan A costs $20 plus $5 per gigabyte. Plan B costs $50 plus $2 per gigabyte. Find the number of gigabytes where the plans cost the same.
Using the same plans (A: $20 + 5g$; B: $50 + 2g$), decide which is cheaper before and after 10 GB.
Sam starts with $200 and saves $30 per week. Alex starts with $80 and saves $50 per week. When will they have the same amount?
What to write in your book
- Always define your variable before writing the models (e.g. "Let $g$ = gigabytes").
- Set the two equations equal and solve algebraically.
- Verify with substitution: substitute the break-even value into both models — outputs must match.
- Decision habit: state which model is better before the break-even point, and which is better after.
Quick check: Sam: $S = 200 + 30w$. Alex: $A = 80 + 50w$. After the break-even point (week 6), who has more savings?
Common errors · the 3 traps that cost marks
What to write in your book
- Always check your answer: substitute the break-even $x$-value into both models — the outputs must be equal.
- The question usually wants a decision, not just a number — state which is cheaper and for what range.
- The model with the lower rate wins for large inputs; the model with the lower fixed cost wins for small inputs.
- A break-even point is only useful if you explain what happens before and after it.
Match each term to its meaning:
Quick-fire practice · 4 calculations
Company A charges $40 plus $12 per hour. Company B charges $70 plus $6 per hour. Find the break-even time.
Using Question 1, decide which company is cheaper for 3 hours and for 8 hours.
Two savings plans are $150 + 20w$ and $30 + 35w$. Find when they are equal.
Explain in one sentence why the cheaper option can change as the input increases.
Two truths and a lie: Three statements about break-even points are below. Which one is false?
The plans break even at 10 GB. Below 10 GB, Plan A is cheaper. Above 10 GB, Plan B is cheaper because it has the lower per-gigabyte rate.
Earlier you wrote what information you needed before deciding. Now that you have found the break-even point, explain why the answer depends on the number of gigabytes.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Company A charges $35 plus $10 per hour. Company B charges $65 plus $4 per hour. Find the break-even time. (4 marks)
Q2. Using Question 1, decide which company is cheaper for 3 hours and for 8 hours. (4 marks)
Q3. Explain what a break-even point means in a model comparison question. (2 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: Company A: $C = 40 + 12t$; Company B: $C = 70 + 6t$. Set equal: $40 + 12t = 70 + 6t \Rightarrow 6t = 30 \Rightarrow t = 5$ hours.
Drill 2: At 3 hr: A = $76, B = $88 → A is cheaper. At 8 hr: A = $136, B = $118 → B is cheaper.
Drill 3: $150 + 20w = 30 + 35w \Rightarrow 120 = 15w \Rightarrow w = 8$ weeks.
Drill 4: The cheaper option changes because the model with the lower rate eventually overtakes the model with the lower fixed cost as input increases.
Q1 (4 marks): Let $t$ = hours [1]. Company A: $A = 35 + 10t$; Company B: $B = 65 + 4t$ [1]. Set equal: $35 + 10t = 65 + 4t \Rightarrow 6t = 30 \Rightarrow t = 5$ hours [1]. Interpretation: At 5 hours both companies charge $85 [1].
Q2 (4 marks): At 3 hr: A = $35 + 30 = \$65$; B = $65 + 12 = \$77$ → Company A is cheaper [2]. At 8 hr: A = $35 + 80 = \$115$; B = $65 + 32 = \$97$ → Company B is cheaper [2].
Q3 (2 marks): A break-even point is the input value where both linear models produce the same output [1]. It marks the boundary between which option is cheaper — one option is better below the break-even point and the other is better above it [1].
Compare fixed costs and rates, solve the equality, then state which option wins on each side. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering break-even and linear model comparison questions. Pool: lesson 13.
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