Mathematics Standard • Year 11 • Module 1 • Lesson 13

Comparing Linear Models and Break-Even Points

Build fluency comparing two linear models, setting them equal to find the break-even input, and justifying which option is cheaper on each side.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 What is a break-even point in a comparison question?

Q1.2 To find a break-even point, you set Model A ______________ Model B.

Q1.3 Which option is cheaper for very small inputs — the one with the lower starting cost or the one with the lower rate? Circle one.

starting cost / rate

Q1.4 Which option is cheaper for very large inputs — lower starting cost or lower rate? Circle one.

starting cost / rate

Stuck? Revisit lesson § Key Method — Break-Even Method.

2. Worked example — phone-plan break-even

Follow each line of working. Every step has a reason on the right.

Problem. Plan A costs $20 plus $5 per gigabyte. Plan B costs $50 plus $2 per gigabyte. Find the break-even number of gigabytes and state which plan is cheaper above and below this value.

Step 1 — Write each cost equation in terms of gigabytes g.

A = 20 + 5g B = 50 + 2g

Reason: each plan has a fixed start (intercept) and a rate (gradient).

Step 2 — Set the two costs equal at the break-even point.

20 + 5g = 50 + 2g

Reason: at the break-even point both plans charge the same amount.

Step 3 — Solve for g.

5g − 2g = 50 − 20 ⇒ 3g = 30 ⇒ g = 10

Reason: collect the g terms on one side and the constants on the other.

Step 4 — Test a value below and above the break-even to decide which plan wins.

At g = 5: A = 20 + 5(5) = 45, B = 50 + 2(5) = 60 ⇒ A cheaper.

At g = 15: A = 20 + 5(15) = 95, B = 50 + 2(15) = 80 ⇒ B cheaper.

Reason: testing both sides of the break-even confirms the direction of the answer.

Conclusion. The break-even point is at g = 10 GB. Plan A is cheaper for less than 10 GB; Plan B is cheaper for more than 10 GB.

3. Faded example — fill in the missing steps

Sam starts with $200 and saves $30 per week. Alex starts with $80 and saves $50 per week. When do they have the same savings? Fill in each blank. 4 marks

Step 1 — Write each savings equation:

Sam: S = __________ + __________ w

Alex: A = __________ + __________ w

Step 2 — Set them equal:

__________ + __________ w = __________ + __________ w

Step 3 — Solve for w:

120 = __________ w ⇒ w = __________

Step 4 — Interpret in a sentence:

After __________ weeks, both people have the same total savings.

Stuck? Revisit lesson § Worked Example 3 — Compare savings plans.

4. Graduated practice — break-even comparisons

Show your working in the space below each part. Always end with a sentence about what the answer means.

Foundation — set-up only (4 questions)

Q	Problem	Answer
4.1 1	Write the equation to set "A = 30 + 5h" equal to "B = 60 + 2h" at the break-even.
4.2 1	Solve 3g = 30 for g.
4.3 1	For A = 20 + 5(4) and B = 50 + 2(4), which is cheaper?
4.4 1	For A = 20 + 5(12) and B = 50 + 2(12), which is cheaper?

Standard — find and interpret the break-even (6 questions)

Write both equations, solve, and conclude with which option wins below and above the break-even.

4.5 Company A charges $40 plus $12 per hour. Company B charges $70 plus $6 per hour. Find the break-even time in hours. 2 marks

4.6 Using Q4.5, decide which company is cheaper for 3 hours and for 8 hours. 2 marks

4.7 Two savings plans are S = 150 + 20w and A = 30 + 35w. Find when they are equal. 2 marks

4.8 Phone plan X: $15 + $0.20 per minute. Phone plan Y: $30 + $0.05 per minute. Find the break-even minute count. 2 marks

4.9 Gym A: $80 join + $25 per month. Gym B: $20 join + $40 per month. Find the break-even number of months. 2 marks

4.10 Two delivery apps: P = 5 + 3k and Q = 8 + 2k for a trip of k km. Find the break-even distance. 2 marks

Extension — interpret and justify (2 questions)

4.11 Use the table to find the break-even input, then explain which model is cheaper before and after it. 3 marks

Hours h: 1 2 3 4 5

Tutor A 70 100 130 160 190

Tutor B 100 120 140 160 180

4.12 Explain in one or two sentences why the cheaper option in a comparison question can change as the input grows larger. Use the words starting cost and rate. 3 marks

Stuck on 4.11? Each tutor's column is linear — find each equation first (e.g. Tutor A increases by 30 per hour from 70 at h = 1, so A = 40 + 30h).

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I wrote 30 + 5h = 60 + 2h (with the = sign in the middle).

For 4.2 I divided both sides by 3 to get g = 10 (not 90).

For 4.3 I calculated A = 40 and B = 58, then circled A as cheaper.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Break-even meaning

The input value at which two models give the same output (e.g. equal cost, equal savings, equal distance).

Q1.2 — Method

Set Model A equal to Model B (or just "= "). The break-even point is the input where both sides match.

Q1.3 — Small inputs

Lower starting cost wins for very small inputs (the fixed fee dominates).

Q1.4 — Large inputs

Lower rate wins for very large inputs (the per-unit charge dominates).

Q3 — Faded savings example

Step 1: Sam: S = 200 + 30w; Alex: A = 80 + 50w.
Step 2: 200 + 30w = 80 + 50w.
Step 3: 200 − 80 = 50w − 30w ⇒ 120 = 20w ⇒ w = 6.
Step 4: After 6 weeks, both people have the same total savings (both have $380).

Q4.1 — Set them equal

30 + 5h = 60 + 2h.

Q4.2 — Solve 3g = 30

g = 30/3 = 10.

Q4.3 — At input 4

A = 20 + 5(4) = 40. B = 50 + 2(4) = 58. A is cheaper ($40 < $58).

Q4.4 — At input 12

A = 20 + 5(12) = 80. B = 50 + 2(12) = 74. B is cheaper ($74 < $80).

Q4.5 — Companies A vs B

40 + 12h = 70 + 6h ⇒ 6h = 30 ⇒ h = 5 hours.

Q4.6 — At 3 hours and 8 hours

3 h: A = 40 + 12(3) = 76; B = 70 + 6(3) = 88. A cheaper.
8 h: A = 40 + 12(8) = 136; B = 70 + 6(8) = 118. B cheaper.

Q4.7 — Savings plans

150 + 20w = 30 + 35w ⇒ 120 = 15w ⇒ w = 8 weeks (both reach $310).

Q4.8 — Phone plans X vs Y

15 + 0.20m = 30 + 0.05m ⇒ 0.15m = 15 ⇒ m = 100 minutes.

Q4.9 — Gym A vs Gym B

80 + 25m = 20 + 40m ⇒ 60 = 15m ⇒ m = 4 months.

Q4.10 — Delivery apps P vs Q

5 + 3k = 8 + 2k ⇒ k = 3 km.

Q4.11 — Tutor A vs Tutor B (table)

From the table: A increases by 30 per hour starting at 70 when h = 1 ⇒ A = 40 + 30h; B increases by 20 per hour starting at 100 when h = 1 ⇒ B = 80 + 20h.
Set equal: 40 + 30h = 80 + 20h ⇒ 10h = 40 ⇒ h = 4 hours (both cost $160 — confirmed in the table).
Below 4 hours, Tutor A is cheaper (lower hourly rate doesn't yet outweigh the higher start cost). Above 4 hours, Tutor B is cheaper (the lower rate dominates).

Q4.12 — Why the cheaper option can switch

The option with the lower starting cost wins for small inputs because the fixed fee dominates the total. As the input grows, the per-unit cost (rate) is repeated many times, so the option with the lower rate eventually overtakes — making it cheaper above the break-even point.