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 + 30 w; Alex: A = 80 + 50 w.
Step 2: 200 + 30w = 80 + 50w.
Step 3: 200 − 80 = 50w − 30w ⇒ 120 = 20 w ⇒ 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.