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VCE VCE VM Numeracy (Year 12) · Unit 3 — Number, Shape, Measures and Relationships · 12m read

Relationships — Formulas, Graphs and Comparing Options

Relationships show how one quantity depends on another — how your pay depends on hours, or how a plan’s cost depends on usage. They help you make smart decisions.

Learning objective: Use formulas, tables and graphs to model relationships such as costs, pay and phone plans, and decide which option is best.

By the end of this lesson you can:

  • I can substitute into formulas and rearrange simple ones.
  • I can create tables and graphs of linear relationships.
  • I can find a break-even point and choose the best option.

A graph is a decision map. Where two lines cross is the point where the better option switches.

Formulas and linear models

  • A FORMULA connects quantities: pay = hourly rate × hours; cost = fixed fee + rate × usage.
  • LINEAR relationships make straight-line graphs: y = (rate) × x + (starting value).
  • TABLES of values help you see patterns and plot graphs.
  • BREAK-EVEN: where two options cost the same — set the formulas equal or read where the graphs cross.

Worked example (Medium)

Plumber A charges a $90 call-out fee plus $60 per hour. Plumber B charges $120 per hour with no call-out fee. When is each cheaper?

  1. Formulas. A = 90 + 60h; B = 120h.
  2. Break-even. 90 + 60h = 120h → 60h = 90 → h = 1.5.
  3. Decide. Under 1.5 h, B is cheaper; over 1.5 h, A is cheaper.

Answer: Plumber B is cheaper for jobs under 1.5 hours; Plumber A is cheaper for longer jobs.

Common mistakes

  • Comparing only one scenario The best option can change with usage. Fix: Find the break-even point.
  • Forgetting fixed costs Call-out fees or monthly fees change the comparison. Fix: Include every part of the formula.

Practice questions

  1. Using pay = $26.50 × hours, how much is earned for 18 hours?

    1. $450.00
    2. $477.00
    3. $497.00
    4. $265.00

    (1 marks)

  2. Phone plan X: $20/month + $0.10 per MB over the limit. Plan Y: $35/month unlimited. How many extra MB make Y the better deal?

    (3 marks)

Summary

  • Formulas model real situations.
  • Linear relationships graph as straight lines.
  • Break-even points show when the best option changes.
  • Explain recommendations clearly.

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