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VCE Foundation Mathematics (Year 12) · Number and Arithmetic · 10m read

Formulas and Expressions

Ever wondered how a simple swap of numbers into a formula can unlock real-world answers—like calculating how far you'll travel or what something will cost? Formulas are mathematical shortcuts that let you plug in your own numbers to get instant results, and it's all about following the correct order of steps (doing multiplication and division before addition and subtraction). In Foundation Maths, you're typically given the formula already, so your job is to substitute your values carefully and calculate accurately to grab those marks.

Learning objective: Substitute values into given formulas and evaluate expressions following the order of operations.

By the end of this lesson you can:

  • I convert percentages to decimals before substituting (e.g. 4% = 0.04).
  • I show the substitution step separately for method marks.
  • I check units are consistent across the formula before calculating.

A formula is a recipe card. Put in the ingredients (values), follow the steps, and out comes the answer — whether it's pay, area or a phone bill.

What are formulas and expressions

A formula is a rule written using letters (variables) and symbols showing how quantities relate. An expression is any combination of numbers, variables and operations.

In VCAA Foundation Maths you don't build formulas from scratch — the question gives you the formula. The marks are for: reading it correctly, substituting the right numbers, and showing your arithmetic. The most common error is misconverting a percentage: 4% is 0.04, not 0.4.

Substitution essentials

  • Replace every letter: Every occurrence of a variable gets replaced with its value.
  • Convert percentages: Always convert % to decimals before substituting.
  • Follow BIDMAS: Order of operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
  • Check units: Units must be consistent. Minutes and hours can't live in the same formula.

Common foundation formulas

  • Distance, speed, time: D = S \times T
  • Simple interest: I = P \times R \times T
  • Linear cost: C = mh + c
  • Celsius to Fahrenheit: F = 1.8C + 32

Quick reference

| Formula | Use |
|---|---|
| Pay = rate × hours | Wages |
| A = l × w | Area |
| Cost = fixed + rate × units | Phone/electricity bills |
| Speed = distance ÷ time | Travel |
| Substitute | Replace letters with numbers |

Worked example (Easy)

Simple interest: I = PRT, where P is principal, R is rate and T is time in years. Find the interest on $5,000 at 4% per year for 3 years.

  1. Identify values. Convert % to decimal. P = 5000, R = 0.04, T = 3
  2. Substitute. Write out the substitution cleanly. I = 5000 × 0.04 × 3
  3. Calculate. Work left to right. 5000 × 0.04 = 200; 200 × 3 = 600

Answer: Interest = $600.

Practice questions

  1. Using D = S × T, find the distance travelled at 85 km/h for 2.5 hours.

    1. 212.5 km
    2. 34 km
    3. 187.5 km
    4. 200 km

    (2 marks)

  2. The formula C = 1.5H + 20 gives cost in dollars for H hours of service. Find the cost of 8 hours.

    1. $32
    2. $42
    3. $28
    4. $52

    (2 marks)

  3. Using simple interest I = PRT, find the total repaid on a $3,500 loan at 6% for 2 years.

    1. $3,920
    2. $420
    3. $3,780
    4. $4,130

    (3 marks)

  4. A plumber charges C = 75 + 45H. A job takes 4.5 hours. What is the total cost?

    (2 marks)

  5. Using A = πr², find the area of a circle with radius 4 cm.

    1. ≈ 50 cm²
    2. ≈ 13 cm²
    3. ≈ 25 cm²
    4. 16 cm²

    (2 marks)

  6. Convert temperature using F = 1.8C + 32. Find F when C = 25.

    1. 77
    2. 57
    3. 47
    4. 90

    (2 marks)

Worked example (Medium)

The body mass index formula is BMI = weight ÷ height², where weight is in kilograms and height is in metres. Find the BMI of a person who is 78 kg and 1.75 m tall, then state the category given: 18.5–24.9 healthy, 25.0–29.9 overweight, 30.0+ obese.

  1. Identify values. Convert if needed (already SI). weight = 78, height = 1.75
  2. Square the height. Indices are step 2 of BIDMAS. 1.75² = 3.0625
  3. Substitute. Use the formula. BMI = 78 ÷ 3.0625
  4. Calculate. Round to 1 dp. ≈ 25.5
  5. Classify. Compare against the bands. 25.0 ≤ 25.5 < 30.0 → overweight

Answer: BMI ≈ 25.5 — overweight.

Common mistakes

  • Substituting 4 for 4% Percentages must become decimals before substitution. 4% is 0.04, not 4. Fix: Convert: divide percentage by 100 BEFORE substituting.
  • Skipping BIDMAS in formulas C = 1.5H + 20 with H = 8 is 32, but students often add first and get 12. Fix: Always do multiplication / division before addition / subtraction.
  • Mixing units (minutes and hours) A formula written for hours but fed minutes gives an answer 60× wrong. Fix: Check units of every variable before substituting; convert to a common unit first.

Going deeper — rearranging formulas

Sometimes the unknown isn't the subject of the formula. Rearranging means using the same balance ideas as solving equations to make the variable you want the subject.

Example: I = PRT, but you know I, P, T and need R. Divide both sides by PT to get R = I ÷ (PT).

Two rules cover almost every Foundation rearrangement:
• Whatever you do to one side you must do to the other.
• Reverse the operations: undo addition / subtraction first, then multiplication / division, then powers.

Summary

  • A formula links variables; substitute every letter with its value before calculating.
  • Percentages must be converted to decimals first (4% → 0.04).
  • Follow BIDMAS — Brackets, Indices, ÷ and ×, then + and −.
  • Check that all variables share consistent units before substituting.

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