Mathematics Standard • Year 11 • Module 1 • Lesson 8
Formula and Equation Synthesis
Apply the strategy decision — substitute, solve, rearrange, build — to mixed practical problems from Australian everyday contexts.
Problem 1 — Gym membership total
A gym charges $20 joining + $15 per class attended. The total cost formula is C = 20 + 15n, where n is the number of classes.
Set up: What are we solving for? Which strategy will you use?
(i) A member did 4 classes. Calculate the total cost. 1 mark
(ii) Another member paid $110 in total. Find how many classes they attended. 2 marks
(iii) Explain in one sentence why (i) and (ii) need different strategies. 1 mark
Stuck? (i) is "output unknown" so substitute. (ii) is "input unknown" so solve an equation.Problem 2 — Sydney to Goulburn road trip
The distance from Sydney to Goulburn is about 195 km. Use the formula d = st (distance = speed × time).
Set up: What are we solving for? Which strategy?
(i) If the driver averages 65 km/h, how long does the trip take? 2 marks
(ii) The driver wants to complete the trip in 2.5 hours. What average speed is needed? 2 marks
(iii) The speed limit on the M5 is 110 km/h. Comment briefly on whether the speed in (ii) is legal and reasonable. 1 mark
Stuck? Both parts need rearranging d = st. (i) rearrange to t = d/s; (ii) rearrange to s = d/t.Problem 3 — Plumber callout
A plumber charges $90 callout plus $75 per hour on site. The cost formula is C = 90 + 75h.
Set up: What are we solving for? Which strategy?
(i) Calculate C for a 3-hour job. 1 mark
(ii) A receipt shows C = $315. Find h. 2 marks
(iii) Check your answer in (ii) is reasonable by substituting h back into the formula. 1 mark
Stuck on (iii)? Revisit lesson § Worked Example 4 — Check reasonableness. Plug your h back in and confirm C matches.Problem 4 — Phone-saving plan
Mia saves the same amount each week for a new phone. After 0, 1, 2 and 3 weeks her balance is $80, $130, $180, $230.
Set up: What are we solving for? Which strategy?
(i) Write a formula B = ... linking balance B and weeks w. 2 marks
(ii) Use your formula to predict the balance after 12 weeks. 1 mark
(iii) The phone costs $1130. After how many weeks can she buy it? 2 marks
Stuck? Start at week 0 (the "starting value"), then look at how much B changes per week (the "rate"). Use B = start + rate × w.Problem 5 — Rectangular vegetable plot
A rectangular vegetable plot uses the formula A = bh (area = base × height). A gardener has a fixed base of 8 m and wants different areas.
Set up: What are we solving for? Which strategy?
(i) Calculate A when h = 3.5 m. 1 mark
(ii) Rearrange A = bh to make h the subject. 1 mark
(iii) The gardener wants A = 60 m². Use your rearranged formula to find h. 2 marks
(iv) Write one sentence explaining why rearranging first is safer than substituting first when the unknown is not the subject. 1 mark
Stuck? Revisit lesson § Worked Example 2 — Rearrange before substituting.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Gym membership
Set up. (i) substitute; (ii) solve an equation.
(i) C = 20 + 15(4) = 20 + 60 = $80.
(ii) 20 + 15n = 110 → 15n = 90 → n = 6 classes.
(iii) (i) gives the output for a known input (substitute); (ii) gives a known output and asks for the input, so we set up and solve an equation.
Problem 2 — Road trip
Set up. Both parts rearrange d = st first.
(i) t = d/s = 195/65 = 3 hours.
(ii) s = d/t = 195/2.5 = 78 km/h.
(iii) 78 km/h is below the 110 km/h M5 limit, so it's legal. It is a reasonable, safe highway speed.
Problem 3 — Plumber callout
Set up. (i) substitute; (ii) solve.
(i) C = 90 + 75(3) = 90 + 225 = $315.
(ii) 90 + 75h = 315 → 75h = 225 → h = 3 hours.
(iii) Substitute back: 90 + 75(3) = 315 ✓. Matches the receipt, so the answer is reasonable.
Problem 4 — Phone-saving plan
Set up. Build the formula from the table, then substitute / solve.
(i) Starting balance = $80 (at w = 0). Per-week rate = $50 (130 − 80 = 50). Formula: B = 80 + 50w.
(ii) B(12) = 80 + 50(12) = 80 + 600 = $680.
(iii) Solve 80 + 50w = 1130 → 50w = 1050 → w = 21. She can buy the phone after 21 weeks.
Problem 5 — Vegetable plot
Set up. Substitute for (i); rearrange then substitute for (iii).
(i) A = 8 × 3.5 = 28 m².
(ii) Divide both sides by b: h = A / b.
(iii) h = 60 / 8 = 7.5 m.
(iv) Rearranging first makes the unknown the subject, so substitution gives the answer directly. Substituting before rearranging often leaves you with an equation you still have to solve, which is more error-prone.