Mathematics Standard • Year 11 • Module 1 • Lesson 3
Equations from Worded Problems
Apply the define–translate–solve–interpret process to real Australian scenarios — events, mobile bills, formal hire, savings goals and small-business invoices.
Problem 1 — School formal venue hire
A school formal committee has booked a venue that charges a $400 hall fee plus $65 per student attending. The committee has a budget of $4,955.
Set up: What are we solving for?
(i) Let s be the number of students who can attend. Write an equation that matches the budget exactly. 1 mark
(ii) Solve the equation for s. 2 marks
(iii) Write a one-sentence interpretation that includes the number and what it means. 1 mark
Stuck? Revisit lesson § Four-Step Translation Process — total = fixed fee + per-student rate × number of students.Problem 2 — Mobile phone bill check
A mobile plan costs $40 per month plus $0.18 per minute of overseas calls. This month's bill came to $76.
Set up: What are we solving for?
(i) Let m be the number of overseas minutes. Write the equation. 1 mark
(ii) Solve for m. 2 marks
(iii) If the customer remembers making only 150 minutes of calls, is the bill correct? Calculate what the bill should be and explain in one sentence whether they have been overcharged. 2 marks
Stuck? Substitute m = 150 into 40 + 0.18m and see if you get $76.Problem 3 — Jumping castle hire
A jumping castle company charges a $120 delivery fee plus $45 per hour. A parent paid $345 in total for one party.
Set up: What are we solving for?
(i) Let h be the number of hours hired. Write the equation. 1 mark
(ii) Solve for h. 2 marks
(iii) A competitor charges a $80 delivery fee plus $55 per hour. For the same total of $345, how many hours would the competitor's deal give? Which deal is better-value for THIS hire? 2 marks
Stuck? Build a second equation 80 + 55h = 345 and solve. "Better value" = more hours for the same dollar.Problem 4 — Saving for a laptop
Anika has saved $180 so far and adds $25 from her casual job each week. She wants to reach $580 to buy a laptop.
Set up: What are we solving for?
(i) Let w be the number of weeks. Write an equation that models reaching the target. 1 mark
(ii) Solve for w. 2 marks
(iii) Anika hears about a $50 weekly bonus for the next 4 weeks. How many fewer weeks will it take her to reach $580 if she gets the bonus for those first 4 weeks? 2 marks
Stuck? Bonus pay for 4 weeks is 4 × $50 = $200 extra. Reduce the gap to $580 by that amount before dividing by $25.Problem 5 — Tradie invoice check
A plumber's invoice has the following items:
Callout fee: $110.00
Labour: ____ hours at $95/hour = $ ____________
Parts: $74.50
Invoice total: $469.00
Set up: What are we solving for?
(i) Let h = the number of labour hours billed. Write an equation that uses all of the line items. 2 marks
(ii) Solve for h. 2 marks
(iii) The customer believes the job only took 2.5 hours. What total should the invoice have been? State whether (and by how much) they have been overcharged. 2 marks
Stuck? Equation: 110 + 95h + 74.50 = 469. Combine constants first (110 + 74.50 = 184.50) to simplify.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — School formal
Set up. Form a budget equation in s and solve for the number of students.
(i) 400 + 65s = 4955.
(ii) Subtract 400: 65s = 4555. Divide by 65: s = 70.
(iii) The committee can afford to invite 70 students within the $4,955 budget.
Problem 2 — Phone bill
Set up. Form an equation for the monthly bill and solve for minutes.
(i) 40 + 0.18m = 76.
(ii) Subtract 40: 0.18m = 36. Divide by 0.18: m = 200 minutes.
(iii) If only 150 min: 40 + 0.18(150) = 40 + 27 = $67. The bill should have been $67, so they have been overcharged by $9 ($76 − $67), or equivalently billed for 50 minutes they don't remember making.
Problem 3 — Jumping castle
Set up. Form a total-cost equation and solve for hours; repeat for the competitor.
(i) 120 + 45h = 345.
(ii) Subtract 120: 45h = 225. Divide by 45: h = 5 hours.
(iii) Competitor: 80 + 55h = 345 ⇒ 55h = 265 ⇒ h ≈ 4.82 hours. The original company is better value for this $345 hire because it gives 5 hours vs about 4.8 hours from the competitor.
Problem 4 — Saving for a laptop
Set up. Form an equation for the cumulative savings reaching the target.
(i) 180 + 25w = 580.
(ii) Subtract 180: 25w = 400. Divide by 25: w = 16 weeks.
(iii) Bonus adds 4 × $50 = $200 over the first 4 weeks. New target gap = 580 − 180 − 200 = $200, paid off at $25/week needs 200/25 = 8 more weeks. Total = 4 + 8 = 12 weeks. That is 4 fewer weeks than the original 16.
Problem 5 — Plumber invoice
Set up. Form one equation including callout + labour + parts = total, solve for hours, then re-check at the customer's claimed hours.
(i) 110 + 95h + 74.50 = 469.
(ii) Combine constants: 184.50 + 95h = 469 ⇒ 95h = 284.50 ⇒ h = 284.50 / 95 = 2.9947... ≈ 3 hours (likely rounded from 3 h on the invoice).
(iii) At 2.5 hours: 110 + 95(2.5) + 74.50 = 110 + 237.50 + 74.50 = $422.00. The customer was charged $469 vs $422, an overcharge of $47 (which is roughly 0.5 hours × $95 — i.e. the disputed half-hour of labour).