Mathematics Standard • Year 11 • Module 1 • Lesson 2
Solving One-Step and Two-Step Equations
Apply inverse operations to real Australian scenarios — taxi fares, gym fees, pay packets, savings goals, and event hire pricing.
Problem 1 — Cinema booking (form an equation and solve)
A cinema booking costs $5 plus $12 per ticket. The total is $41.
Set up: What are we solving for?
(i) Let t be the number of tickets. Write an equation that models this booking. 1 mark
(ii) Solve the equation to find the number of tickets. Show every balancing step. 2 marks
(iii) Check your answer by substituting back into the original equation. 1 mark
Stuck? Revisit lesson § Revisit the Ticket Problem — the equation is 5 + 12t = 41.Problem 2 — Sydney taxi fare
A taxi fare is $7 flagfall plus $3 per kilometre. The total fare for a trip was $31.
Set up: What are we solving for?
(i) Let k be the number of kilometres. Write an equation for this fare. 1 mark
(ii) Solve for k. 2 marks
(iii) A different taxi has a $5 flagfall plus $3.50 per kilometre. For the same $31 total fare, would the trip be longer or shorter? Solve and explain in one sentence. 2 marks
Stuck? Subtract the flagfall first, then divide by the per-km rate.Problem 3 — Gym membership target
A gym charges a $60 joining fee plus $22 per week. Sam has budgeted $390 for joining and his first weeks of membership.
Set up: What are we solving for?
(i) Let w be the number of weeks Sam can afford. Write the equation. 1 mark
(ii) Solve for w. 2 marks
(iii) State the maximum whole number of weeks Sam can afford, and explain in one sentence why you rounded that way. 2 marks
Stuck? "Can afford" means total cost ≤ budget — if w comes out as 15 exactly, that's the max; if it comes out 15.3 you round DOWN to 15.Problem 4 — Casual pay packet
A casual worker is paid a $30 weekly base allowance plus $24 per hour worked. This week she received $222 in total.
Set up: What are we solving for?
(i) Let h be the number of hours worked. Write the equation. 1 mark
(ii) Solve for h and check by substitution. 2 marks
(iii) The following week the worker is paid $342 in total at the same rates. Without re-doing all the working, explain in one sentence how you would adjust the equation to find the new hours. 1 mark
Stuck? The base allowance stays the same — only the right-hand side total changes.Problem 5 — Party hire bracketed pricing
A party-hire company quotes the cost C of an event using C = 5(n + 12), where n is the number of guests above 12 (so n = 0 if exactly 12 guests turn up).
A customer was charged $135. Find how many guests they booked for above 12, and the total guest count.
Set up: What are we solving for?
(i) Write the equation that matches the situation. 1 mark
(ii) Solve for n in two ways: (a) divide both sides by 5 first; (b) expand the brackets first. 3 marks
(iii) State the total guest count and write a conclusion sentence. 1 mark
Stuck? Revisit lesson § Worked Example 3 — Solve an equation with brackets.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Cinema booking
Set up. Form an equation in t for the total, then solve and check.
(i) 5 + 12t = 41 (or 12t + 5 = 41).
(ii) Subtract 5: 12t = 36. Divide by 12: t = 3.
(iii) Check: 5 + 12(3) = 5 + 36 = 41 ✓. Three tickets were booked.
Problem 2 — Taxi fare
Set up. Form an equation for the total fare and solve for distance.
(i) 7 + 3k = 31.
(ii) Subtract 7: 3k = 24. Divide by 3: k = 8 km.
(iii) Second taxi: 5 + 3.50k = 31 ⇒ 3.50k = 26 ⇒ k = 26/3.50 ≈ 7.43 km. The trip is shorter because the higher per-km rate ($3.50 vs $3.00) eats the lower flagfall faster.
Problem 3 — Gym membership
Set up. Form an equation for total cost and solve for the number of weeks.
(i) 60 + 22w = 390.
(ii) Subtract 60: 22w = 330. Divide by 22: w = 15.
(iii) Sam can afford 15 weeks exactly. Since 22w = 330 divides cleanly, no rounding is needed. (If the answer had been, say, 15.3, we would round DOWN to 15 because going over would exceed the budget.)
Problem 4 — Casual pay packet
Set up. Form an equation for weekly pay and solve for hours.
(i) 30 + 24h = 222.
(ii) Subtract 30: 24h = 192. Divide by 24: h = 8 hours. Check: 30 + 24(8) = 30 + 192 = 222 ✓.
(iii) Replace the right-hand side with 342: 30 + 24h = 342 ⇒ h = 312/24 = 13 hours. Only the total changes; the base and hourly rate stay the same.
Problem 5 — Party hire
Set up. Form a bracketed equation and solve two ways for the extra-guest count.
(i) 5(n + 12) = 135.
(ii) (a) Divide first. n + 12 = 27 ⇒ n = 15. (b) Expand first. 5n + 60 = 135 ⇒ 5n = 75 ⇒ n = 15. Both methods give n = 15.
(iii) Total guests = 12 + n = 12 + 15 = 27. The booking was for 27 guests.