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.