Mathematics Standard • Year 12 • Module 6 • Lesson 1
Introduction to Project Networks — Problem Set
Apply project-network vocabulary and minimum-time analysis to realistic Australian scheduling scenarios: a kitchen renovation, a music festival, a school assignment, a cafe opening and a car-service workshop.
Problem 1 — Kitchen renovation (Sydney)
A Sydney homeowner is renovating a kitchen. The contractor lists the tasks below in days.
Demo old kitchen DM(2, —)
Plumbing rough-in PR(3, DM)
Electrical rough-in ER(2, DM)
Cabinet install CI(4, PR and ER)
Benchtop BT(2, CI)
Set up: What are we solving for?
(i) Which activity is the start activity and which activities can run in parallel after DM finishes? 1 mark
(ii) Find every path from DM to BT, including the path-sum of durations in days. 2 marks
(iii) State the minimum number of days needed to finish the kitchen renovation, and clearly name the controlling path. 2 marks
Stuck? Revisit lesson § Activities and Events — CI cannot start until both PR and ER finish.Problem 2 — Music festival stage build
A Melbourne event company is building a festival stage. The activities (hours) and dependencies are:
Truss assembly T(6, —)
Lighting rig L(4, T)
Sound rig S(5, T)
Backdrop B(3, T)
Final cue check C(2, L and S and B)
Set up: What are we solving for?
(i) Which activities can run in parallel after Truss assembly? 1 mark
(ii) Calculate the minimum total time (in hours) to be ready for the cue check. Show the controlling path. 2 marks
(iii) The lighting designer asks: "If I slow down the lighting rig by 1 hour, will we miss the cue check?" Decide yes or no, and justify in one sentence using the path-sums. 2 marks
Stuck? Revisit lesson § Dummy Activities and parallel tasks — three activities all starting after T form three parallel paths to C.Problem 3 — Group assignment (Year 12 study)
A study group plans a 12-day Maths Standard project. Task list in days:
Choose topic CH(1, —)
Research RE(3, CH)
Survey design SD(2, CH)
Run survey RS(4, SD)
Analyse results AN(2, RE and RS)
Write report WR(3, AN)
Set up: What are we solving for?
(i) List every path from CH to WR and write its total duration. 2 marks
(ii) Find the minimum project time and identify the path that controls it. 2 marks
(iii) The group has 12 calendar days. Decide whether they can finish on time without changing any task, and explain in one sentence. 1 mark
Stuck? Revisit lesson § Precedence Tables — AN must wait for BOTH RE and RS to finish.Problem 4 — Pop-up cafe opening
A new pop-up cafe in Brisbane plans its opening week. Tasks in days:
Lease signed LS(1, —)
Fit-out FO(5, LS)
Equipment delivery ED(3, LS)
Council inspection CI(1, FO and ED)
Staff training ST(2, CI)
Soft launch SL(1, ST)
Set up: What are we solving for?
(i) Which two activities can run in parallel and which activity is a merge point? 1 mark
(ii) Find the minimum number of days from signing the lease to the soft launch. Show the controlling path. 2 marks
(iii) The owner asks whether speeding up Equipment delivery by 1 day will bring the soft launch forward. Decide yes or no, with one short reason. 2 marks
Stuck? Compare path-sums LS → FO → CI → ST → SL and LS → ED → CI → ST → SL — only the longer one controls the finish.Problem 5 — Workshop car service (mechanical apprentice rota)
A regional NSW mechanic has scheduled the following tasks for a single car service. Durations are in minutes:
Diagnostic DG(15, —)
Oil change OC(20, DG)
Brake check BC(25, DG)
Tyre rotate TR(10, DG)
Road test RT(20, OC and BC and TR)
Invoice IV(5, RT)
Set up: What are we solving for?
(i) List every path from DG to IV and its total time in minutes. 2 marks
(ii) Find the minimum total service time in minutes, and name the controlling path. 2 marks
(iii) The customer asks if the car can be ready in 60 minutes from drop-off. Decide yes or no, then explain in one sentence using the controlling-path total. 2 marks
Stuck? Revisit lesson § Strategy — list every path, sum durations, take the maximum. Then compare to 60.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Kitchen renovation
Set up. We are finding the minimum number of days from demolition (DM) to benchtop (BT) by listing every path and taking the maximum.
(i) Start activity = DM. After DM, the parallel pair is PR and ER (both depend only on DM and neither depends on the other).
(ii) Two paths from DM to BT:
DM → PR → CI → BT = 2 + 3 + 4 + 2 = 11 days.
DM → ER → CI → BT = 2 + 2 + 4 + 2 = 10 days.
(iii) Minimum project time = max(11, 10) = 11 days, controlled by DM → PR → CI → BT. (Common error: just adding all activity durations — wrong, because PR and ER overlap.)
Problem 2 — Music festival stage build
Set up. We are finding the time from truss to cue check across three parallel branches that all merge into C.
(i) After Truss assembly the three parallel branches are L, S and B.
(ii) T → L → C = 6 + 4 + 2 = 12. T → S → C = 6 + 5 + 2 = 13. T → B → C = 6 + 3 + 2 = 11. Min total = max(12, 13, 11) = 13 hours, controlled by T → S → C.
(iii) No — slowing lighting (L) by 1 hour makes the L-path 13 hours, which only ties the sound path. The cue check still happens at 13 hours total. (L was not on the controlling path; it had 1 hour of slack before — exactly enough to absorb the slowdown.)
Problem 3 — Group assignment
Set up. We need the minimum days to complete WR, given AN must wait for both RE and RS.
(i) Path A: CH → RE → AN → WR = 1 + 3 + 2 + 3 = 9. Path B: CH → SD → RS → AN → WR = 1 + 2 + 4 + 2 + 3 = 12.
(ii) Minimum project time = max(9, 12) = 12 days, controlled by CH → SD → RS → AN → WR.
(iii) Yes — but only just. 12 = 12, so the group finishes exactly on day 12 if every task runs on time. Any delay on the controlling path would push them over the deadline.
Problem 4 — Pop-up cafe opening
Set up. We are finding the minimum days from LS to SL, then deciding whether speeding up ED helps.
(i) Parallel pair after LS: FO and ED. Merge point: CI (which needs both).
(ii) LS → FO → CI → ST → SL = 1 + 5 + 1 + 2 + 1 = 10 days. LS → ED → CI → ST → SL = 1 + 3 + 1 + 2 + 1 = 8 days. Minimum = max(10, 8) = 10 days, controlled by LS → FO → CI → ST → SL.
(iii) No — ED is not on the controlling path (it has 2 days of slack already). Speeding ED up changes the ED-path from 8 to 7 days but the project is still controlled by FO's 10-day path. The owner should crash FO instead.
Problem 5 — Workshop car service
Set up. We are finding the minimum total service time (in minutes) when three after-diagnostic tasks all merge into the road test.
(i) Three paths from DG to IV:
DG → OC → RT → IV = 15 + 20 + 20 + 5 = 60.
DG → BC → RT → IV = 15 + 25 + 20 + 5 = 65.
DG → TR → RT → IV = 15 + 10 + 20 + 5 = 50.
(ii) Minimum service time = max(60, 65, 50) = 65 minutes, controlled by DG → BC → RT → IV (the brake-check branch is the slowest).
(iii) No — the controlling path is 65 minutes, which is more than the customer's 60-minute target. The car cannot be ready in 60 minutes unless the brake check is shortened. (Common error: adding all activity times to get 95 minutes; that ignores parallel work.)