The idea, step by step
The chain that cannot be delayed
The critical path is a longest-duration directed path from the start event to the finish event. Its duration equals the earliest project completion. If an activity on that controlling chain takes longer, the finish usually moves later unless a change elsewhere compensates for it.
An activity is critical when its total float is zero. To form a critical path, follow connected zero-float arrows from start to finish. A collection of disconnected zero-float labels is not by itself a path. Dummies may appear on a critical route even though they have no duration.
There can be more than one
Different branches can have the same maximum duration. In that case the project has multiple critical paths. Shortening only one may not change the project finish because another remains at the old maximum.
In the displayed project, A–D–E–F takes 4+4+2+3=13 days. A–C–F takes 12, while B–dummy–E–F takes 8. The critical path is therefore A–D–E–F. The result is about elapsed project time, not the total amount of work.
| Activity | Duration | Immediate predecessors |
|---|---|---|
| A | 4 | None |
| B | 3 | None |
| C | 5 | A |
| D | 4 | A |
| E | 2 | B and D |
| F | 3 | C and E |
A common trap
Calling the shortest route critical because it looks most efficient.
Instead: The project must wait for the slowest required chain. The longest complete route controls its minimum completion time.
See the thinking
Compare complete paths
A project has complete path durations 17, 14, 17 and 11 hours. State the minimum completion time and the number of critical paths.
- Every branch must finish.
- The largest complete-path duration is 17.
- Two paths have that duration.
Two parallel routes
A and B start together. A takes 2 days then C takes 3; B takes 4 days then D takes 1. A final task E takes 2 days after C and D. Find the critical paths.
- A–C–E lasts 2+3+2=7.
- B–D–E lasts 4+1+2=7.
- Both routes control the same finish.
Check your understanding
One critical path takes 20 days and another takes 20. If an activity found only on the first is shortened by 2 days, must the project finish at 18?
Show answer
No; the other critical path still requires 20 days.
All controlling paths must be considered when planning a time reduction.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Find the critical path and its duration for the displayed network.3 marks
- Complete paths have durations 9, 15, 12 and 15 weeks. Find the completion time and number of critical paths.2 marks
- A project starts with P (3 hours) and Q (5 hours) in parallel. R (4 hours) follows P and S (2 hours) follows Q. T (1 hour) follows both R and S. Find all critical paths.3 marks
- A path contains a zero-duration dummy. Can it still be critical?2 marks
- A network shows three connected zero-float activities beginning at the project start but not reaching the finish. Is that list enough to name a complete critical path?2 marks
- The critical path has duration 24 hours. A non-critical path takes 21. If an activity appearing only on the non-critical path is extended by 4 hours, what is the revised completion time?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find the critical path and its duration for the displayed network.3 marks
A–C–F=12; A–D–E–F=13; B–dummy–E–F=8.
The maximum is 13.
A–D–E–F, 13 days. - Complete paths have durations 9, 15, 12 and 15 weeks. Find the completion time and number of critical paths.2 marks
Take the largest complete-path duration.
Both 15-week routes are controlling.
15 weeks and two critical paths. - A project starts with P (3 hours) and Q (5 hours) in parallel. R (4 hours) follows P and S (2 hours) follows Q. T (1 hour) follows both R and S. Find all critical paths.3 marks
3+4+1=8.
5+2+1=8.
P–R–T and Q–S–T, both 8 hours. - A path contains a zero-duration dummy. Can it still be critical?2 marks
A critical route is determined by total duration and dependency structure.
A dummy may be needed to connect that controlling route.
Yes. - A network shows three connected zero-float activities beginning at the project start but not reaching the finish. Is that list enough to name a complete critical path?2 marks
A critical path must run continuously from the start event to the finish event.
Continue tracing the controlling arrows.
No. - The critical path has duration 24 hours. A non-critical path takes 21. If an activity appearing only on the non-critical path is extended by 4 hours, what is the revised completion time?3 marks
The changed path becomes 21+4=25.
The new maximum is max(24,25)=25, so the critical path changes.
25 hours.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: discrete models, project scheduling and assignment. Activity-on-arrow conventions are used for project networks; technology may verify calculations but does not replace interpreting the model.
Official curriculum source ↗