QCE General Mathematics - Unit 4 - Networks and decision mathematics 1
Critical Path Analysis | QCE General Mathematics
Learn activity-on-arc project networks, forward and backward scans, earliest and latest starting times, critical paths and float times.
Updated 2026-05-18 - 5 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Construct a project network diagram (activity on arc) to represent the durations and interdependencies of activities that must be completed during the project (excluding dummy activities).
- Use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) for each activity in the project.
- Use ESTs and LSTs to locate the critical path/s for a project.
- Use the critical path to determine the minimum time for a project to be completed.
- Calculate float times for non-critical activities.
- Solve small-scale practical problems involving critical path analysis.
Critical path analysis is used to plan projects where some activities must happen before others. The aim is to find the shortest possible project duration and identify which activities cannot be delayed.
Original Sylligence diagram for general critical path.
Activity-on-arc networks
In an activity-on-arc network, activities are shown on arcs and events are shown as nodes. The number on an activity is its duration. The network must respect prerequisites. If activity $C$ cannot begin until $A$ and $B$ finish, the diagram must show that dependency.
Forward scan
The forward scan finds earliest event times. Start at $0$ and move through the network. At a merge point, take the maximum incoming time because all prerequisite activities must be finished.
Backward scan
The backward scan finds latest event times. Start at the final project time and move backwards. At a split point, take the minimum outgoing time so the project is not delayed.
Critical path and float
The critical path is a chain of activities with zero float. Delaying a critical activity delays the whole project. Float is the amount of time a non-critical activity can be delayed without increasing the project duration.
Worked example
EST and LST conventions
For each event node, the earliest start time is found by the forward scan and the latest start time is found by the backward scan. In many diagrams, these are written near the node as a pair of values.
At a merge, use the maximum because every incoming prerequisite must be complete. At a split in the backward scan, use the minimum because the project must still finish on time along every outgoing path.
Current syllabus note on dummy activities
Older worked examples may include dummy activities to show dependencies. The 2025 General Mathematics syllabus excludes dummy activities, so Sylligence notes should focus on activity-on-arc networks that do not require them.
Float formula
For an activity from event $i$ to event $j$ with duration $d$:
$ \text{float}=\text{LST at }j-\text{EST at }i-d $
Zero-float activities are critical. A critical path is any complete start-to-finish chain of zero-float activities.
Depth: activity networks
Critical path analysis models a project as activities with durations and dependencies. An activity can start only after all prerequisite activities have finished.
Key quantities:
| Quantity | Meaning | |---|---| | EST | earliest start time | | EFT | earliest finish time | | LST | latest start time without delaying the project | | LFT | latest finish time without delaying the project | | float | time an activity can be delayed without delaying the project |
The forward pass finds earliest times. The backward pass finds latest times.
Forward and backward pass
For an activity:
$ EFT=EST+\text{duration} $
If an activity has several prerequisites, its EST is the largest EFT among the prerequisites.
For the backward pass:
$ LST=LFT-\text{duration} $
If an activity leads into several later activities, its LFT is the smallest LST among the following activities.
Float and critical activities
Float can be calculated as:
$ \text{float}=LST-EST $
or:
$ \text{float}=LFT-EFT $
Critical activities have zero float. The critical path is the chain of critical activities that determines the minimum project time.
Interpreting project changes
If a critical activity is delayed by 2 days, the whole project is usually delayed by 2 days unless another change reduces the critical path. If a non-critical activity is delayed by less than or equal to its float, the project finish time is unchanged.
The 2025 syllabus excludes dummy activities, so notes and questions should focus on dependency tables and activity networks that do not require dummy arcs.
Depth: dependency table workflow
If the project is given as a dependency table, draw the network before calculating times.
| Activity | Depends on | Duration | |---|---|---:| | A | none | 3 | | B | A | 4 | | C | A | 2 | | D | B, C | 5 |
Here, D cannot start until both B and C are finished. Its EST is therefore the larger of B's EFT and C's EFT, not the smaller or the average.
This "wait for all prerequisites" rule is one of the main ideas in CPA.