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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WDM11 · Decision Mathematics 1Critical path analysisActivity networks and critical paths

Critical path analysis · Subtopic 1 of 1

Critical Path Analysis

A project's activities and their dependencies form a network. A forward pass gives earliest times, a backward pass latest times. Activities with zero total float form the critical path, whose length is the minimum project duration. Total float $=$ latest finish $-$ earliest start $-$ duration. A lower bound for workers is $\left\lceil\frac{\text{total activity time}}{\text{duration}}\right\rceil$ .

Edexcel YMA01 WDM11Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Draw activity networks from precedence tables (with dummies where needed).
Find early and late times, the critical path and floats.
Draw Gantt charts and resource histograms; find a lower bound for workers.

Pause and think

To make breakfast you boil the kettle (3 min), make toast (4 min) and butter it after it's toasted (1 min). What's the shortest time?

Section 2 of 14

Core idea

Start hereBuilding a house

Foundations before walls, walls before roof; but plumbing and electrics can run at the same time. Some tasks can slip without delaying the move-in date; others can't. Critical path analysis finds which.

The big idea

Forward pass (max) gives earliest event times; backward pass (min) gives latest. Activities with zero total float form the critical path, and its length is the project's minimum duration.

1. Early and late times

Activities: A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D.

Worked exampleForward and backward passes

Find the duration, critical activities and floats.

Goal: Duration and floats.
Strategy: Max forwards, min backwards.

$D:\ \max(3,4)=4\to9$
D waits for both A and B.
$E:\ \max(5,9)=9\to12$
E waits for C and D.
$C:\ \text{latest finish }9\Rightarrow\text{float }9-3-2=4$
C only needs to finish by E's start.
$A:\ \text{latest finish }\min(7,4)=4\Rightarrow\text{float }1$
A must be done for D at 4.

AnswerDuration

12

; critical path B–D–E; floats A

1

, C

4

✓ CheckCritical durations

4+5+3=12

✓.

\text{total float}=\text{latest finish}-\text{earliest start}-\text{duration}

float: how long the activity can be delayed without delaying the project.

2. Resources

Lower bound for workers

$\text{total work}=\sum d=17$
Each activity needs one worker for its duration.
$\text{workers}\times T\ge17$
$w$ workers over $T=12$ days provide at most $12w$ worker-days.
$w\ge\left\lceil\tfrac{17}{12}\right\rceil=2$
Round up.

CPA workflow

Drawing diagram…

From data to schedule.

Common misconceptionMax and min Forward: max of incoming. Backward: min of outgoing. Swapping them is the commonest error.

Key takeawayDummies Dummy activities (duration 0, dashed) show dependence without work, e.g. when D needs A and B but C needs only A.

Pause and think

If C takes 6 days instead of 2, what changes?

Summary

Forward: max; backward: min.
Critical: zero float.
Workers $\ge\lceil\sum d/T\rceil$ .

Critical path: Longest chain of activities; sets the duration.
Total float: Possible delay without delaying the project.
Gantt chart: Bar chart of activity timings.

HistoryPolaris and DuPont. CPA and PERT were developed in 1957–58 for DuPont chemical plants and the US Navy's Polaris missile, cutting project times dramatically.

Fun factPit stops. An F1 pit stop of about 2 seconds is a CPA masterpiece: around 20 people working in parallel on a planned critical path.

Section 3 of 14

Key formulae

Forward pass

e_j=\max(e_i+d_{ij})

Backward pass

l_i=\min(l_j-d_{ij})

Float and workers

\text{float}=l_j-e_i-d;\quad\text{workers}\ge\left\lceil\tfrac{\sum d}{T}\right\rceil

TrickCheck the critical path. On a critical activity, early and late times at both ends match and the duration fills the gap exactly.

Section 4 of 14

Worked examples

Example 1 · Critical path0 / 3 steps

A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D. Find the duration and critical path.

Example 2 · Float0 / 2 steps

Find the floats of C and A.

Section 5 of 14

Explore the graph

Resource histogram (all activities start early)

Loading the interactive graph…

Try this

Workers needed if everything starts as early as possible: 2 until time 5, then 1.
The line at $\frac{17}{12}$ is the average workload: at least 2 workers are needed.

Total activity time $17$ over $12$ days.

In the real worldConstruction. Building firms track critical paths daily: a delay to a critical activity delays the whole building; float lets other tasks slip safely.

Section 6 of 14

Common mistakes

IncorrectForward pass uses min

CorrectUse max

An event can't happen until all preceding activities finish.

IncorrectBackward pass uses max

CorrectUse min

Must be early enough for every following activity.

IncorrectLower bound rounded down

CorrectRound up

You can't have part of a worker.

Activity with float 0

Me with no time to spare all week.

Section 7 of 14

Exam tip

Exam tipDummies. Use a dummy when two activities share some but not all predecessors, or to stop two activities having the same start and end events.

Section 8 of 14

Guided practice

SA-DM-CP-G012 marksL2 StandardNon-calculatorCritical path

Activities: A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D. Find the minimum project duration.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-DM-CP-0012 marksL2 StandardNon-calculatorCritical path

Find the total float of activity C.

Salam Academy Original Practice · Proposed mark scheme

SA-DM-CP-0022 marksL3 ExamNon-calculatorCritical path

Find the total float of activity A.

Salam Academy Original Practice · Proposed mark scheme

SA-DM-CP-0032 marksL2 StandardNon-calculatorScheduling

Each activity needs one worker. Find a lower bound for the number of workers needed to finish in

12

days.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-DM-CP-0042 marksL3 ExamNon-calculatorCritical path

Activity B is delayed by

2

days. Find the new minimum duration.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Forward pass: max.
Backward pass: min.
Critical: zero float.
Workers $\ge\lceil\frac{\sum d}T\rceil$ .

Section 12 of 14

Quick quiz

Q1 · SA-DM-CP-Q011 markL1 FoundationNon-calculatorCritical path

A critical activity is one that

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Decision 1 · Critical path analysis
Expected: Activity-on-arc networks with dummies; early and late event times; critical activities; total float; Gantt (cascade) charts; resource histograms; lower bound for workers; scheduling.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: YMA01 · WDM11 9FM0 · Option

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).