Mathematics Standard • Year 12 • Module 6 • Lesson 7

Gantt Charts — Skill Drill

Build fluency in turning ES/EF/LF data into a clear Gantt chart — bars, float extensions and critical paths shown correctly.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 On a Gantt chart, the horizontal axis shows ____________ and the vertical axis shows ____________.

Q1.2 An activity has ES = 4 and EF = 9. Its solid bar starts at day ____ and ends at day ____.

Q1.3 Float on a Gantt chart is shown by extending the bar from ____ to ____ with a lighter or dashed shade.

Stuck? Revisit lesson § Gantt Chart Basics — bar position runs from ES to EF; float extends from EF to LF.

2. Worked example — drawing a Gantt chart from ES/EF/LF data

Each step explains exactly what to draw and why.

Problem. Draw a Gantt chart for: A(ES=0, EF=3, LF=3), B(ES=3, EF=7, LF=7), C(ES=3, EF=5, LF=7), D(ES=7, EF=10, LF=10). Show float on any non-critical activity.

Step 1 — Set up axes.

Horizontal: time from 0 to 10 (project duration is 10). Vertical: list activities A, B, C, D top-to-bottom.

Reason: max EF in the table is 10, so the chart must extend to day 10.

Step 2 — Identify critical vs non-critical.

A: LF − EF = 3 − 3 = 0 (critical). B: 7 − 7 = 0 (critical). C: 7 − 5 = 2 (float = 2). D: 10 − 10 = 0 (critical).

Step 3 — Draw bars from ES to EF.

A: solid bar from day 0 to day 3. B: solid bar from day 3 to day 7. C: solid bar from day 3 to day 5. D: solid bar from day 7 to day 10.

Step 4 — Extend float on non-critical activities.

C: lighter / dashed extension from day 5 to day 7 (the 2-day float). A, B, D have no extension.

Conclusion. Critical path A → B → D is visible as a continuous chain of solid bars from 0 to 10. C runs in parallel with B but has 2 days of slack.

3. Faded example — fill in the missing steps

Network data: A(ES=0, EF=4, LF=4), B(ES=4, EF=7, LF=7), C(ES=4, EF=6, LF=7), D(ES=7, EF=9, LF=9). Fill in each blank line. 4 marks

Step 1 — Project duration: Project duration = max EF = ____ days. Time axis runs 0 to ____.

Step 2 — Float for each activity:

A: LF − EF = 4 − 4 = ____. B: 7 − 7 = ____. C: 7 − 6 = ____. D: 9 − 9 = ____.

Step 3 — Bar positions (ES to EF):

A: ____ to ____. B: ____ to ____. C: ____ to ____. D: ____ to ____.

Step 4 — Float extensions:

Only ____ has float (1 day). Extend its bar from day ____ to day ____ in a lighter shade. All other bars are solid only.

Conclusion. Critical path is A → ____ → ____ . Project duration = ____ days.

Stuck? Revisit lesson § Worked Example — bar runs ES to EF, then a lighter extension shows the EF-to-LF float.

4. Graduated practice — Gantt chart skills

Sketch each chart in the space below, or describe each bar as "X: ES to EF, float = n" so your method is visible.

Foundation — single-activity reads (4 questions)

Q	Problem	Answer
4.1 1	Activity X: ES = 2, EF = 6. Where does its solid bar start and end?
4.2 1	Activity Y: EF = 8, LF = 11. How many days of float does Y have?
4.3 1	Activity Z: ES = 5, EF = 5. What special name is given to a zero-duration activity drawn as a diamond?
4.4 1	An activity has float = 0. Is it critical or non-critical?

Standard — typical HSC difficulty (6 questions)

Show working: list each bar as "activity: ES to EF, extension EF to LF if any".

4.5 Draw a Gantt chart for: A(0,3,3), B(3,6,6), C(3,5,6), D(6,9,9). Identify the critical path. 2 marks

4.6 Activities have ES/EF/LF: P(0,2,2), Q(2,5,5), R(2,4,5), S(5,8,8). State the float for each. 2 marks

4.7 Looking at the chart from 4.5, which activities are in progress on day 4? 2 marks

4.8 Draw a Gantt chart for: A(0,2,2), B(2,5,5), C(2,4,5), D(5,8,8), E(4,6,8). Show float as extensions. 3 marks

4.9 On the chart from 4.8, which day(s) show three activities running simultaneously? 2 marks

4.10 A Gantt chart shows 4 critical activities laid end-to-end from day 0 to day 14, with no gaps. What is the project duration, and how much float do the critical activities have? 2 marks

Extension — chart-reading + resources (2 questions)

4.11 Activities P, Q, R run as bars 0–4, 3–8, 5–10 respectively, and each requires 2 workers. Build a resource histogram (workers vs day) and state the peak demand. 3 marks

4.12 A Gantt chart shows: A(0,5,5) critical, B(5,9,9) critical, C(5,7,9) float 2, D(7,10,12) float 2. The site has only 2 workers and every activity needs 1 worker. Suggest how float can be used to keep peak demand at or below 2. 3 marks

Stuck on 4.11? List each day 0 → 10 and count how many of P, Q, R cover that day; multiply by 2 workers.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I put the bar's left edge at ES = 2 and its right edge at EF = 6 (not at LF).

For 4.2 I calculated float as LF − EF = 11 − 8 = 3 days.

For 4.4 I said "critical" because zero float means the activity has no slack.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Gantt chart axes

Horizontal: time (days/weeks). Vertical: activities.

Q1.2 — Bar position

Solid bar from day 4 to day 9 (ES to EF).

Q1.3 — Float extension

Extend from EF to LF.

Q3 — Faded example

Step 1: Project duration = 9 days; axis 0 to 9.
Step 2: Floats A = 0, B = 0, C = 1, D = 0.
Step 3: A: 0 to 4. B: 4 to 7. C: 4 to 6. D: 7 to 9.
Step 4: Only C has float. Extend from day 6 to day 7.
Conclusion: Critical path = A → B → D. Duration = 9 days.

Q4.1 — Bar from ES = 2, EF = 6

Solid bar from day 2 to day 6.

Q4.2 — Float for Y

Float = LF − EF = 11 − 8 = 3 days.

Q4.3 — Zero-duration activity

Milestone (drawn as a diamond on the Gantt chart).

Q4.4 — Float = 0

Critical. Zero float means no slack — any delay slips the project.

Q4.5 — Chart and critical path

A: solid 0–3. B: solid 3–6. C: solid 3–5, float extension 5–6. D: solid 6–9. Critical path: A → B → D (continuous solid bars from 0 to 9).

Q4.6 — Floats for P, Q, R, S

P: 2 − 2 = 0. Q: 5 − 5 = 0. R: 5 − 4 = 1. S: 8 − 8 = 0.

Q4.7 — Activities in progress on day 4

From 4.5: A finishes day 3, B runs 3–6, C runs 3–5, D starts day 6. On day 4: B and C are both in progress.

Q4.8 — Chart with five activities

A: solid 0–2. B: solid 2–5. C: solid 2–4, extension 4–5 (float = 1). D: solid 5–8. E: solid 4–6, extension 6–8 (float = 2). Critical path A → B → D.

Q4.9 — Three activities running together

Day 4 → day 5 has B (2–5), C-solid ends at 4 but C-extension runs to 5, and E starts day 4 (solid 4–6). If we count only solid bars: B and E both solid on day 4 to 5; C solid only on 2–4. So day 4 → 5 has B and E in active work plus C in float — three bars present. (Two are solid, one is float.)

Q4.10 — End-to-end critical bars

Project duration = 14 days. Critical activities always have float = 0 by definition.

Q4.11 — Resource histogram for P, Q, R

Day-by-day workers required:
Days 0–3: P only → 2 workers.
Days 3–4: P + Q → 4 workers.
Days 4–5: Q only → 2 workers (P just finished).
Days 5–8: Q + R → 4 workers.
Days 8–10: R only → 2 workers.
Peak demand = 4 workers (days 3–4 and 5–8).

Q4.12 — Using float to smooth resource demand

Current peak: B (5–9) and C (5–7) overlap → 2 workers needed days 5–7. D (7–10) and B (5–9) overlap → 2 workers needed days 7–9. Peak = 2 — already at the limit.
Suggestion: Use C's float (2 days) to delay C to start at day 7, finishing at day 9. Then C runs alongside D (days 7–9) but B has already finished by day 9. Recheck day-by-day to confirm new peak stays at 2. Float smoothing means shifting non-critical activities within their slack window so resource demand never exceeds capacity.