---
title: "CPM network, 9 activities (A=5 to I=2)"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:cpm-network-nine-act-a75"
source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-nine-act-a75
site: BCT Engineering
---

# CPM network, 9 activities (A=5 to I=2)

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2075 Asoj, 12 marks
- Video: [Use forward and backward pass to determine project duration and critical path](https://www.youtube.com/watch?v=4oDLMs11Exs) — Engineer4Free
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-nine-act-a75
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Draw the CPM network diagram and compute EST, EFT, LST, LFT, TF, FF, Int.F and Ind.F from the information given below. Also compute the project duration and mark the critical path.

| Activity | Duration (week) | Predecessor |
|---|---|---|
| A | 5 | - |
| B | 4 | - |
| C | 0 | A |
| D | 6 | A |
| E | 7 | B, C |
| F | 8 | B, C |
| G | 6 | D, E |
| H | 3 | F |
| I | 2 | G, H |

## Short answer

Project duration is 20 weeks; critical path is A-C-E-G-I. EST/EFT/LST/LFT computed by forward/backward pass; total floats: A,C,E,G,I zero, B and D 1, F and H 2.

## Answer

Notation: EST/EFT = earliest start/finish, LST/LFT = latest start/finish, TF = total float, FF = free float, Int.F = interfering float, Ind.F = independent float.

#### Network diagram (activity-on-arrow)

Each activity is an arrow with its duration in brackets (weeks). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only.

```text
(1) --A(5)--> (2)
(1) --B(4)--> (3)
(2) --C(0)--> (3)
(2) --D(6)--> (4)
(3) --E(7)--> (4)
(3) --F(8)--> (5)
(4) --G(6)--> (6)
(5) --H(3)--> (6)
(6) --I(2)--> (7)
```

The network has 7 events and 0 dummy activities. Start event is (1) and the end event is (7).

Activity C has zero duration, so it is treated as a normal activity of 0 weeks (it only passes the logic on).

#### Forward pass (EST and EFT)

Rule: EST = largest EFT of the preceding activities (0 for start activities); EFT = EST + D.

- A: EST = 0; EFT = 0 + 5 = 5
- B: EST = 0; EFT = 0 + 4 = 4
- C: EST = EFT of A = 5; EFT = 5 + 0 = 5
- D: EST = EFT of A = 5; EFT = 5 + 6 = 11
- E: EST = max(B:4, C:5) = 5; EFT = 5 + 7 = 12
- F: EST = max(B:4, C:5) = 5; EFT = 5 + 8 = 13
- G: EST = max(D:11, E:12) = 12; EFT = 12 + 6 = 18
- H: EST = EFT of F = 13; EFT = 13 + 3 = 16
- I: EST = max(G:18, H:16) = 18; EFT = 18 + 2 = 20

Project duration = largest EFT = **20 weeks**.

#### Backward pass (LFT and LST)

Rule: LFT = smallest LST of the succeeding activities (project duration for end activities); LST = LFT - D.

- I: LFT = project duration = 20; LST = 20 - 2 = 18
- H: LFT = LST of I = 18; LST = 18 - 3 = 15
- G: LFT = LST of I = 18; LST = 18 - 6 = 12
- F: LFT = LST of H = 15; LST = 15 - 8 = 7
- E: LFT = LST of G = 12; LST = 12 - 7 = 5
- D: LFT = LST of G = 12; LST = 12 - 6 = 6
- C: LFT = min(E:5, F:7) = 5; LST = 5 - 0 = 5
- B: LFT = min(E:5, F:7) = 5; LST = 5 - 4 = 1
- A: LFT = min(C:5, D:6) = 5; LST = 5 - 5 = 0

#### Floats

- **Total float** $TF = LST - EST = LFT - EFT$
- **Free float** $FF = \min(\text{EST of successors}) - EFT$
- **Interfering float** $IntF = TF - FF$
- **Independent float** $IndF = \max\left(0,\ \min(\text{EST of successors}) - \max(\text{LFT of predecessors}) - D\right)$

#### Result table

| Activity | D | EST | EFT | LST | LFT | TF | FF | Int.F | Ind.F |
|---|---|---|---|---|---|---|---|---|---|
| A | 5 | 0 | 5 | 0 | 5 | 0 | 0 | 0 | 0 |
| B | 4 | 0 | 4 | 1 | 5 | 1 | 1 | 0 | 1 |
| C | 0 | 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| D | 6 | 5 | 11 | 6 | 12 | 1 | 1 | 0 | 1 |
| E | 7 | 5 | 12 | 5 | 12 | 0 | 0 | 0 | 0 |
| F | 8 | 5 | 13 | 7 | 15 | 2 | 0 | 2 | 0 |
| G | 6 | 12 | 18 | 12 | 18 | 0 | 0 | 0 | 0 |
| H | 3 | 13 | 16 | 15 | 18 | 2 | 2 | 0 | 0 |
| I | 2 | 18 | 20 | 18 | 20 | 0 | 0 | 0 | 0 |

Event times (TE from the forward pass, TL from the backward pass):

| Event | TE (earliest) | TL (latest) | Slack |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 5 | 5 | 0 |
| 3 | 5 | 5 | 0 |
| 4 | 12 | 12 | 0 |
| 5 | 13 | 15 | 2 |
| 6 | 18 | 18 | 0 |
| 7 | 20 | 20 | 0 |

#### Critical path and project duration

- Critical activities (TF = 0): A, C, E, G, I
- Critical path: A - C - E - G - I (= 5+0+7+6+2 = 20 weeks)

**Answer: project duration = 20 weeks; critical path = A - C - E - G - I**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Activity E follows B (EFT 4) and C (EFT 5). What is the EST of E?
   A. 4
   B. 5
   C. 9
   D. 0
   Answer: B. EST is the largest EFT of the predecessors: max(4, 5) = 5. (Key point: Forward pass rule)

2. What is the project duration?
   A. 18 weeks
   B. 22 weeks
   C. 20 weeks
   D. 16 weeks
   Answer: C. The largest EFT is that of I: EST 18 + 2 = 20 weeks. (Key point: Project duration)

3. Which is the critical path (all TF = 0)?
   A. B - F - H - I
   B. A - D - G - I
   C. A - C - F - H - I
   D. A - C - E - G - I
   Answer: D. A (5) + C (0) + E (7) + G (6) + I (2) = 20 weeks, and these have zero total float. (Key point: Critical path)

4. Activity F has TF = 2 and FF = 0. What is its interfering float?
   A. 2
   B. 0
   C. 1
   D. 4
   Answer: A. Int.F = TF - FF = 2 - 0 = 2. (Key point: Interfering float)

## Close variants set in other exams

- [CPM network, 9 activities (A=3 to I=1)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-b82) — Project Engineering, chapter 3, asked 5×
- [CPM network, 13 activities (A to M)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-thirteen-act-k82) — Project Engineering, chapter 3, asked 5×
- [CPM network, 13 activities (A=2 to M=7)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-thirteen-act-b81) — Project Engineering, chapter 3, asked 5×

## Related in Project Engineering

- [CPM network, 12 activities (A=10 to L=8)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-twelve-act-as67) — Project Engineering, chapter 3
- [CPM network, 12 activities (A=8 to L=3)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-twelve-act-b66) — Project Engineering, chapter 3
- [CPM network, 10 activities (A=4 to J=9)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-as70) — Project Engineering, chapter 3
- [CPM network, 10 activities with successors](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-a74) — Project Engineering, chapter 3
- [CPM network, 10 activities (A=10 to J=4)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-k79) — Project Engineering, chapter 3
- [CPM network, building project (A to K)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-eleven-act-s73) — Project Engineering, chapter 3

## Related in other subjects

- [CPM schedule, critical path and float](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Acpm-network-critical-path-float) — Industrial Engineering and Management, chapter 3