Question 8
Activities P and Q are the critical activities for a project.
| Activity | Duration | Prerequisite activity |
|---|---|---|
| P | 3 | – |
| Q | 6 | P |
What are the earliest starting time (EST) and latest starting time (LST) for Activity Q?
That’s it. Step through the hints below to see why. In 2023, 49.1% of Queensland students chose (A).
Not quite. The EST of 3 is right. One way to get an LST of 6 is to use Q’s duration as if it were a start time. Q is critical, so it has no float and its latest start equals its earliest start. Starting Q at 6 would push the finish from 9 to 12. In 2023, 27.3% of Queensland students chose (B).
Not quite. 6 is how long Q takes, not when it starts. One way to get (C) is to read both times off the Duration column. Q can start the moment P finishes, at time 3. In 2023, 7.4% of Queensland students chose (C).
Not quite. One way to get (D) is to work with Q’s duration and its finish time, $3+6=9$, instead of its start times. Q finishes at 9, but the question asks when it can start, and that is 3. In 2023, 15.3% of Queensland students chose (D).
Slide Q along the timeline. Q has to wait for P, and the project should finish as early as possible. Find every start time that does both.
Forward scan. P has no prerequisite, so it starts at 0 and finishes at 3. Q needs P, so the earliest Q can start is the moment P finishes.
$$\text{EST}_Q=0+3=3$$
The 6 in the table is how long Q takes. It never goes into Q’s own start time.
Backward scan. The project’s minimum completion time is the length of the critical path, P then Q.
$$\begin{aligned}\text{minimum time}&=3+6=9\\\text{LST}_Q&=9-6=3\end{aligned}$$
Q must finish by 9 and takes 6, so it must start by 3.
The question gave you a shortcut. Q is a critical activity, and a critical activity has zero float.
$$\text{float}=\text{LST}-\text{EST}=0\ \Rightarrow\ \text{LST}=\text{EST}$$
So the two numbers in the answer must be equal. That leaves (A) and (C), and (C) uses Q’s duration as a start time.
$$\begin{aligned}\text{EST}_Q&=\text{EST}_P+\text{duration}_P=0+3=3\\\text{minimum completion time}&=3+6=9\\\text{LST}_Q&=9-\text{duration}_Q=9-6=3\end{aligned}$$
Check: the float of Q is $3-3=0$, as it must be for a critical activity.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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