Calculation questions

Calculations make up only a small share of PMP questions, but they are among the few questions with an answer you can be certain of. Candidates commonly report seeing a handful of earned value, schedule, communication, estimating and risk calculations, often wrapped in a short scenario or shown on a graphic. This lesson collects the formulas you need, works them on Project Compass numbers, and shows the traps that wrong options are built from.

You will have an on-screen calculator and a whiteboard or scratch surface. There is no formula sheet, so the formulas below should be memorized well enough to write them down quickly when needed. Many candidates jot the key ones down early in the exam, during the first minutes or at a break, so they are ready when needed.

Earned value management

Earned value compares the planned value of work (PV), the earned value of work actually completed (EV) and the actual cost of that work (AC). The budget at completion (BAC) is the total planned value. From these come variances, indices and forecasts.

Negative variances and indices below 1.0 are unfavorable: behind schedule or over budget. A useful memory aid is that earned value always comes first: every variance starts with EV, and every index puts EV on top.

# Common PMP calculations, using Project Compass month 6 figures ($000).
BAC, PV, EV, AC = 2400, 1440, 1290, 1400
SV, CV = EV - PV, EV - AC
SPI, CPI = EV / PV, EV / AC
EAC = BAC / CPI                          # if current cost performance continues
ETC, VAC = EAC - AC, BAC - EAC
TCPI = (BAC - EV) / (BAC - AC)           # efficiency needed to finish on BAC
print(f"SV {SV}  CV {CV}  SPI {SPI:.2f}  CPI {CPI:.2f}")
print(f"EAC {EAC:.0f}  ETC {ETC:.0f}  VAC {VAC:.0f}  TCPI {TCPI:.3f}")
print(f"EAC = AC + (BAC - EV) = {AC + (BAC - EV)}  (if remaining work goes to plan)")
print()
# Communication channels when the core team grows from 8 to 11 people.
ch = lambda n: n * (n - 1) // 2
print(f"channels 8 -> 11 people: {ch(8)} -> {ch(11)}, added {ch(11) - ch(8)}")
# PERT estimate for the ceiling installation (days).
O, M, P = 8, 10, 18
print(f"PERT expected {(O + 4 * M + P) / 6:.1f} days, sigma {(P - O) / 6:.2f}")
# Expected monetary value of two risks and an opportunity ($000).
risks = [(0.30, -40), (0.10, -90), (0.25, +12)]
print(f"EMV total {sum(p * i for p, i in risks):.1f}")

SV -150  CV -110  SPI 0.90  CPI 0.92
EAC 2605  ETC 1205  VAC -205  TCPI 1.110
EAC = AC + (BAC - EV) = 2510  (if remaining work goes to plan)

channels 8 -> 11 people: 28 -> 55, added 27
PERT expected 11.0 days, sigma 1.67
EMV total -18.0

At month 6, Project Compass had PV $1,440,000, EV $1,290,000 and AC $1,400,000 against a BAC of $2,400,000. The schedule variance is −$150,000 and the cost variance −$110,000; SPI is 0.90 and CPI about 0.92, so the project is behind schedule and over budget. If the current cost performance continues, the estimate at completion is about $2,605,000, a variance at completion of about −$205,000. To finish on the original budget, the remaining work would need a cost performance of about 1.11, which is unlikely given a CPI of 0.92.

The scenario usually tells you which formula to use. Words such as the problem has been fixed or the overrun was caused by a single event point to AC + (BAC − EV). Words such as the trend is expected to continue point to BAC / CPI. If the question says nothing, BAC / CPI is the usual default.

Critical path and float

Schedule calculations ask for the critical path, the project duration or the float of an activity. Use the forward pass to find early start and early finish, and the backward pass to find late start and late finish. Total float is the difference between late start and early start; activities with zero total float are on the critical path.

# Forward and backward pass for a small practice network (durations in days).
acts = {  # name: (duration, predecessors)
    "A": (3, []), "B": (5, ["A"]), "C": (2, ["A"]),
    "D": (4, ["B"]), "E": (6, ["C"]), "F": (3, ["D", "E"]),
}
ES, EF = {}, {}
for a in acts:                                   # listed in a valid order
    d, pre = acts[a]
    ES[a] = max((EF[p] for p in pre), default=0)
    EF[a] = ES[a] + d
end = max(EF.values())
LS, LF = {}, {}
for a in reversed(list(acts)):
    succ = [s for s in acts if a in acts[s][1]]
    LF[a] = min((LS[s] for s in succ), default=end)
    LS[a] = LF[a] - acts[a][0]
print(f"{'Act':4}{'ES':>4}{'EF':>4}{'LS':>4}{'LF':>4}{'Float':>7}")
for a in acts:
    print(f"{a:4}{ES[a]:>4}{EF[a]:>4}{LS[a]:>4}{LF[a]:>4}{LS[a] - ES[a]:>7}")
print("critical path:", "-".join(a for a in acts if LS[a] == ES[a]), f"({end} days)")

Act   ES  EF  LS  LF  Float
A      0   3   0   3      0
B      3   8   3   8      0
C      3   5   4   6      1
D      8  12   8  12      0
E      5  11   6  12      1
F     12  15  12  15      0
critical path: A-B-D-F (15 days)

In the practice network, the path A-B-D-F takes 15 days and has zero float, so it is critical. Activities C and E each have one day of float, because the path A-C-E-F takes 14 days. If activity E slipped by two days, it would use up its one day of float and delay the project by one day, making A-C-E-F the new critical path. Exam questions often test exactly this: what happens to the end date when a non-critical activity slips by more than its float?

Free float is the time an activity can slip without delaying the early start of any successor. In the network, E's free float is one day, because F can start on day 12 even if E finishes on day 12. C's free float is zero, because E starts as soon as C finishes, although C still has one day of total float shared along its path.

Communication channels

The number of communication channels among n people is n(n − 1)/2. Questions usually ask how many channels are added when people join. Count the project manager if the question includes them in the team.

When the Project Compass core team grew from 8 to 11 people, the channels rose from 28 to 55, an increase of 27. Three extra people almost doubled the channels. That is why adding people to a late project can slow it down, at least at first, as more coordination is needed.

Estimating

Three-point estimates use optimistic (O), most likely (M) and pessimistic (P) values. The beta or PERT estimate weights the most likely value four times; the triangular estimate takes a simple average. The standard deviation approximates uncertainty.

For the ceiling installation, with O = 8, M = 10 and P = 18 days, the PERT estimate is 11 days and σ is about 1.67 days. The triangular estimate would be 12 days. If a question asks for the three-point estimate without saying which, check the options; the PERT value is usually the one intended. Remember that for a path, expected durations add and variances (σ squared) add, not standard deviations.

Risk calculations

Expected monetary value multiplies each risk's probability by its impact and sums the results, with threats negative and opportunities positive. Decision trees use EMV to compare options.

For two threats, a 30 % chance of losing $40,000 and a 10 % chance of losing $90,000, and an opportunity with a 25 % chance of saving $12,000, the EMV is −$12,000 − $9,000 + $3,000 = −$18,000. In a decision tree, the value of each option is its cost plus the EMV of its outcomes; choose the option with the best expected value. PMP-131 worked a decision tree for the intrusive ceiling survey on Project Compass.

Other calculations you may meet

Several of these, such as net present value and internal rate of return, are more often tested as concepts than as full calculations: which option to choose given the figures, or what a value above 1.0 means. You should be able to interpret them quickly even if you are not asked to compute them.

A routine for calculation questions

• Identify what is asked: a variance, an index, a forecast, a float, a count.
• Write down the formula before entering numbers.
• Pull exactly the values needed from the scenario or graphic, checking units and whether figures are in thousands.
• Calculate, then check the sign and size against the story. A project described as over budget should have a negative CV and a CPI below 1.
• Compare with the options. If your answer is not there, look for a likely slip, such as dividing the wrong way or using PV instead of EV.