Adding a Sixth Attendee Under a Fixed Cap
A field office is sending agents to a three-day regional conference. Registration is $375 per attendee. Lodging is $142 per room per night for two nights, and agents may share rooms. Per diem is $59 for a full day and 75 percent of that for each travel day, with day one and day three counted as travel days. Two government vehicles make the 214-mile round trip at $0.67 per mile; adding passengers does not change mileage. The office has $5,000. Five agents are approved. A sixth agent asks to attend, and the office must decide without exceeding the cap.
1. Compute the total cost for five agents, itemized, and state the remaining balance.
Read the worked answer
Per diem per agent: 0.75 x $59 = $44.25 for each travel day; $59 for the full middle day. $44.25 + $59 + $44.25 = $147.50. For five agents: $147.50 x 5 = $737.50.
Lodging: 5 rooms x 2 nights x $142 = $1,420.00.
Registration: $375 x 5 = $1,875.00.
Mileage: 214 x $0.67 = $143.38 per vehicle; x 2 vehicles = $286.76.
Total: $737.50 + $1,420.00 + $1,875.00 + $286.76 = $4,319.26.
Remaining balance: $5,000.00 - $4,319.26 = $680.74.
The five-agent trip fits comfortably. Note that the two travel days are discounted, not billed at the full rate - using $59 for all three days would overstate the per diem by $14.75 per agent and make the margin look smaller than it is.
2. Compute the marginal cost of the sixth agent under two lodging arrangements, determine whether the trip can be made to fit, and give a recommendation that accounts for what the cap does and does not cover.
Read the worked answer
Marginal cost with a separate room: $147.50 per diem + (2 x $142 = $284) lodging + $375 registration = $806.50. Mileage is unchanged, since passengers do not affect the two vehicles.
Total: $4,319.26 + $806.50 = $5,125.76 - over the cap by $125.76.
Marginal cost with one shared room (six agents in five rooms): $147.50 + $142 + $375 = $664.50.
Total: $4,319.26 + $664.50 = $4,983.76 - under the cap by $16.24.
Recommendation: approve the sixth agent conditional on the room share. The margin is thin - under 0.5 percent of the cap - so it works only if the numbers hold.
What the cap does not cover, and the reason for caution:
- The $5,000 is a planning figure, not necessarily an obligated amount; confirm whether funds are already committed.
- A shared room is more expensive per head than the five-agent baseline? No - it is cheaper, but it creates a dependency: if the roommate cancels, the cost rises by $142 and the plan breaks.
- Incidental costs (meals beyond per diem, parking, local transport) are outside these figures entirely.
A defensible decision: approve the sixth agent with the shared room, flag that $16.24 of headroom leaves no room for error, and identify in advance which cost absorbs any overage. Approving on the assumption that sharing will simply work out is how a $5,000 cap becomes a $5,100 obligation.
3. A supervisor proposes cutting registration for the sixth agent by having that agent attend only the final day at a prorated one-third fee. Evaluate the arithmetic and the operational reasoning.
Read the worked answer
Arithmetic: one-third of $375 is $125. The saving would be $375 - $125 = $250, which is enough to cover the $125.76 overage with roughly $124 to spare, and it eliminates the need to force a room share.
Operational reasoning - why it is weak:
- The travel costs do not prorate. The sixth agent still needs lodging, three days of per diem, and a seat in a vehicle already making the trip. Cutting the registration to one-third saves $250 while spending $664.50 in travel to deliver one day of attendance.
- Cost per day of training attended rises sharply. Full attendance costs roughly $806.50 for three days (about $269 per day). Partial attendance costs about $556.50 for one day ($664.50 minus the $125 registration, if the travel were justified at all).
- The office has no stated interest in partial attendance. Nothing in the scenario says a single day has value to the mission.
Assessment: the proposal is arithmetically sound and substantively poor. It solves a budget problem by degrading the thing the budget exists to buy. The room-share option costs $16.24 of headroom and delivers full attendance; it is the better trade. If $16.24 of headroom is genuinely unacceptable, the honest answer is that the sixth agent cannot go, not that the sixth agent should attend a third of a conference.