Fitting the Training Into $480
A six-person team has $480 total for a one-day training session. The meeting room rents for a flat $145 for the day. Lunch costs $12.50 per person, and printed materials cost $4.25 per person. Everyone who attends receives both lunch and materials - there is no option to attend without them. The team lead wants to invite as many people as possible while staying at or under budget. The venue requires a firm headcount five days in advance, and no additional money can be added to the budget after that point.
1. What is the largest number of people the team can invite? Show the setup and check your answer against the nearest whole numbers on either side.
Read the worked answer
Setup: the room fee is paid once; lunch and materials are paid per person.
Cost per person = 12.50 + 4.25 = $17.75
145 + 17.75p ≤ 480
17.75p ≤ 335
p ≤ 18.87
Since people come in whole numbers, p = 18.
Check both sides, because rounding down always needs verifying:
- 18 people: 145 + 18(17.75) = 145 + 319.50 = $464.50 - under budget.
- 19 people: 145 + 19(17.75) = 145 + 337.25 = $482.25 - over budget.
$15.50 is left over at 18 people, which is less than the $17.75 needed for a 19th person, so the answer is confirmed.
2. After the headcount of 18 is locked in, the venue raises the room fee from $145 to $165. The lead suggests "just skip the printed materials for everyone" to fix it. Does that work? Show the arithmetic.
Read the worked answer
Cost at the new room fee: 165 + 18(17.75) = 165 + 319.50 = $484.50, which is $4.50 over budget.
The proposed fix, checked: dropping materials saves 18 × 4.25 = $76.50, so removing materials would work easily - but that is not what the lead is weighing against the alternative.
The tighter fix, checked: if only the materials cost is trimmed rather than eliminated, the room fee plus lunch alone is 165 + 18(12.50) = 165 + 225 = $390. Materials budget available = 480 − 390 = $90, or $5.00 per person - more than the $4.25 needed, so 18 people still fit.
The important point: the plan is not automatically broken by a $20 increase. The lead should re-run the total rather than guess, and the margin is only $5.50 total, so any further increase breaks it. Dropping to 17 people (165 + 301.75 = $466.75) is the safe fallback.
3. Instead, the lead is told that 22 people must attend. Using the original $145 room fee, what is the highest per-person lunch cost that keeps the total at or under $480?
Read the worked answer
Work backward from the fixed costs.
Materials: 22 × 4.25 = $93.50
Room fee: $145.00
Remaining for lunch: 480 − 145 − 93.50 = $241.50
Per person: 241.50 ÷ 22 = $10.977...
Because rounding up would break the budget, round down: $10.97 per person.
Check: 22(10.97 + 4.25) = 22(15.22) = $334.84; plus 145 = $479.84, under budget by $0.16.
Rounding to $10.98 instead gives 22(15.23) = 335.06 + 145 = $480.06, which is over. The direction of rounding is the whole problem here, not the division.