Numerical Reasoning - Staffing Recommendation From 84 Days of Call Data
Chief Alvarez hands you a one-page staffing summary for Station 7 covering the last 12 weeks (84 days) and asks for a recommendation on adding a second transport unit:
- Incidents run by Station 7: 1,260.
- Medic 7, the ALS ambulance, was dispatched on 735 of them; Engine 7 ran the other 525.
- Department planning guideline: one transport unit should not average more than 8 incidents per 24-hour day; a sustained average above 8 supports a second unit.
- Station 7 staffs three 24-hour groups of four members. During the period, a member had to be replaced on one out of every 12 shifts, at $38.00 per hour for the full 24-hour shift.
His note clipped to the folder reads: 'Give me numbers, not adjectives. If you recommend a unit, tell me what assumption your recommendation rests on.'
1. Compute Medic 7's average incidents per 24-hour day for the period, state whether the guideline is met, and express the gap as a percentage.
Read the worked answer
Set up the denominator first. Three 24-hour groups rotating means exactly one group is on duty on any given calendar day, so 84 days equals 84 shifts and 84 unit-days.
Average: 735 ÷ 84 = 8.75 incidents per 24-hour day.
Quick check by a second route: 84 × 8 = 672, and 672 + 63 = 735, so the average is 8 + (63 ÷ 84) = 8 + 0.75 = 8.75. The arithmetic agrees.
Guideline: 8.75 is above 8, so the guideline is not met.
Size of the gap: 0.75 ÷ 8 = 0.09375, or about 9.4% above the threshold - or, framed the other way, 0.75 ÷ 8.75 = 8.6% of Medic 7's current volume would have to move elsewhere to bring it to exactly 8.
That second number is the one worth carrying into the recommendation. 'Almost 9% over the line' and 'needs to shed 8.6% of its calls' describe the same data, but only the second one tells the chief how big the fix has to be before it counts.
2. What did the replacements cost Station 7 during the 12-week period, and what would that same rate cost over a full year?
Read the worked answer
Period cost: replacements occurred on 1 of every 12 shifts, and there were 84 shifts.
- 84 ÷ 12 = 7 replacement shifts
- Each replacement: 24 hours × $38.00 = $912.00
- 7 × $912 = $6,384.00
Annualized: a year contains 365 ÷ 12 ≈ 30.4 such shifts. At the same rate, 30.4 × $912 ≈ $27,725 per year, call it $27,700.
Two cautions to state rather than bury. First, the assumption that a replacement is paid for all 24 hours is conservative - a member who fills a shift is usually paid for the hours worked, and this figure treats every fill as a full shift. Second, overtime is not automatically an argument for a new unit: the $6,384 is overtime driven by absence, and a second unit adds a permanent 4-person crew, which is a far larger recurring cost. Presenting overtime as savings would be a reasoning error the chief would catch in about four seconds.
3. Rewrite the recommendation in two or three sentences that satisfy the chief's note, including the assumption it rests on.
Read the worked answer
Model version:
'Medic 7 averaged 8.75 incidents per 24-hour day over 84 days, about 9% above the 8-per-day guideline, and 7 overtime replacements cost $6,384 during the period. I recommend the second unit only if dispatch data show at least 9% of Medic 7's current volume would transfer to it; at 40% transfer Medic 7 falls to 5.25 per day and the new unit would average 3.5, but below about 9% transfer Medic 7 stays over the guideline and the new unit does not pay for itself.'
Why this shape works:
- Every claim is a number or is labeled as an assumption. 'Overworked' is an adjective; '8.75 against a guideline of 8' is a finding.
- It states the break-even. 8.75 × 0.60 = 5.25; 8.75 × 0.40 = 3.5. Both come from the same 40% assumption and show the chief the plan's best case.
- It names the thing that could be wrong. The 40% figure is not in the data - it is a guess about dispatch behavior. A recommendation that hides its load-bearing assumption invites the chief to approve a unit on numbers nobody verified.
If the dispatch data do not support at least ~9% transfer, the honest recommendation is a peak-hour unit or a mutual-aid transport agreement, not a 24-hour second ambulance justified by a 0.75-call margin.