Mental math in a case is not a contest to divide 4,817 by 23 in your head. It is whether you can state the equation, keep units, round on purpose, and say what the number means before anyone asks. Speed without a setup is how people drop a thousand and still sound confident.
This page owns the live habit. Tricks sit on shortcuts. Training loops sit on practice. Formulas sit on the sheet.
The four-beat that interviewers actually grade
- Equation in English. “Admissions per year equals beds × occupancy × days in year, divided by average length of stay.”
- Units. Beds × (unitless) × days / days-per-stay = stays / year.
- Friendly numbers, announced. “I will use 365 ≈ 360 to keep it clean, then note I am ~1% low.”
- So-what. “That is 31,000 admissions. If ALOS rose 0.3 days, we lose about 2,000 stays — that is the bed problem, not marketing.”
If you skip (1), a right number still looks lucky. If you skip (4), you did a worksheet.
Worked example: occupancy is not “how busy we are”
Prompt. St. Brigid’s is a 420-bed district hospital. Occupancy 86%. Average length of stay 4.2 days. The CEO says “we are full; we need a tower.” The CFO asks how many inpatient admissions per year that occupancy implies, and what happens if ALOS is cut to 3.9 days without adding beds.
Setup (out loud).
Annual occupied-bed-days = 420 beds × 0.86 × 365 days/year.
Admissions / year = occupied-bed-days / ALOS.
Occupied-bed-days. 420 × 0.86. 420 × 0.8 = 336; 420 × 0.06 = 25.2; 361.2 occupied beds on an average night. × 365. 360 × 365: 360 × 400 = 144,000 minus 360 × 35 = 12,600 → 131,400. Plus 1.2 × 365 ≈ 440. ~131,800 occupied-bed-days / year.
Shortcut: 420 × 365 = 153,300 bed-days available; × 0.86. 153,300 × 0.8 = 122,640; 153,300 × 0.06 = 9,198; 131,838. Same.
Admissions. 131,838 / 4.2. 4.2 × 31,000 = 130,200. Remainder 1,638 / 4.2 ≈ 390. ≈ 31,400 admissions / year.
Cut ALOS to 3.9, same beds and occupancy. Occupied-bed-days unchanged if occupancy stays 86% — wait. If ALOS falls and demand is there, occupancy falls unless you fill with new admissions. The CEO’s question is capacity: max admissions if you still run 86%.
Max admissions = 131,838 / 3.9. 3.9 × 33,800 = 131,820. ≈ 33,800 admissions.
Delta = 2,400 extra stays / year (~8%) without a tower, if you can discharge faster and backfill. A tower is $200m+. ALOS work is process. The so-what is the case.
If demand is not waiting in the ED, cutting ALOS just creates empty beds (occupancy falls). Then the tower is even more wrong. Ask which world we are in. Mental math does not replace that question; it makes the two worlds numeric:
| World | Occupancy | ALOS | Admissions |
|---|---|---|---|
| Today | 86% | 4.2 d | ~31,400 |
| Faster discharge, demand waits | 86% | 3.9 d | ~33,800 |
| Faster discharge, no extra demand | ~80% | 3.9 d | ~31,400 |
Same arithmetic identity. Different CEO sentence.
Habits that keep you honest
Round one factor, not all of them the same way. 365 → 360 is fine if you say it. Also rounding 4.2 → 4.0 and 86% → 90% stacks error.
Keep 000s as words. “One hundred thirty-two thousand bed-days” beats a stray 1318.
Percent of a percent. Occupancy 86% is not “86.” Write 0.86 once.
Sanity. 420 beds × 365 / 4.2 ≈ 36,500 if 100% full. 86% of that is ~31,400. If you got 3,140, you dropped a zero. If you got 314,000, you invented a city.
When grinding mental math is the wrong move
The exhibit already computed it. Read the title.
The identity is NPV of a 12-year uneven stream. Use a stated 10% and a range; see NPV.
You do not know the equation. Stop. Write it. Wrong-fast is still wrong.
The mistake unique to “mental math” as a performance
Silent speed. Candidates race to a number and the interviewer cannot tell 31,400 from 3,140 until the recommendation is already airborne. The unique fail is skipping the occupied-bed-day identity and multiplying 420 × 86 × 4.2, which is beds × occupancy × ALOS — not admissions. Units would have caught it: beds × days / stay-days = stays, so ALOS belongs in the denominator.
What to say
“Occupied-bed-days are 420 times 0.86 times 365, about 132,000. Divide by 4.2 day stays, about 31,000 admissions. Cutting ALOS to 3.9 at the same occupancy is about 2,400 more stays if patients are waiting. I used 365, not 360, so I am not leaving a percent on the table.”
That is case math. A muttered 31,388 is not better.
Equation first, then timed reps
Quick-math drills the four-beat: setup, units, round, so-what.