Hospital beds in a metro city
Norm-based vs demand-based — build it both ways and reconcile.
Estimate the number of hospital beds in a 10-million metro. This one rewards a two-way build: a quick norm-based answer (beds per 1,000 people), then a demand-based check (admissions × stay length ÷ occupancy). Reconciling the two is the differentiator.
Path A (norm)
Metro bed density ~2.2 per 1,000 (above the national ~1.4 because metros serve referral inflows) → 10M × 2.2/1,000 = 22,000.
Path B (demand)
Hospitalization incidence ~7%/yr → 700K admissions × ~4-day average stay = 2.8M bed-days/yr ÷ 365 ≈ 7,700 occupied beds… ÷ 70% target occupancy ≈ ~11,000? No — recompute: 2.8M ÷ 365 ≈ 7,670 average occupied; at 70% occupancy, installed = 7,670 ÷ 0.7 ≈ ~11,000 city-resident beds; add ~50–100% referral inflow from surrounding districts (metros treat far more than their own population) → ~20,000+.
Reconcile
The two paths agree only after counting referral inflow — that is the insight: metro hospitals serve a catchment 1.5–2× the city itself.
How to defend it
The deliberate "mistake-and-repair" in Path B mirrors real interviews: if your two paths disagree, hunt the structural reason (here, referral catchment) instead of fudging a number. Naming why they diverged earns more credit than never diverging.