The Bacteria Colony

Half full at 9:50 AM (10 minutes before full). Quarter full at 9:40 AM (20 minutes before full). With 2 starting bacteria: full at 9:50 AM (saves one 10-minute doubling period). The backward reasoning from full → half → quarter is the key technique.

Exponential growth problems are most elegantly solved by working backwards from the known endpoint. The colony doubles every 10 minutes, so going backward in time halves the population every 10 minutes. Step 1: At 10:00 AM: dish is 100% full. The colony started with 1 bacterium at 9:00 AM and doubled every 10 minutes — that is 6 doublings in 60 minutes: 1→2→4→8→16→32→64 units. So the dish holds 64 units. Step 2: Half full: 64/2 = 32 units. Looking at the doubling sequence, 32 units occurs one step before 64 — which is 10 minutes before 10:00 AM. Half full at 9:50 AM. Step 3: Quarter full: 64/4 = 16 units. Two steps before full — 20 minutes before 10:00 AM. Quarter full at 9:40 AM. Step 4: Starting with 2 bacteria: the sequence becomes 2→4→8→16→32→64. That is 5 doublings to reach 64. At 10 minutes per doubling, it takes 50 minutes total. Starting at 9:00 AM: full at 9:50 AM. Step 5: The insight: starting with 2 instead of 1 saves exactly one doubling period (10 minutes), moving the full time from 10:00 AM to 9:50 AM. Step 6: General principle: starting with k times as many organisms shortens the time to fill by log₂(k) doubling periods. Key Insight: Backward reasoning through exponential growth converts a hard forward computation into a trivial subtraction. 'What was the state one doubling period ago?' is always the fastest path in these problems. The counterintuitive result — that the dish was half full just 10 minutes before it was completely full — illustrates why exponential growth feels suddenly explosive near its end.