Why Two Percentage Cuts Do Not Simply Add

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Why Two Percentage Cuts Do Not Simply Add

A home security system logs 18 false alarms a week, then stacks two fixes: pet-immune sensors that cut alarms 45% and sensor relocation that cuts a further 20%. The tempting shortcut adds them to 65% off, leaving 6.3 alarms a week. This audit runs the fixes the way they actually fire — in sequence — and asks whether the real combined cut is that 65%, or the 56% that leaves 7.9 alarms still standing.

Companion to the chapter Smart Home — every number here comes from that chapter.

Ada: The security example starts at 18 false alarms per week, then stacks two fixes: pet-immune sensors cut alarms by 45%, and sensor relocation cuts them by a further 20%. The seductive shortcut is to add the percentages — 45 plus 20 is 65% off, leaving 18 x (1 - 0.65) = 6.3 per week. That answer is wrong, and the reason is worth seeing. Let me apply the fixes the way they actually run: in sequence.

The second fix never touches the original 18; it acts on whatever the first fix leaves behind. After pet-immune filtering, 18 x (1 - 0.45) = 18 x 0.55 = 9.9 alarms remain. Relocation then removes 20% of that remainder: 9.9 x (1 - 0.20) = 9.9 x 0.80 = 7.92, about 7.9 per week.

The two survival fractions multiply — 0.55 x 0.80 = 0.44 — so the real combined reduction is 1 - 0.44 = 0.56, a 56% cut, not 65%. The additive shortcut overstates the removal and lands at 6.3 instead of 7.9.

Design meaning: sequential mitigations compound multiplicatively on the alarms still standing, so stacking “minus 45%” and “minus 20%” can never reach “minus 65%” — any false-alarm roadmap that adds its percentages will promise fewer alerts than the hardware can actually deliver.

Every number above is taken from the chapter’s own material and re-derived step by step.