Why Two Percentage Cuts Do Not Simply Add
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.
Run Calculate for 18 false alarms/week, then apply a 45% sensor cut followed by a 20% relocation cut.
The result falls first to 9.9 and then 7.92 alarms/week, not the 6.3 predicted by adding percentages.
Sequential reductions multiply the surviving 55% by 80%, leaving 44% of the original 18 alarms rather than subtracting 65 points once.
See the relationship before changing it
The figure reads from left to right. The blue input is first alarm cut. The middle card names the page’s rule. The green output is alarms left. The arrow matters: change the input, apply the rule once, then read the result with its unit.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 45 %.
- 2
Name the relationship. remaining = 18 x (1 - first cut) x (1 - 0.20)
- 3
Substitute with units. 18 x 0.55 x 0.80 = 7.92 alarms/week
- 4
Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.
Predict, then change first alarm cut
Try Predict how alarms left responds when first alarm cut moves. Calculate first alarm cut; compare alarms left with that prediction.
Observe Return to 45 %. Recheck alarms left with first alarm cut at its chapter value.
Explain The second cut acts on the alarms left by the first cut.
Check yourself
What should you do before trusting a moved-slider result?
What does this small model leave out?
Technical boundaries
For the alarm sequence, excluded from this fixed arithmetic are occupancy changes, weather, tariff variation, device standby loads, or interaction between the two saving measures.
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.