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3 Responses
Lovely question! Percent changes are shapeshifters: each one rescales the whole price as it currently is. That means they multiply, not add.
Let’s walk it calmly with a friendly number like 100.
– Start: 100
– Up 30%: multiply by 1.30 → 100 × 1.30 = 130
– Down 15%: multiply by 0.85 → 130 × 0.85 = 110.5
– Up 10%: multiply by 1.10 → 110.5 × 1.10 = 121.55
So the final price is 121.55, which is a net increase of 21.55%. Not 25%.
In factor form, you did exactly what’s needed:
1.30 × 0.85 × 1.10 = 1.2155
Net percent change = (1.2155 − 1) × 100% = 21.55%.
Why can’t we just do 30 − 15 + 10? Because each change applies to a new base. Adding 30 − 15 + 10 = 25% ignores the compounding. If you expand the product, you can see the “interaction” terms that make the difference:
(1 + 0.30)(1 − 0.15)(1 + 0.10)
= 1 + (0.30 − 0.15 + 0.10) + (0.30)(−0.15) + (0.30)(0.10) + (−0.15)(0.10) + (0.30)(−0.15)(0.10)
= 1 + 0.25 − 0.0345
= 1.2155.
Those extra terms subtract 3.45% from the naive 25%.
General rule:
– Convert each change p% to a factor 1 + p/100 (use 1 − q/100 for a decrease).
– Multiply all the factors.
– Subtract 1 and convert back to a percent for the net change.
Bonus sanity check: the order doesn’t matter for the final result because multiplication of these factors is commutative, but adding the percentages will generally be wrong unless the percentages are tiny and you’re only approximating.
You can’t add those percentages, because each change applies to the new price, not the original. A clean way is to use factors: start at 100. After +30% you have 100 × 1.30 = 130; then −15% gives 130 × 0.85 = 110.5; then +10% gives 110.5 × 1.10 = 121.55. So the overall factor is 1.30 × 0.85 × 1.10 = 1.2155, which is a 21.55% increase, not 25%. Think of it like resizing a photo: enlarge by 30%, shrink by 15%, then enlarge by 10%-because each step uses the current size as its base, the final result isn’t the same as just adding the percentages.
You can’t add and subtract the percentages because each change applies to the new price, not the original one. The right method is to multiply by the corresponding factors: “up p%” means multiply by 1 + p, and “down q%” means multiply by 1 − q (with p and q in decimal form). For your sequence, multiply by 1.30, then 0.85, then 1.10. Using a simple starting value of 100 to keep it clear: 100 × 1.30 = 130, then 130 × 0.85 = 110.5, then 110.5 × 1.10 = 121.55. That final value is 1.2155 times the start, i.e., a 21.55% overall increase-not 25%. In general, the overall percentage change is (product of all factors − 1) × 100%.