SECONDARY MATHEMATICS · PERCENTAGE DEEP STRUCTURE
A 20% increase followed by a 20% decrease does not return a quantity to where it started. The percentages are applied to different bases.
Percentage change is directional. The starting value matters because it defines the 100% base for that change. When the quantity changes, the base for the next percentage move may also change.
A simple counterexample
Start with 100. Increase by 20%: 100×1.2=120. Now decrease 120 by 20%: 120×0.8=96. The final value is 4% below the original.
The increase was 20, but the decrease was 24 because 20% of 120 is larger than 20% of 100.
Why opposite percentages do not cancel
An increase of p% multiplies by 1+p. A decrease of p% multiplies by 1−p, where p is written as a decimal. Their combined multiplier is (1+p)(1−p)=1−p², which is less than 1 for any nonzero p between 0 and 1.
For p=0.2, the combined multiplier is 1−0.04=0.96.
The reverse percentage is different
If a quantity rises by 25%, the multiplier is 1.25. To return exactly to the original, multiply by 1/1.25=0.8, which is a 20% decrease, not 25%.
Likewise, if a quantity falls by 20%, returning to the original requires dividing by 0.8, which multiplies by 1.25: a 25% increase.
Equal absolute changes can have unequal percentage changes
A quantity rising from 80 to 100 increases by 20. That is 20/80=25%. Falling from 100 back to 80 decreases by the same absolute 20, but the percentage decrease is 20/100=20%.
The absolute change is symmetric. The percentage change is not because the denominator changes.
Worked example: rise then fall
A price rises by 10% and then falls by 10%. Combined multiplier=1.1×0.9=0.99. Final price is 99% of the original, a 1% net decrease.
Worked example: fall then rise
A value falls by 30% and then rises by 30%. Multiplier=0.7×1.3=0.91. Final value is 9% below the original.
Finding the exact recovery percentage
If a quantity falls by 40%, it becomes 60% of the original. To recover, multiply by 1/0.6=1.666…, which means an increase of 66 2/3% from the reduced value.
This is why a 50% loss requires a 100% gain to recover: after falling from 100 to 50, the increase of 50 is 100% of the new base 50.
Graphs of percentage change
If a quantity is repeatedly multiplied by a fixed percentage multiplier, equal time intervals produce multiplicative rather than additive change. A graph may curve rather than form a straight line. The difference between additive and multiplicative change is central to growth modelling.
Common misconceptions
“+20% and −20% cancel.” False because they use different bases.
“If you lose 30%, you need 30% to recover.” False. Recovery is measured from the reduced base.
“Same absolute change means same percentage change.” False unless the bases are the same.
A first-principles teaching sequence
Start with 100 so the percentage movement is visible. Then repeat with a different starting value to show that the principle does not depend on the convenient base. Move from arithmetic to multipliers, and only then derive the general expression (1+p)(1−p).
Next, ask recovery questions in both directions. Students should explain which value is now 100% before calculating.
Diagnostic checkpoints
- Can the student name the base for each percentage move?
- Can they use multipliers instead of adding percentages?
- Can they find a recovery percentage after a loss?
- Can they explain why equal absolute changes can produce unequal percentages?
Practice set
1. +15%, then −15% → multiplier 1.15×0.85=0.9775, net decrease 2.25%.
2. −25%, then what increase restores the original? 1/0.75=1.333…, so 33 1/3%.
3. 60→75 is what percentage increase? 25%. 75→60 is what percentage decrease? 20%.
4. A value doubles and then halves. Does it return to the start? Yes, because multipliers 2×0.5=1. This is different from “+100% then −100%”, where the latter sends the value to zero.
Continue
Build from The Percentage Base, then continue to Reverse Percentage Through Equations and Repeated Percentage Change. Return to the Secondary Mathematics Master Index.