All writing · 4 min read
Everyday numbers · By Yosef Gerson
A 20% drop needs more than a 20% recovery.
A shop cuts a $100 price by 20%. The sale ends, and someone raises the new price by 20%. The price is back to normal. Right?
Not quite. The discount takes it to $80. The increase adds $16, bringing it to $96. The price is still four dollars below where it started.
The arithmetic is not the trap. The changing starting point is.
Twenty percent of what?
A percentage is a fraction of a reference amount. In the first step, 20% means 20% of $100: $20. In the second, it means 20% of $80: $16.
The labels match. The amounts do not.
To get from $80 back to $100, you need an extra $20. Relative to the new starting point, that is $20 ÷ $80 = 25%.
Why multiplying works better than adding
A 20% decrease leaves 80% of the original amount, so multiply by 0.8. A 20% increase produces 120% of the current amount, so multiply by 1.2.
100 × 0.8 × 1.2 = 96
Adding −20% and +20% gives zero, but that treats both percentages as though they refer to the same amount. Multiplication keeps track of the changing base.
This also explains a surprising detail: reversing the order does not fix it. Raise $100 by 20% and you get $120. Reduce that by 20% and you again get $96.
The larger the drop, the steeper the return
After a 10% drop, you need about 11.1% to return. After a 25% drop, you need 33⅓%. After a 50% drop, you need 100%: what remains must double.
For a drop of d, written as a decimal from 0 up to, but not including, 1, the required recovery is d ÷ (1 − d). A 20% drop gives 0.2 ÷ 0.8 = 0.25, or 25%.
You do not need to memorize the formula. Ask how much was lost, then divide that amount by what remains.
Try the idea
Change the drop. Watch the recovery.
Start with 100. Choose a percentage decrease to see how much is left, and the rise needed to get back.
A drop of 20% leaves 80 from every 100. To restore the missing 20, divide 20 by 80. The required rise is 25%.
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Not every “20” is a percentage change
Suppose a survey result moves from 40% to 60%. It rose by 20 percentage points. Relative to the original 40%, it rose by 50%.
Both descriptions can be correct. They answer different questions. Percentage points compare the two rates directly; percentage change measures the difference against the starting rate.
A useful habit
Whenever a percentage appears in a headline, a discount, or an AI explanation, pause for one question:
What amount is this percentage being measured against?
If the starting point changed, calculate each step separately. If the starting point is missing, the claim is not yet clear enough to use.