Math & logic · ENGLISH
A 20% drop and a 20% rise do not cancel.
The missing question is always: a percentage of what?
Find the base first
A percentage expresses a part per hundred. To calculate 20% of 80, multiply 80 by 0.20 to get 16. The number 80 is the base. A percentage without an identified base is not enough to calculate an amount.
Follow one price
Start with a price of 100 units. A 20% reduction removes 20, leaving 80. Increasing that new price by 20% adds 16, bringing it to 96. The percentages match; the bases do not.
To return from 80 to 100, you need an increase of 20 out of 80. That is 25%. This is arithmetic about an example price, not a recommendation about buying or investing.
Points versus relative change
Suppose a quiz success rate rises from 40% to 50%. It rose by 10 percentage points. Relative to the original 40%, that is a 25% increase: the change of 10 divided by 40. Both statements describe the same change, using different comparisons.
Mixing these descriptions can make a small-looking number seem dramatic or a dramatic-looking number seem small. Write down the original and new values before choosing the wording.
In the game
A fair percentage prompt names the starting quantity and any intermediate change. “20% more” needs something to be more than. A multi-step price question deserves more time than recognising one quarter as 25%.
The useful habit is to restart the calculation whenever the base changes. Rather than carrying the first amount in your head and adding percentage signs together, identify the number each percentage applies to. That small pause prevents a large class of everyday mistakes.
Reverse the calculation carefully
A price of 80 rises by 25% to 100. To return from 100 to 80 requires a 20% decrease, not 25%. The first change adds 20 out of 80; the second removes 20 out of 100. The amount is the same but the fraction of the starting value is different.
A compact way to calculate successive changes is to multiply factors. A 20% decrease multiplies by 0.8; a 20% increase multiplies by 1.2. Together they multiply by 0.96. Starting at 50 therefore ends at 48. The order of these two idealised changes does not alter their product.
Percentage points describe a different difference
If a proportion rises from 20% to 25%, it rises by five percentage points. Its relative increase is 5 ÷ 20 = 25%. Calling that a “5% increase” confuses the two descriptions. State the before and after values when a short label could hide the base.
For a check, use an easy starting amount such as 100, carry each intermediate result forward, and ask whether the final value makes sense. This models the stated percentages; it does not include real-world complications such as fees or rounding.
TRY THE IDEA · NO TIMER · NO RANKING
Same percentage. Different starting point.
Start at 100. Reduce it, then increase the new amount by the same percentage.
- Start 100.00
- After reduction 80.00
- After increase 96.00
The rise applies to 80.00, not to the original 100. Try 0%, 20% and 50% to see the difference.
Sources & further reading
Sources support the factual background. The worked examples and game connections are written for 3 SECONDS.