The three percentage problems
Almost every percentage question in daily life is one of these three wearing a disguise. "What's the tip on $84?" is the first. "I got 43 out of 60, what's my grade?" is the second. "I paid $63 after a 30% discount, what was the original price?" is the third — and it is the one people most often get wrong, because they try to add 30% back on instead of dividing.
Mental shortcuts worth knowing
- Move the decimal. 10% of any number is that number with the decimal moved one place left. 10% of 847 is 84.7. Everything else builds from there: 5% is half of that, 20% is double it.
- Percentages are commutative. X% of Y always equals Y% of X. 4% of 75 is awkward; 75% of 4 is obviously 3. This trick is close to magic once you start using it.
- Build from 1%. Divide by 100, then multiply. 1% of 340 is 3.40, so 7% is 23.80.
- Halves and doubles. 15% = 10% + half of 10%. 25% = a quarter. 33⅓% = a third.
Where percentages show up, and where they mislead
Percentages compress a comparison into one number, and that compression is exactly where the errors hide.
- Discounts and tax. A 20% discount followed by 8% tax is not a 12% change — the two apply to different bases and must be worked in order.
- Tips and service charges. Check whether service is already included before adding more.
- Interest. The same 5% means very different things monthly, annually, simple or compounded.
- Exam marks and grades. Straightforward part-of-whole, but weighted components need each weight applied before summing.
- Statistics in the news. The most common manipulation is quoting a relative change without the absolute one. “Doubles your risk” sounds alarming; if the underlying risk moved from 1 in 100,000 to 2 in 100,000, it is not.
The classic trap
A shop raises a price by 20%, then a month later reduces it by 20%. Customers assume they are back where they started. A $100 item becomes $120, then $96 — a net 4% fall. Sequential percentage changes multiply: 1.20 × 0.80 = 0.96. The larger the swings, the wider the gap.
A quick sanity check that catches most mistakes: before calculating, ask “percent of what?” Almost every percentage error is a wrong base rather than wrong arithmetic.
Percentage points are not percent
A distinction that matters, and that news reports mangle constantly. If an interest rate rises from 4% to 6%, that is a rise of two percentage points — but a 50% increase. Both statements are true and they describe the same event with wildly different emotional weight.
Related trap: a 50% fall followed by a 50% rise does not return you to where you started. $100 falls to $50, then rises to $75. Percentage changes multiply, and they are not symmetric.