KnowThisKey Takeaways
Key Takeaways
- 1"Percent" literally means "per hundred" — 40% is exactly 40 out of every 100, no more complicated than that.
- 2Percentages, fractions, and decimals are three notations for the same value: 25% = 1/4 = 0.25.
- 3"Percentage points" and "percent change" are different measurements — mixing them up produces genuinely wrong numbers, not just sloppy wording.
The concept
Interest rates rise from 4% to 6%. How would you correctly describe this change?
Worked examples
Example 1: Finding a percentage of a number (baseline case)
Example 2: Working backward from a percentage (edge/variation case)
Example 3: A price after two sequential discounts (applied case)
Common mistakes
Common Mistakes
Adding two sequential percentage discounts together instead of applying them one after another.
→ Multiply the remaining fraction at each step (× 0.8 then × 0.9), never just add the percentages — 20% + 10% off is not the same as 20% off then 10% off.
Confusing percentage points with percent change.
→ A change from 10% to 15% is 5 percentage points, but a 50% relative increase — state which one you mean, since they're both correct answers to different questions.
Forgetting to convert a percentage to a decimal before multiplying.
→ Always divide by 100 first (or move the decimal point two places left) before using a percentage in a calculation.
Common misconception
“A discount followed by an equal-sized markup (or vice versa) brings you back to the original price.”
It doesn't, because each percentage applies to a different base. A $100 item marked up 50% becomes $150; discounting that $150 by 50% gives $75, not the original $100 — the markup and discount were applied to different starting amounts.
Try it yourself
What to do next
What to do next
- Practice converting between percentages, decimals, and fractions until the three feel interchangeable rather than like separate topics.
- Next time you see a sale with two stacked discounts, calculate the real combined price by multiplying, not adding, the percentages.
- When you read a news headline about a rate "rising by X%," check whether X is a percentage-point change or a relative percent change.