Percentages
The idea
A percentage is a fraction with a fixed denominator of 100 — 45% literally means 45 out of every 100. That fixed base is what makes percentages so useful: prices, test scores, and battery levels all become comparable because they are rescaled onto the same 0-to-100 ruler. You already know most of this from fractions and decimals: 45% = 45/100 = 0.45, the same number in three outfits.
The reliable move is to convert to a decimal and multiply: 30% of 250 is 0.30 × 250 = 75. The most common trap is treating a percentage like a fixed amount. A 10% rise followed by a 10% drop does not return you to the start, because the second 10% is taken of a different, larger number. Whenever a percentage appears, ask: a percentage of what?
Worked example
A jacket costs $80. The store raises the price by 25%, and a month later puts the new price on sale at 25% off. What is the final price?
- Find the raised price first: 25% of $80 is 0.25 × 80 = $20, so the new price is 80 + 20 = $100.
- The discount applies to the NEW price, not the original: 25% of $100 is 0.25 × 100 = $25.
- Subtract the discount from the raised price: 100 − 25 = $75.
- Sanity-check the surprise: the final price sits below the original $80 because the 25% increase was computed on 80 while the 25% decrease was computed on the larger 100 — equal percentages of different bases are different amounts.
Answer. The final price is $75 — lower than the original $80, because the two 25% changes acted on different bases.
Check your understanding
- Why does a 25% increase followed by a 25% decrease not cancel out — and would the order of the two changes matter?
- How would you explain to a friend that 45%, 0.45 and 45/100 are the same number?
- What happens when the 'whole' changes partway through a percentage problem, and how do you keep track of which base each percentage uses?
- If a quantity doubles, what percentage increase is that — and why is the answer not 200%?
Build the foundations first
Percentages builds on these concepts. If any feel shaky, start there.