Percentages
Percentages are just fractions in disguise
Percent means "out of 100." Once you see percentages as fractions, calculations become much simpler.
See It in Action
The Problem
A jacket costs $80. It is on sale for 15% off. What is the sale price?
Common Mistake
15% of 80 = 12. Sale price = 80 − 15 = 65.
The percentage was calculated correctly ($12), but the student subtracted 15 (the percentage number) instead of $12 (the actual discount). Always subtract the dollar amount, not the percentage number.
Correct Approach
15% of $80 = 0.15 × 80 = $12. This is the discount. Sale price = $80 − $12 = $68.
Answer: $68
$68 + 15% of $68 ≠ $80 (that would be adding 15% back to the discounted price, not the original). Quick check: $12 is 15% of $80, and $80 − $12 = $68. ✓
The Core Concept
Percent literally means "per hundred." So 35% means 35 out of every 100, which is the same as the fraction 35/100, which simplifies to 7/20, which as a decimal is 0.35. All four are the same number written differently.
The most common calculation is "find X% of Y." The reliable method: convert the percentage to a decimal (divide by 100) and multiply. 30% of 240 = 0.30 × 240 = 72. Or use fractions: 30% = 30/100 = 3/10, so 3/10 × 240 = 72. Both work.
For percentage increase and decrease, the formula is: new amount = original × (1 ± percentage as decimal). A 20% increase on 150 = 150 × 1.20 = 180. A 15% decrease on 200 = 200 × 0.85 = 170. Understanding this lets you also reverse it: if a price after a 25% increase is $250, the original was $250 ÷ 1.25 = $200.
Common Mistakes to Avoid
Subtracting the percentage number instead of the calculated amount
"15% off $80" gives a discount of $12, not $15. Always calculate the actual dollar/unit amount first.
Finding the percentage of the wrong value
"What percentage is 12 out of 48?" means 12/48 × 100 = 25%. Students often flip numerator and denominator, getting 48/12 × 100 = 400% — which is clearly wrong.
Thinking 100% of something equals 100
100% of 240 is 240, not 100. One hundred percent means the whole thing, whatever that whole thing is.
Confusing percentage increase and finding the new total
A 20% increase on 150 means the NEW amount is 180, not 20% (which is 30). The increase is 30, the new total is 180.
Try It Yourself
A school has 600 students. 45% are boys. How many girls are there?
Hint: Find 45% of 600 to get the number of boys. Then subtract from 600. (Or: girls = 100% − 45% = 55% of 600.)
Key Tips
Convert percentages to decimals by dividing by 100: 35% → 0.35, 8% → 0.08.
Common ones to memorise: 50% = 1/2, 25% = 1/4, 10% = 1/10, 20% = 1/5.
To find X% of Y: multiply Y × (X ÷ 100). That's it.
Ready to practise Percentages?
Reading is the start. The Maths Gym has exercises designed around this skill — with instant feedback and progress tracking.