Percentages
Per cent means per hundred. 50% means 50 out of 100, or half, or 0.5. Master conversions between percentages, fractions, and decimals.
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If a test has 50 questions and you get 80% correct, how many did you get right? And what fraction is 80%? Write your thinking.
Percentage means per hundred. 50% = $ rac{50}{100}$ = 0.50. To convert a percentage to a fraction, write it over 100 and simplify. To convert to a decimal, divide by 100. To convert a fraction or decimal to a percentage, multiply by 100.
Think of a percentage as a fraction with denominator 100. 25% = $ rac{25}{100}$ = $ rac{1}{4}$ = 0.25. 100% = $ rac{100}{100}$ = 1 (the whole thing). Percentages over 100% mean more than the whole: 150% = 1.5. Percentages under 1% are tiny amounts: 0.5% = $ rac{0.5}{100}$ = $ rac{1}{200}$.
Know
- % means per hundred
- Common % ↔ fraction ↔ decimal conversions
- How to find a percentage of an amount
Understand
- Why 100% = 1 and 50% = 1/2
- How percentages compare parts to wholes
- Percentages greater than 100%
Can Do
- Convert between %, fractions, and decimals
- Find a percentage of any amount
- Express one quantity as a % of another
Wrong: 75% of 80 = 75 × 80 = 6000. No! 75% = 0.75, so 0.75 × 80 = 60.
Right: 75% of 80 = $ rac{75}{100}$ × 80 = 0.75 × 80 = 60. Always convert % to decimal or fraction first.
Wrong: 20% = 1/5 = 0.05. No! 1/5 = 0.20, not 0.05. 0.05 = 5% = 1/20.
Right: 20% = $ rac{20}{100}$ = $ rac{1}{5}$ = 0.20. Double-check by converting back: 0.20 × 100 = 20%.
% → fraction: write over 100, simplify. % → decimal: divide by 100. Fraction → %: make denominator 100, or ×100. Decimal → %: ×100.
Convert 60% to a fraction and decimal. Fraction: $ rac{60}{100}$ = $ rac{3}{5}$ (HCF = 20). Decimal: 60 ÷ 100 = 0.6. Convert 0.125 to a %: 0.125 × 100 = 12.5%. Convert $ rac{7}{20}$ to a %: $ rac{7}{20}$ = $ rac{35}{100}$ = 35%.
Convert the percentage to a fraction or decimal, then multiply by the amount. Mental shortcuts: 10% = divide by 10, 50% = halve, 25% = quarter, 1% = divide by 100.
Find 35% of $80. Method: 35% = $ rac{35}{100}$ = 0.35. 0.35 × 80 = $28. Mental shortcut: 10% = $8, so 30% = $24, and 5% = $4. $24 + $4 = $28. Both methods agree!
To express one quantity as a percentage of another: (1) Write as a fraction, (2) Multiply by 100, (3) Add the % sign.
Express 18 out of 40 as a percentage. $ rac{18}{40}$ × 100. Simplify: $ rac{18}{40}$ = $ rac{9}{20}$. $ rac{9}{20}$ × 100 = $ rac{900}{20}$ = 45%. Or convert: $ rac{18}{40}$ = 0.45. 0.45 × 100 = 45%.
Find 15% of $120.
Convert 15% to a decimal: 15% = 0.15. Or use mental method: 10% = $12, 5% = $6.
Method 1: 0.15 × 120 = 18. Method 2: 10% + 5% = $12 + $6 = $18.
Check: 15% is a bit more than 10%, and $18 is a bit more than $12. Reasonable! Also: 10% of $18 = $1.80, and 15% of $120 = $18. 1.80/18 = 0.10, 18/120 = 0.15. Correct.
15% of $120 = $18
In a class of 25 students, 17 are girls. What percentage are girls?
Write as a fraction: $\frac{17}{25}$.
Convert to %: $\frac{17}{25}$ = $\frac{68}{100}$ = 0.68. 0.68 × 100 = 68%. Or: $\frac{17}{25}$ × 100 = $\frac{1700}{25}$ = 68.
Check: Just over half (50%) of 25 is 12.5. 17 is more than 12.5, so 68% makes sense. $\frac{17}{25}$ is close to $\frac{18}{25}$ = 72% and $\frac{16}{25}$ = 64%. 68% is right in between.
68% of the class are girls
A shirt costs $80. It is reduced by 25% in a sale. What is the sale price?
Method 1: Find discount, subtract. 25% of $80 = 0.25 × 80 = $20. Sale price = $80 − $20 = $60.
Method 2: Pay 75% (100% − 25%). 75% of $80 = 0.75 × 80 = $60. Same answer!
Check: 25% = 1/4, so discount is $80/4 = $20. Sale = $80 − $20 = $60. Correct! Method 2 is faster for quick mental calculations.
Sale price = $60
Mistake: 15% of 200 = 15 × 200 = 3000. No! You forgot to divide by 100. 15% = 0.15, so 0.15 × 200 = 30.
Fix: Always convert % to decimal first: 15% = 0.15. Then 0.15 × 200 = 30. Estimate: 10% = 20, so 15% ≈ 30.
Mistake: 7 out of 20 = 7/20 = 35 (no % sign). A percentage always needs the % symbol!
Fix: 7/20 = 0.35 = 35%. Multiply by 100 and add the % sign.
Mistake: 120% of 50 = 50 − (0.20 × 50) = 40. No! 120% means MORE than the whole, not less.
Fix: 120% = 1.20. 1.20 × 50 = 60. Over 100% always gives a bigger answer.
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to build your percentage fluency. Work each, then reveal the answer.
-
1 Find 20% of $150.
20% = 0.20 = 1/5. 150/5 = 30. Or: 0.20 × 150 = 30.$30 -
2 Convert $\frac{3}{25}$ to a percentage.
3/25 = 12/100 = 0.12. 0.12 × 100 = 12%.12% -
3 Express 9 out of 30 as a percentage.
9/30 = 3/10 = 0.30 = 30%.30% -
4 Find 12.5% of 240.
12.5% = 1/8. 240/8 = 30. Or: 0.125 × 240 = 30.30
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. (a) Find 35% of $240. Show both the decimal method and the mental 10% method. (b) A team won 16 out of 25 games. What percentage did they win?
Q7. Convert each to the other two forms: (a) 0.6, (b) $\frac{7}{25}$, (c) 45%. Show your working.
Q8. A student says: “To find 15% of an amount, I divide by 15.” Explain why this is wrong, and show the correct method for finding 15% of $300.
Quick Check
1. B — 25% = 1/4, 120/4 = 30.
2. A — 0.08 × 100 = 8%.
3. C — 14/40 = 7/20 = 35%.
4. B — 0.40 × 90 = 36.
5. D — 1.5 × 40 = 60.
Show Your Working Model Answers
Q6 (4 marks): (a) Decimal: 0.35 × 240 = 84 [1]. Mental: 10% = 24, 30% = 72, 5% = 12, 35% = 72 + 12 = 84 [1]. (b) 16/25 = 64/100 = 64% [2].
Q7 (3 marks): (a) 0.6 = 3/5 = 60% [1]. (b) 7/25 = 0.28 = 28% [1]. (c) 45% = 9/20 = 0.45 [1].
Q8 (3 marks): Dividing by 15 gives 1/15, not 15% [1]. 15% = 15/100 = 0.15 [0.5]. 0.15 × 300 = 45 [1]. Or 10% = 30, 5% = 15, so 15% = 45 [0.5].
The Successive Percentage Puzzle
A $400 phone increases in price by 20%, then decreases by 20%. A student says it returns to $400. Is the student correct? Calculate the final price. Then find the overall percentage change from the original price. Why is it not zero?
Reveal solution
Increase: $400 × 1.20 = $480. Decrease: $480 × 0.80 = $384. Final = $384, not $400! Overall change: ($400 − $384)/$400 = 16/400 = 4% decrease. The decrease applies to a larger amount ($480) than the increase ($400), so the decrease “wins.” The overall effect is (1.20 × 0.80) = 0.96 = 96%, a 4% decrease.
% means /100
Always start here
% → decimal
÷ 100 (dot 2 left)
Decimal → %
× 100 (dot 2 right)
% of amount
(% as decimal) × amount
A as % of B
(A/B) × 100
100% = 1
50%=1/2, 25%=1/4
Interactive: Percentage Change Predictor
Predict the new price after a percentage increase or decrease. Watch out for the gotcha — percentages apply to the CURRENT price!
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