To learn Percentages quickly and correctly, remember what the symbol means: per cent means out of one hundred. A percentage is another way of describing a fraction or proportion, not a special calculation unrelated to fractions. With that one foundation, you can find percentages of amounts, work out discounts, calculate percentage changes and check whether an answer makes sense. These eduKateSG Quick Notes teach a small set of reusable methods rather than a long collection of shortcuts.
- Translate: 25% means 25/100 = 1/4 = 0.25.
- Find a percentage of an amount: multiply the amount by percentage ÷ 100.
- Find a part as a percentage: part ÷ whole × 100%.
- Change a value: apply the increase or decrease to the original base.
- Reverse a percentage: use the remaining or new percentage multiplier.
- Check: compare the result with 0%, 50% and 100% landmarks.
First, build a mental picture of one hundred equal parts. If 25 out of 100 squares are shaded, 25% is shaded. Divide numerator and denominator by 25: 25/100 = 1/4. The same amount is 0.25. Switching between percentage, fraction and decimal is the most useful basic skill. For example, 50% = 1/2, 10% = 1/10 and 5% = 1/20. These familiar anchors make quick mental checks possible.
Second, separate the rate from the whole. To find 15% of 80, take 15/100 × 80 = 12. You can also use 10% of 80 (which is 8) plus 5% (which is 4). To express 12 out of 80 as a percentage, calculate 12 ÷ 80 × 100% = 15%. The two problems are related, but one asks for an amount and the other asks for the percentage. Read the wording before deciding which operation you need.
Third, percentage change always needs a base. If a price rises from $50 to $60, the change is $10, measured against the original $50. Thus percentage increase = 10 ÷ 50 × 100% = 20%. If it later returns to $50, the percentage decrease from $60 is 10 ÷ 60 × 100% = 16⅔%, not 20%. The amount of change is the same, but the starting whole has changed. Keep asking: “20% of what?”
Worked example: a discount
A book has a marked price of $50 and is discounted by 20%. Twenty per cent of $50 is 0.20 × 50 = $10. Subtract the discount: $50 − $10 = $40. Alternatively, you pay 80% of the original, so 0.80 × $50 = $40. Both methods agree. A 20% discount must make the final price smaller than the original.
Worked example: finding the original amount
A jacket costs $72 after a 20% discount. The final price represents 80% of the original. Let the original price be P: 0.8P = 72. Divide by 0.8: P = $90. Check by taking 20% of $90, which is $18; $90 − $18 = $72. Do not add 20% of $72 to $72 — that uses the reduced price as the wrong base.
Choose the right formula
- Percentage of a number: (percentage ÷ 100) × whole.
- What percentage is a part? (part ÷ whole) × 100%.
- Percentage increase: (new − original) ÷ original × 100%.
- Percentage decrease: (original − new) ÷ original × 100%.
- New amount after r% increase: original × (1 + r/100).
- New amount after r% decrease: original × (1 − r/100).
Why adding and subtracting the same percentage does not undo a change
Start with $100. Increase by 10%: you get $110. Decrease that new amount by 10%: you remove $11, leaving $99, not $100. The percentages act on different bases. For an exact reversal, find the original by dividing the final value by the appropriate multiplier.
Common mistakes and corrections
- Using 15 instead of 0.15: 15% is 15/100, not 15.
- Confusing part and whole: label both before forming a percentage.
- Measuring change against the final amount: standard percentage change uses the original as the denominator.
- Adding a discount instead of subtracting it: check the direction of the answer.
- Rounding too early: keep full calculator precision, then round as requested.
Mini-test: three short checks
- Find 35% of 200.
- An item costs $60 before a 15% discount. Find the sale price.
- After a 25% increase, an amount is $100. Find the original amount.
Answers and reasoning
1. 0.35 × 200 = 70. 2. The discount is $9, so the sale price is $51. 3. The new amount is 125% of the original: 100 ÷ 1.25 = $80. Check that 25% of $80 is $20 and $80 + $20 is $100.
How to revise percentages without wasting time
Practise one direct percentage, one change and one reverse-percentage question. For each answer, write the base underneath the percentage. If a question is difficult, convert the percentage into a decimal multiplier and check the direction. Tomorrow, repeat with different numbers and no worked example in sight.
Keep learning: your next small step
Build the next foundation with Fractions, Ratios, Mathematics Word Problems, Mathematics. These Quick Notes work best when you study one idea, answer one question alone and return the next day without opening the example.
The longer school-learning context is Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. That original reference remains unchanged.