Percentage means "per hundred". Almost every word problem about profit, interest, data, or population is a percentage problem wearing a costume. The fastest students stop computing percentages and start recognising them.
The fraction table you should know cold
Converting a percentage to a fraction turns slow multiplication into instant cancellation.| Percentage | Fraction |
|---|---|
| 50% | 1/2 |
| 25% | 1/4 |
| 10% | 1/10 |
| 33.33% | 1/3 |
| 12.5% | 1/8 |
| 20% | 1/5 |
| 16.66% | 1/6 |
| 6.25% | 1/16 |
Successive change of a% then b%
Net % change = a + b + (a × b) / 100 (use a minus sign for a decrease).
How to Approach It
Every percentage question hides a single decision: what is the base — the quantity the percentage is taken 'of'? Get the base right and everything after it is mechanical.
- Pin down the base. Underline the quantity the percentage refers to. 'x% of A' makes A the base; 'B is x% more than C' makes C the base. Most wrong answers come from using the wrong base.
- Convert to a multiplier or fraction. Turn the percentage into ×(1±x/100) for increases and decreases, or into a familiar fraction such as 25% → 1/4.
- Chain changes in sequence. For two changes one after another, never just add them — apply the multipliers in turn, or use a + b + ab/100. A rise followed by an equal fall never returns you to the start.
- Re-read exactly what's asked. The question may want the final value, the size of the change, or the original amount. Report the precise quantity asked for.
Techniques & Methods
- Multiplier method. A rise of x% is ×(1+x/100); a fall is ×(1−x/100). Chain them in one line. e.g. +20% then −10% → ×1.2×0.9 = ×1.08 → net +8%.
- Successive-change formula. Two changes a% then b% combine to a + b + ab/100. e.g. +10% then +10% → 21%.
- Base-change rule. 'A is x% more than B' means B is x/(100+x)×100 % less than A. e.g. 25% more → 20% less.
- Constant-product (expenditure). If price rises x%, cut quantity by x/(100+x)×100 to hold the bill fixed. e.g. Price +25% → cut use by 20%.
The Edge
When a value goes up by x% and then down by x%, it never returns to the start — it always falls by (x^2)/100 percent. Salary +10% then −10%? Net = −1%. Price +20% then −20%? Net = −4%. Memorising this kills a whole class of "net change" questions instantly.Worked example
A's salary is 25% more than B's. By what percent is B's salary less than A's?
- Spot the trap: 'more' and 'less' use different bases. A is 25% more than B (base B), but the question asks how much less B is than A (base A).
- Put numbers on it: let B = 100, so A = 125.
- Compare B to A using A as the base: gap = 125 − 100 = 25, and 25/125 = 1/5 = 20%.
- The shortcut x/(100+x)×100 = 25/125×100 = 20% gives the same answer in one line.
Answer: B earns 20% less than A
Worked example
If the price of sugar rises 25%, by what percent must a family cut consumption to keep its sugar bill unchanged?
- The bill = price × quantity, and we want it unchanged, so the quantity must fall to cancel the rise.
- Set price and quantity to 100 each → bill 10,000. After +25%, price = 125.
- To hold the bill: new quantity = 10,000 ÷ 125 = 80.
- Quantity fell from 100 to 80 — a 20% drop. Shortcut: 25/125×100 = 20%.
Answer: Cut consumption by 20%
Worked Drills
Worked example
40 is what percent of 250?
- Percent = 40/250 × 100.
- = 16%.
Answer: 16% (option c)
Worked example
Two successive discounts of 20% and 10% equal a single discount of:
- Net change = −20 + (−10) + (−20)(−10)/100 = −30 + 2 = −28%.
- So a single discount of 28%.
Answer: 28% (option b)
Worked example
A salary is increased by 20% and then decreased by 20%. The net change is:
- Equal up-then-down by x% gives a fall of x^2/100.
- = 20^2/100 = 4% decrease.
Answer: 4% decrease (option b)
Worked example
In a two-candidate election, the winner gets 65% of the votes and wins by 4500 votes. The total votes polled:
- Winning margin = 65% − 35% = 30% of total = 4500.
- Total = 4500 / 0.30 = 15000.
Answer: 15000 (option b)
Worked example
A's income is 25% more than B's. B's income is what percent of A's?
- Let B = 100, so A = 125.
- B/A = 100/125 = 80%.
Answer: 80% (option b)
Worked example
A salary is cut by 20%. To restore the original, it must be raised by:
- After −20%, value = 80% of original.
- To go from 80 back to 100: rise = 20/80 × 100 = 25%.
Answer: 25% (option c)
Worked example
The price of an article falls 20% while sales rise 25%. The effect on revenue is:
- Revenue multiplier = 0.8 × 1.25 = 1.
- Revenue is unchanged.
Answer: No change (option a)
Worked example
30% of a number is 135 more than 20% of it. The number is:
- 30% − 20% = 10% of the number = 135.
- Number = 135 / 0.10 = 1350.
Answer: 1350 (option b)
Worked example
Pass mark is 40%. A student scores 200 and fails by 40 marks. The maximum marks are:
- Pass mark = 200 + 40 = 240 marks = 40% of maximum.
- Maximum = 240 / 0.40 = 600.
Answer: 600 (option c)
⚠ Watch out
- "x% more" and "x% less" use different bases — never assume they cancel.
- Two successive discounts of 20% and 10% are not a 30% discount; they equal 28%.
- A percentage point change is not the same as a percentage change.
Takeaways
- Find the base first. Most errors come from taking the percentage 'of' the wrong quantity.
- Use multipliers to chain. ×(1±x/100) lets successive changes collapse into one line.
- Up-then-down by x% loses x²/100. The value never returns to its start.
- Memorise the fraction table. 12.5% → 1/8, 16.66% → 1/6 turn arithmetic into cancellation.