In my final result, Maths was one of the subjects my maximum marks came from, and percentage sits under most of that syllabus. It is not just one chapter. Profit and loss, discount, simple interest, ratio and data interpretation all run on percentages. If you are fast here, you are fast in half the Maths section. I practised from R.S. Aggarwal, and this page is the summary I wish I had for revision.
What this topic covers
The fraction table: learn it by heart
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% |
| 1/3 | 33.33% | 1/10 | 10% |
| 1/4 | 25% | 1/11 | 9.09% |
| 1/5 | 20% | 1/12 | 8.33% |
| 1/6 | 16.67% | 1/16 | 6.25% |
| 1/7 | 14.28% | 1/20 | 5% |
| 1/8 | 12.5% | 1/25 | 4% |
Why it matters: 12.5% of 640 is just 640 ÷ 8 = 80. No multiplication needed.
Core percentage formulas
- Successive change: for changes of a% and b%, net change = a + b + ab/100 (use minus for decrease).
- "More than" and "less than": if A is r% more than B, then B is r/(100 + r) × 100 % less than A. If A is r% less than B, then B is r/(100 − r) × 100 % more than A.
- Price up, keep expenditure same: if price rises by r%, consumption must fall by r/(100 + r) × 100 %.
- Population or depreciation: after n years at r% a year, value = P × (1 ± r/100)ⁿ.
Core profit and loss formulas
- Profit % = Profit ÷ CP × 100. Loss % = Loss ÷ CP × 100. Always on cost price.
- SP = CP × (100 + Profit%) / 100, or CP × (100 − Loss%) / 100.
- Discount is on the marked price: SP = MP × (100 − Discount%) / 100.
- Successive discounts d1% and d2% = d1 + d2 − d1·d2/100.
- Same SP, x% profit on one and x% loss on the other: always a loss of x²/100 %.
- False weight: gain % = error ÷ (true value − error) × 100.
How it is asked in the exam
Example 1: successive change
A price rises by 20% and then falls by 10%. What is the net change?
20 − 10 + (20 × −10)/100 = 10 − 2 = 8% increase.
Example 2: "more than / less than"
A's salary is 20% more than B's. By what percent is B's salary less than A's?
20/120 × 100 = 16.67%. (1/6 from the table.)
Example 3: expenditure
The price of sugar rises by 25%. By how much must a family cut consumption to keep spending the same?
25/125 × 100 = 20%.
Example 4: find CP, then new SP
By selling an article for ₹540, a shopkeeper loses 10%. At what price should he sell it to gain 10%?
- CP = 540 × 100/90 = ₹600.
- New SP = 600 × 110/100 = ₹660.
Example 5: CP of some = SP of others
The cost price of 20 articles equals the selling price of 16 articles. Find the profit %.
Profit % = (20 − 16)/16 × 100 = 25%.
Example 6: marked price and discount
An article marked at ₹1,000 is sold at a 10% discount. The cost price is ₹750. Find the profit %.
- SP = 1,000 × 90/100 = ₹900.
- Profit = 900 − 750 = ₹150.
- Profit % = 150/750 × 100 = 20%.
Example 7: false weight
A shopkeeper sells at cost price but uses a 900 g weight instead of 1 kg. Find his gain %.
100/900 × 100 = 11.11%. (1/9 from the table.)
Points to remember
- Two successive discounts of 20% and 10% are not 30%. They equal 20 + 10 − 2 = 28%.
- 10% profit and 10% loss on the same SP is never "no profit no loss". It is a 1% loss.
- When a question gives profit on SP (it sometimes does, to trap you), convert first: profit of 20% on SP is 25% on CP.
- Most answers here are "nice" fractions. If you get 16.66, look for 1/6. If you get 14.28, look for 1/7.
Common mistakes
- Calculating discount on cost price. Discount is on marked price.
- Adding successive percentages directly.
- Using SP in the profit % denominator.
- Spending 3 minutes on one question. If the table doesn't give you a quick path within a minute, mark it and move on, then come back.
Practise straight after reading
List the types on this page (successive change, more/less than, expenditure, CP-SP, CP of x = SP of y, discount, false weight, same SP). Solve at least 15 questions of each type from your Maths book, then 100 to 200 mixed questions from the Maths quizzes on this site. Write the fraction table on a card and revise it every day for the first week; after that, 1-7-30 is enough.