Percentage
This guide summarizes the key concepts from your Percentage PDF, making the topic simple with step-by-step explanations, formulas, conversion tables, solved examples, cheatsheets, practice questions (with answers), and visual aids.
1. What is a Percentage?
- Definition: A percentage is a fraction with a denominator of 100, denoted by the symbol %.
- Example: $ 10% = \frac{10}{100} = 0.1 $1.
2. Key Formulas and Conversions
A. Basic Percentage Formula
- To find $ y% $ of $ x $:
Example: $ 25% $ of $ 200 = 200 \times \frac{25}{100} = 50 $12.
B. Expressing Fractions as Percentages
- $ \frac{a}{b} $ as a percent:
Example: $ \frac{1}{4} \times 100 = 25% $12.
C. Expressing Percentages as Fractions
D. Increase/Decrease by a Percentage
- Increase by $ x% $: Multiply by $ \frac{100 + x}{100} $
- Decrease by $ x% $: Multiply by $ \frac{100 - x}{100} $1.
E. Percentage Change
- Increase: $ \frac{Final Value - Initial Value}{Initial Value} \times 100 $
- Decrease: $ \frac{Initial Value - Final Value}{Initial Value} \times 100 $1.
3. Cheatsheet: Common Conversions
| Fraction | Percentage |
|---|---|
| 1/2 | 50% |
| 1/3 | 33.33% |
| 1/4 | 25% |
| 1/5 | 20% |
| 1/6 | 16.66% |
| 1/7 | 14.28% |
| 1/8 | 12.5% |
| 1/9 | 11.11% |
| 1/10 | 10% |
| 1/11 | 9.09% |
| 1/12 | 8.33% |
| 1/13 | 7.69% |
| 1/14 | 7.14% |
| 1/15 | 6.66% |
| 1/16 | 6.25% |
| 1/20 | 5% |
| 1/25 | 4% |
4. Visuals & Graphics
A. Percentage as a Pie Chart
[100% = Whole Circle]
|---25%---|---25%---|---25%---|---25%---|Each quarter represents 25%.
B. Bar Representation
|--------------------------| 100%
|-----------| 40%A bar filled to 40% of its length.
5. Important Concepts and Tricks
- Expressing One Quantity as a Percent of Another:
- If A is X% more than B, then B is $ \frac{X}{100+X} \times 100% $ less than A.
- If A is X% less than B, then B is $ \frac{X}{100-X} \times 100% $ more than A.1
6. Solved Examples
Example 1: Find the Number from a Percentage
Q: If $ 40% $ of $ P = 100 $, find $ P $.
$$ P \times \frac{40}{100} = 100 \implies P = \frac{100 \times 100}{40} = 250 $$Example 2: Fraction to Percentage
Q: Express $ \frac{1}{5} $ as a percentage.
$$ \frac{1}{5} \times 100 = 20\% $$Example 3: Percentage Increase
Q: A number increases from 50 to 65. What is the percentage increase?
$$ \frac{65-50}{50} \times 100 = \frac{15}{50} \times 100 = 30\% $$Example 4: Percentage of Marks
Q: A student scores 476 out of 500 marks. What is the percentage?
$$ \frac{476}{500} \times 100 = 95.2\% $$7. Practice Questions with Answers
| Q# | Question | Answer |
|---|---|---|
| 1 | A man distributes 10%, 18%, and 22% of his salary to his three children who spend 40%, 60%, and 25% of that amount respectively. The difference between the total amount left with the children and man is Rs. 1015. What is the salary of the man? | Rs. 5000 |
| 2 | Salary of A is 37.5% of the total salary of A and B. B saves 60% of his salary and total savings of A and B is 50% of their total income. Their average expenditure is Rs 16,000. What is the total salary of A and B? | Rs. 64,000 |
| 3 | In a class, 25% passed both English and Hindi, 37.5% failed both, 60% failed Hindi. The difference between those who passed English and Hindi is 15. What is the total number of students? | 200 |
| 4 | Out of total students, 100/3% are in hostel A, rest in hostel B. If 20 move from B to A, hostel A becomes 50% of total. If 20 move from A to B, what percent are in A? | 16.67% |
| 5 | AB de Villiers scores 86 runs in 16 balls, only in boundaries. What is the maximum percent of runs scored by hitting fours? | 23.25% |
| 6 | A mobile phone is sold at 20% discount, plus 10% cashback. If Suraj pays Rs. 36,000, what is the original price? | Rs. 50,000 |
| 7 | 25% of females and 20% of males hold positions above level 2. If ratio is 3:2, what percent work below level 2? | 77% |
| 8 | Dishonest salesman buys x% more grains, uses 800g for 1000g, sells at 10% above cost, earns 65% profit. Find x. | 20% |
| 9 | Passing marks for class IX and X are 30% and 45%. A class X student scores 1225, fails by 125 marks. Find passing marks for IX. | 900 |
| 10 | 40% of novels sold at 3/4 cost price. How much above cost price should the rest be sold to get 20% profit overall? | 50% |
| 11 | Jittu spends 1/5 on notebook, 25% of rest (Rs. 12) on marker. Find money lost. | Rs. 80 |
| 12 | 20% of village died from diabetes, 2000 from lung cancer (33.33% of smokers). Diabetes deaths 1200 more than lung cancer. What percent were smokers? | 37.5% |
| 13 | Rakul spends 10% on rent, 14% on car, 12% on kids’ school, 15% and 10% of rest on groceries and vacation. He saves Rs. 5,18,400 yearly. What is his monthly salary? | Rs. 90,000 |
| 14 | Two villages had same population 2 years ago. Rampur decreases at R% p.a., Jamnagar increases at R% p.a. Difference now is 1000R. What was the population 2 years ago? | 25,000 |
| 15 | Rama’s deposit is 100% more than Ajay, 75% more than Jatin. Rama’s deposit is what percent of Ajay + Jatin? | 93.33% |
| 16 | In a school, 40% are in high school or above (boys:girls = 7:3), rest in junior high or below (boys:girls = 7:5). Ratio of boys in high school to junior high? | 4:5 |
| 17 | Ramu scored 92%, Naveen 56%, Samarth 634 marks, average 643. What percent did Samarth get? | 72.45% |
| 18 | A, B, C speed ratio 5:4:3. C finishes in 20 min. When B finishes, A’s watch shows 7:27pm. When C finishes, his watch shows 7:30pm, B’s shows 7:16pm. At start, what was the difference between A and B’s watch? | 16 min |
8. Advanced Percentage Tricks
- Successive Percentage Change: If a value is increased by $ a% $ and then by $ b% $, the net change is:
Example: Increase by 20%, then 10%: $ 20 + 10 + \frac{20 \times 10}{100} = 32% $ increase.
- Population Change Formula: For population $ P $, increasing at $ R% $ per annum for $ n $ years:
For decrease, use minus sign.
9. Percentage in Data Interpretation
Percentages are widely used in pie charts, bar graphs, and tables. For example, if a sector in a pie chart is 90°, it represents $ \frac{90}{360} \times 100 = 25% $ of the total3.
10. Summary Table: Percentage Formulas
| Purpose | Formula |
|---|---|
| Find y% of x | $ x \times \frac{y}{100} $ |
| Convert fraction to percent | $ \frac{a}{b} \times 100 $ |
| Convert percent to fraction | $ x% = \frac{x}{100} $ |
| Increase by x% | $ Number \times \frac{100+x}{100} $ |
| Decrease by x% | $ Number \times \frac{100-x}{100} $ |
| % Increase/Decrease | $ \frac{Change}{Original} \times 100 $ |
| Successive % change | $ a + b + \frac{ab}{100} $ |
Conclusion
- Percentages are essential for exams and daily life.
- Learn the formulas, practice the conversion tables, and use the solved examples.
- Practice the above questions to master the topic.
Visualize percentages as parts of a whole (pie/bar charts) and always relate them to 100 for easy calculation!
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