Percentages
Practice MCQsPercentages represent proportions or ratios expressed in parts per hundred, often used to compare relative quantities or express a portion of a whole. Percentages are commonly used in various contexts such as finance, statistics, and everyday life to convey information about growth rates, discounts, probabilities, and distribution of values.
Percentages are one of the most important topics in quantitative aptitude. A percentage represents a number out of 100. Percentage concepts are widely used in profit and loss, discounts, simple interest, compound interest, data interpretation, population, marks, salary, tax, and comparison-based questions.
What is Percentage?
The word percentage means “per hundred”. If we say \(25\%\), it means \(25\) out of \(100\).
Percentages help us compare quantities easily, even when the total values are different. For example, scoring 45 marks out of 50 and scoring 90 marks out of 100 both represent the same performance because both are \(90\%\).
| Expression | Meaning | Fraction / Decimal |
|---|---|---|
| 25% | 25 out of 100 | \(\frac{25}{100}=0.25\) |
| 50% | 50 out of 100 | \(\frac{50}{100}=0.5\) |
| 75% | 75 out of 100 | \(\frac{75}{100}=0.75\) |
| 100% | Complete whole | \(\frac{100}{100}=1\) |
“A percentage is simply a convenient way of comparing values on a base of 100.”
Key points
- Percentage means per hundred.
- To convert percentage to fraction, divide by 100.
- To convert fraction to percentage, multiply by 100.
- Percentage increase is calculated on original value.
- Percentage decrease is calculated on original value.
- Successive percentage changes must be applied step by step.
Visual Understanding
These diagrams show how percentages represent parts of a whole and how percentage increase and decrease work.
25% Means One-Fourth
\(25\%\) means one-fourth of the whole.
Percentage Increase
Increase percentage is always calculated on the original value.
Percentage Decrease
Decrease percentage is also calculated on the original value.
Important Formulas and Rules
Percentage to Fraction
Divide by \(100\) to convert percentage into fraction.
Fraction to Percentage
Multiply by \(100\) to convert fraction into percentage.
Value of a Percentage
Used to find a percentage of a given number.
Percentage of a Value
Used when part and whole are known.
Percentage Increase
Increase is calculated on original value.
Percentage Decrease
Decrease is calculated on original value.
New Value After Increase
Used when value increases by \(x\%\).
New Value After Decrease
Used when value decreases by \(x\%\).
Common Fraction and Percentage Values
These values are very useful for quick calculations in exams.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| \(\frac{1}{2}\) | 50% | \(\frac{1}{3}\) | 33.33% |
| \(\frac{1}{4}\) | 25% | \(\frac{3}{4}\) | 75% |
| \(\frac{1}{5}\) | 20% | \(\frac{2}{5}\) | 40% |
| \(\frac{1}{8}\) | 12.5% | \(\frac{3}{8}\) | 37.5% |
| \(\frac{1}{10}\) | 10% | \(\frac{1}{20}\) | 5% |
Common Types of Percentage Questions
Basic Percentage
Find percentage of a number.
- 20% of 500
- 15% of 800
- 40% of 250
- Direct formula
Marks Percentage
Find percentage based on marks obtained.
- Obtained marks
- Total marks
- Pass percentage
- Comparison of scores
Increase and Decrease
Find new value after rise or fall.
- Salary increase
- Population decrease
- Price rise
- Consumption decrease
Successive Percentage
More than one percentage change.
- Increase then decrease
- Two discounts
- Population change
- Apply step by step
Method Bank
Example: Find 20% of 500.
Example: 45 out of 60.
Example: 200 increased by 10%.
Example: 500 decreased by 20%.
Tip: For quick calculations, convert common percentages into fractions.
Successive Percentage Change
Step-by-Step Solving Method
| Step | Action | Example |
|---|---|---|
| Step 1 | Identify the whole or original value. | Total marks = 500 |
| Step 2 | Identify the part or changed value. | Marks obtained = 375 |
| Step 3 | Use the percentage formula. | \(\frac{\text{Part}}{\text{Whole}}\times100\) |
| Step 4 | Substitute values. | \(\frac{375}{500}\times100\) |
| Step 5 | Write answer with percentage sign. | 75% |
Solved Examples
| Question | Method | Answer |
|---|---|---|
| Find 20% of 500. |
\[
20\%\text{ of }500=\frac{20}{100}\times500
\]
\[
=100
\]
|
100 |
| What percentage is 45 out of 60? |
\[
\text{Percentage}=\frac{45}{60}\times100
\]
\[
=75
\]
|
75% |
| A student scores 360 marks out of 500. Find the percentage. |
\[
\text{Percentage}=\frac{360}{500}\times100
\]
\[
=72
\]
|
72% |
| The price of an item increases from ₹400 to ₹500. Find the percentage increase. |
Increase:
\[
500-400=100
\]
\[
\text{Increase Percent}=\frac{100}{400}\times100=25
\]
|
25% |
| The population of a town decreases from 50,000 to 45,000. Find the percentage decrease. |
Decrease:
\[
50000-45000=5000
\]
\[
\text{Decrease Percent}=\frac{5000}{50000}\times100=10
\]
|
10% |
| A number is increased by 20%. If the original number is 300, find the new number. |
\[
\text{New Value}=300\left(1+\frac{20}{100}\right)
\]
\[
=300\times1.2=360
\]
|
360 |
| A number is decreased by 15%. If the original number is 800, find the new number. |
\[
\text{New Value}=800\left(1-\frac{15}{100}\right)
\]
\[
=800\times0.85=680
\]
|
680 |
| A value is increased by 20% and then decreased by 10%. If original value is 100, find final value. |
\[
100\times\frac{120}{100}\times\frac{90}{100}
\]
\[
=108
\]
|
108 |
Note: In successive percentage change, do not add or subtract percentages directly. Apply each change one after another.
Common Traps and Shortcuts
Common Traps
- Using new value instead of original value for percentage change.
- Adding successive percentage changes directly.
- Forgetting to multiply by 100 while finding percentage.
- Confusing percentage with percentage points.
- Writing 20% as 20 instead of \(\frac{20}{100}\).
- Not identifying the base value correctly.
Useful Shortcuts
- 10% means one-tenth.
- 20% means one-fifth.
- 25% means one-fourth.
- 50% means one-half.
- 75% means three-fourths.
- For increase by \(x\%\), multiply by \(\frac{100+x}{100}\).
- For decrease by \(x\%\), multiply by \(\frac{100-x}{100}\).
Practice
A) Multiple Choice Questions
-
Find 15% of 600.
60 75 90 120
-
What percentage is 30 out of 50?
40% 50% 60% 75%
-
A number is increased from 200 to 250. Find the percentage increase.
20% 25% 30% 50%
-
A number 500 is decreased by 20%. Find the new value.
300 350 400 450
-
Which fraction is equal to 25%?
\(\frac{1}{2}\) \(\frac{1}{3}\) \(\frac{1}{4}\) \(\frac{3}{4}\)
B) Solve the Higher-Order Problems
- A student scores 420 marks out of 600. Find the percentage. (Hint: Use \(\frac{\text{Part}}{\text{Whole}}\times100\).)
- The salary of a person increases from ₹40,000 to ₹46,000. Find percentage increase. (Hint: Increase is based on original salary.)
- A population of 80,000 decreases by 12.5%. Find the new population. (Hint: 12.5% = \(\frac{1}{8}\).)
- A value is increased by 25% and then decreased by 20%. If original value is 400, find final value. (Hint: Apply changes successively.)
- If 35% of a number is 140, find the number. (Hint: Let the number be \(x\).)
C) Match the Concept with the Correct Meaning
| Concept | Correct Meaning |
|---|---|
| Percentage | Value per hundred |
| 50% | One-half |
| 25% | One-fourth |
| Percentage Increase | Increase compared with original value |
| Percentage Decrease | Decrease compared with original value |
| Successive Change | Percentage changes applied one after another |
Aptitude Reminder
Percentage means per hundred. In every percentage problem, first identify the base value or whole value. For increase and decrease, the original value is the base. For successive changes, apply each percentage change step by step.
Task: Create five percentage questions using percentage of a number, marks percentage, percentage increase, percentage decrease, and successive percentage change.
Show Suggested Answers
Multiple Choice
-
90
\[ 15\%\text{ of }600=\frac{15}{100}\times600=90 \] -
60%
\[ \frac{30}{50}\times100=60 \] -
25%
Increase \(=250-200=50\).\[ \text{Increase Percent}=\frac{50}{200}\times100=25 \] -
400
\[ 500\left(1-\frac{20}{100}\right)=400 \] -
\(\frac{1}{4}\)
\[ 25\%=\frac{25}{100}=\frac{1}{4} \]
Higher-Order Problems
-
Marks \(=420\), total \(=600\):
\[ \text{Percentage}=\frac{420}{600}\times100=70 \]Answer = 70%.
-
Salary increases from ₹40,000 to ₹46,000:
\[ \text{Increase}=46000-40000=6000 \]\[ \text{Increase Percent}=\frac{6000}{40000}\times100=15 \]Answer = 15%.
-
Population \(=80000\), decrease \(=12.5\%=\frac{1}{8}\):
\[ \text{Decrease}=\frac{1}{8}\times80000=10000 \]\[ \text{New Population}=80000-10000=70000 \]Answer = 70,000.
-
Original value \(=400\), increase \(25\%\), then decrease \(20\%\):
\[ 400\times\frac{125}{100}\times\frac{80}{100} \]\[ =400 \]Answer = 400.
-
Let the number be \(x\).
\[ \frac{35}{100}x=140 \]\[ x=140\times\frac{100}{35}=400 \]Answer = 400.
Concept Matching
- Percentage → Value per hundred
- 50% → One-half
- 25% → One-fourth
- Percentage Increase → Increase compared with original value
- Percentage Decrease → Decrease compared with original value
- Successive Change → Percentage changes applied one after another
Clue Explanation
Percentage calculations depend on the correct base value. For change-based problems, always compare the change with the original value.
Exam tips
- Percentage means value out of 100.
- Use original value as base for increase or decrease.
- Memorize common fractions like 50%, 25%, 20%, and 12.5%.
- For increase, multiply by \(\frac{100+x}{100}\).
- For decrease, multiply by \(\frac{100-x}{100}\).
- Apply successive changes step by step.