Percentages

What is a Percentage?

A percentage is a way to express a number as a fraction of 100. It literally means “per hundred.”

If you say 30%, it means:

$$\frac{30}{100} = 0.3$$

Examples:

• 25% of 200 = $\frac{25}{100} \times 200 = 50$
• 75% = $\frac{3}{4}$

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Visual Intuition

Percentages can be thought of as parts of a whole divided into 100 equal parts.

Imagine: A circle (like a pie chart) split into 100 equal pieces.

• 50% means half the pie is shaded
• 100% means the entire pie is shaded
• 200% means you have two pies

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Key Formulas

Converting Between Forms

Conversion Tips:

• To convert % to decimal: divide by 100
• To convert decimal to %: multiply by 100

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Types of Problems

1. Finding the Percentage of a Number

Formula:

$$\text{Percentage of a number} = \frac{\text{Given \%} \times \text{Total}}{100}$$

Example:
What is 40% of 250?

$$\frac{40 \times 250}{100} = 100$$

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2. Finding What Percentage One Number is of Another

Formula:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

Example:
What percent of 50 is 20?

$$\frac{20}{50} \times 100 = 40\%$$

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3. Percentage Increase and Decrease

Increase:

$$\text{New Value} = \text{Original} \times \left(1 + \frac{\text{Increase \%}}{100}\right)$$

Decrease:

$$\text{New Value} = \text{Original} \times \left(1 - \frac{\text{Decrease \%}}{100}\right)$$

Examples:

• A salary of ₹1000 increased by 20% becomes ₹1200
• A price of ₹500 decreased by 10% becomes ₹450

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4. Successive Percentage Change

Two successive changes of $x\%$ and $y\%$ are not simply additive.

Formula:

$$\text{Net \% Change} = x + y + \frac{xy}{100}$$

Example:
If a price increases by 20% and then by 10%:

$$20 + 10 + \frac{20 \times 10}{100} = 30 + 2 = 32\%$$

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5. Reverse Percentage

Used when final value is given, and you need to find original.

Formula:

$$\text{Original} = \frac{\text{Final Value} \times 100}{100 \pm \text{\% change}}$$

Example:
After a 20% increase, price is ₹120.

$$\text{Original} = \frac{120 \times 100}{120} = \text{Rs.~}100$$

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6. Comparing Two Percentages

Example:
Which is more: 20% of 60 or 30% of 40?

• 20% of 60 = 12
• 30% of 40 = 12
  → They are equal

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Conceptual Tips

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Visual Concept Map

% of number → Value  
One number is what % of another → Ratio form  
Change in % → Compare difference  
Reverse % → Final → Original  
Successive % → Compound effect  

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Sample Practice Questions

Q1. A price is increased by 20% and then decreased by 20%. What is the net change?

Solution:

$$20 - 20 + \frac{-20 \times 20}{100} = 0 - 4 = -4\% \text{ (Net decrease)}$$

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Q2. What is 12.5% of 640?

$$12.5\% = \frac{1}{8} \Rightarrow \frac{640}{8} = 80$$

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Q3. A student scored 30% and failed by 20 marks. Passing marks are 100. What are maximum marks?

Let total = $x$

$$30\% \text{ of } x = 100 - 20 = 80 \Rightarrow \frac{30}{100} \times x = 80 \Rightarrow x = \frac{8000}{30} = 266.67 \Rightarrow 267$$