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Percentage

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Introduction to Percentage

Have you ever seen a sign that says "20% off" during a sale? Or wondered what it means when your exam score is 75%? The word percentage is everywhere in daily life, from shopping discounts to bank interest rates and exam results. But what exactly is a percentage?

Percentage means "per hundred." It is a way to express a part of a whole as a fraction of 100. For example, 45% means 45 parts out of 100 parts. This makes percentages a very useful tool to compare quantities easily, especially when the total amounts are different.

Understanding percentages is essential not only for everyday calculations but also for competitive exams, where questions on percentages are very common. In this chapter, we will learn what percentage means, how it relates to fractions and decimals, and how to solve various problems involving percentages.

Since India uses the metric system and the Indian Rupee (INR) as currency, our examples will often involve measurements in kilograms, litres, or money in INR to make the concepts relatable and practical.

Understanding Percentage

A percentage is a ratio expressed as a fraction of 100. It tells us how many parts out of 100 parts we have.

For example, if you scored 45 marks out of 100 in a test, your score can be written as 45%. But what if the test was out of 80 marks? We can still find the percentage by converting your score to a fraction of 100.

Let's see how percentage relates to fractions and decimals:

Conversion between Fraction, Decimal, and Percentage
Percentage Fraction Decimal
25% \(\frac{25}{100} = \frac{1}{4}\) 0.25
50% \(\frac{50}{100} = \frac{1}{2}\) 0.5
75% \(\frac{75}{100} = \frac{3}{4}\) 0.75
12.5% \(\frac{12.5}{100} = \frac{1}{8}\) 0.125
45% \(\frac{45}{100} = \frac{9}{20}\) 0.45

How to convert:

  • Percentage to fraction: Divide the percentage by 100 and simplify the fraction if possible.
  • Percentage to decimal: Divide the percentage by 100.
  • Fraction to percentage: Convert the fraction to a decimal by division, then multiply by 100.
  • Decimal to percentage: Multiply the decimal by 100.

For example, to convert 45% to fraction and decimal:

  • Fraction: \(\frac{45}{100} = \frac{9}{20}\)
  • Decimal: \(45 \div 100 = 0.45\)
Key Concept

Percentage

A way to express a number as parts per hundred.

Calculating Percentage of a Number

One of the most common tasks is to find what a certain percentage of a number is. For example, what is 15% of 200 INR?

The method involves three simple steps:

graph TD    A[Identify the percentage] --> B[Convert percentage to decimal]    B --> C[Multiply decimal by the number]    C --> D[Get the result]

Why convert to decimal? Because multiplying by a decimal is straightforward and avoids confusion. Remember, 15% means 15 per 100, which is 0.15 in decimal.

1
->

Step 1

Convert percentage to decimal by dividing by 100

2
->

Step 2

Multiply the decimal by the given number

3

Step 3

The result is the required percentage of the number

Formula Bank

Formula Bank

Percentage to Fraction
\[\text{Fraction} = \frac{\text{Percentage}}{100}\]
where: Percentage = given percentage value
Percentage to Decimal
\[\text{Decimal} = \frac{\text{Percentage}}{100}\]
where: Percentage = given percentage value
Finding Percentage of a Number
\[\text{Percentage of Number} = \frac{\text{Percentage}}{100} \times \text{Number}\]
where: Percentage = given percentage, Number = given number
Finding Whole from Percentage
\[\text{Whole} = \frac{\text{Part} \times 100}{\text{Percentage}}\]
where: Part = given part value, Percentage = given percentage
Percentage Increase/Decrease
\[\text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100\]
where: Change = difference between new and original value, Original Value = initial value
Profit Percentage
\[\text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\]
where: Profit = Selling Price - Cost Price, Cost Price = original price
Discount Percentage
\[\text{Discount \%} = \frac{\text{Discount}}{\text{Marked Price}} \times 100\]
where: Discount = Marked Price - Selling Price, Marked Price = original price
Example 1: Finding 15% of 200 INR Easy
Find 15% of 200 INR.

Step 1: Convert 15% to decimal by dividing by 100.

\(15\% = \frac{15}{100} = 0.15\)

Step 2: Multiply the decimal by the number.

\(0.15 \times 200 = 30\)

Answer: 15% of 200 INR is 30 INR.

Example 2: Finding the Whole when 30% is 90 INR Medium
If 30% of an amount is 90 INR, find the total amount.

Step 1: Let the total amount be \(x\).

Step 2: According to the problem, 30% of \(x\) is 90.

\(\frac{30}{100} \times x = 90\)

Step 3: Solve for \(x\).

\(x = \frac{90 \times 100}{30} = \frac{9000}{30} = 300\)

Answer: The total amount is 300 INR.

Example 3: Calculating Percentage Increase in Price Medium
The price of a product increases from 400 INR to 460 INR. Find the percentage increase.

Step 1: Find the increase in price.

Increase = \(460 - 400 = 60\) INR

Step 2: Use the percentage increase formula:

\(\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Price}} \times 100\)

\(= \frac{60}{400} \times 100 = 15\%\)

Answer: The price increased by 15%.

Example 4: Discount Calculation on a Product Priced at 1500 INR with 12% Discount Medium
A product is priced at 1500 INR. If a 12% discount is offered, find the discount amount and the final price.

Step 1: Calculate the discount amount.

Discount = \(\frac{12}{100} \times 1500 = 180\) INR

Step 2: Calculate the final price after discount.

Final Price = \(1500 - 180 = 1320\) INR

Answer: Discount amount is 180 INR and final price is 1320 INR.

Example 5: Profit Percentage when Cost Price is 1200 INR and Selling Price is 1380 INR Medium
A shopkeeper buys an article for 1200 INR and sells it for 1380 INR. Find the profit percentage.

Step 1: Calculate the profit.

Profit = Selling Price - Cost Price = \(1380 - 1200 = 180\) INR

Step 2: Calculate profit percentage using the formula:

\(\text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 = \frac{180}{1200} \times 100 = 15\%\)

Answer: The profit percentage is 15%.

Finding Percentage of a Number

\[\text{Percentage of Number} = \frac{\text{Percentage}}{100} \times \text{Number}\]

Multiply the decimal equivalent of percentage with the number to find the part.

Tips & Tricks

Tip: Convert percentage to decimal by dividing by 100 to simplify calculations.

When to use: When performing multiplication or division involving percentages.

Tip: Use the proportion method (cross multiplication) to find unknown values quickly.

When to use: When given part and percentage or whole and percentage.

Tip: Remember that 1% of a number is the number divided by 100, useful for quick mental calculations.

When to use: When estimating percentage values mentally.

Tip: For percentage increase/decrease, always subtract original value from new value to find change.

When to use: When calculating percentage change.

Tip: In discount problems, calculate discount amount first, then subtract from marked price to find final price.

When to use: When solving retail price discount problems.

Common Mistakes to Avoid

❌ Confusing percentage with absolute value (e.g., treating 15% as 15 instead of 0.15)
✓ Always convert percentage to decimal or fraction before calculations.
Why: Because percentage represents a part per hundred, not the raw number.
❌ Calculating percentage increase/decrease using wrong base (using new value instead of original value)
✓ Always use original value as denominator in percentage change formula.
Why: Percentage change is relative to original value, not the new value.
❌ Forgetting to convert percentage to decimal in multiplication problems.
✓ Divide percentage by 100 before multiplying.
Why: Multiplying by percentage directly inflates the value incorrectly.
❌ Mixing up cost price and selling price in profit/loss calculations.
✓ Identify cost price and selling price clearly before calculation.
Why: Profit or loss depends on difference between selling and cost price.
❌ Ignoring units (like INR) leading to confusion in word problems.
✓ Always include units in calculations and final answers.
Why: Units provide context and prevent misinterpretation.
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