👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Number System and Arithmetic
Study mode

Percentage

Study Practice Revise 🔒 Test

Introduction to Percentage

Have you ever wondered how shops show discounts like 20% off or how your school reports marks as percentages? The word percentage means "per hundred." It is a way to express any number as a part of 100. For example, if you scored 45 marks out of 50 in a test, your score can be expressed as a percentage to understand it better.

Percentages are everywhere-in money, measurements, statistics, and even in daily conversations. Understanding percentages helps you compare quantities easily and solve many real-life problems, especially in competitive exams.

Since percentages are closely related to fractions and decimals, we will start by exploring these connections to build a strong foundation.

Definition and Conversion

What is Percentage?

A percentage is a number expressed as a fraction of 100. The symbol for percentage is %. For example, 25% means 25 out of 100, or \(\frac{25}{100}\).

So,

Percentage = (Part / Whole) x 100

This means if you have a part of something and want to know how much it is out of 100, you convert it into a percentage.

Relation with Fractions and Decimals

Since percentages are parts of 100, they can be easily converted to fractions and decimals.

Fraction Decimal Percentage
\(\frac{1}{2}\) 0.5 50%
\(\frac{1}{4}\) 0.25 25%
\(\frac{3}{4}\) 0.75 75%
\(\frac{1}{5}\) 0.2 20%
\(\frac{7}{10}\) 0.7 70%

To convert from fraction to percentage, multiply the fraction by 100.

To convert from decimal to percentage, multiply the decimal by 100.

To convert from percentage to decimal, divide by 100.

To convert from percentage to fraction, write the percentage over 100 and simplify.

Calculating Percentage

Now that we understand what percentage means, let's learn how to calculate it in different situations.

graph TD    A[Start] --> B[Identify the percentage value (p%) and the number (N)]    B --> C[Convert percentage to decimal: p% = p/100]    C --> D[Calculate percentage of number: (p/100) x N]    D --> E{Is it an increase or decrease?}    E -->|Increase| F[New value = N x (1 + p/100)]    E -->|Decrease| G[New value = N x (1 - p/100)]    F --> H[End]    G --> H[End]

This flowchart shows the steps to calculate:

  • Percentage of a number
  • Percentage increase
  • Percentage decrease

Finding Percentage of a Number

To find \(p\%\) of a number \(N\), use the formula:

\[ \text{Percentage of a number} = \frac{p}{100} \times N \]

For example, to find 20% of 250, calculate \(\frac{20}{100} \times 250 = 50\).

Percentage Increase and Decrease

When a quantity increases by \(p\%\), the new value is:

\[ \text{New Value} = N \times \left(1 + \frac{p}{100}\right) \]

When a quantity decreases by \(p\%\), the new value is:

\[ \text{New Value} = N \times \left(1 - \frac{p}{100}\right) \]

These formulas help you quickly find the new amount after a percentage change.

Worked Examples

Example 1: Finding Percentage of a Number Easy
Calculate 15% of 200 INR.

Step 1: Identify the percentage and the number.

Percentage \(p = 15\%\), Number \(N = 200\) INR.

Step 2: Use the formula:

\(\frac{p}{100} \times N = \frac{15}{100} \times 200\)

Step 3: Calculate:

\(\frac{15}{100} \times 200 = 0.15 \times 200 = 30\)

Answer: 15% of 200 INR is 30 INR.

Example 2: Percentage Increase Medium
Find the new price of an item originally priced at 500 INR after a 12% increase.

Step 1: Identify original price and percentage increase.

Original price \(N = 500\) INR, Increase \(p = 12\%\).

Step 2: Use the formula for percentage increase:

\[ \text{New Price} = N \times \left(1 + \frac{p}{100}\right) = 500 \times \left(1 + \frac{12}{100}\right) \]

Step 3: Calculate:

\[ 500 \times 1.12 = 560 \]

Answer: The new price after 12% increase is 560 INR.

Example 3: Successive Percentage Changes Hard
Calculate the final price after two successive discounts of 10% and 20% on a product priced at 1000 INR.

Step 1: Identify original price and discounts.

Original price \(N = 1000\) INR, First discount \(p = 10\%\), Second discount \(q = 20\%\).

Step 2: Calculate price after first discount:

\[ \text{Price after 1st discount} = 1000 \times \left(1 - \frac{10}{100}\right) = 1000 \times 0.9 = 900 \]

Step 3: Calculate price after second discount on new price:

\[ 900 \times \left(1 - \frac{20}{100}\right) = 900 \times 0.8 = 720 \]

Step 4: Final price after both discounts is 720 INR.

Alternative Step: Use successive percentage change formula:

\[ \text{Net change factor} = \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{20}{100}\right) = 0.9 \times 0.8 = 0.72 \]

\[ \text{Final Price} = 1000 \times 0.72 = 720 \]

Answer: The final price after two successive discounts is 720 INR.

Example 4: Profit and Loss Using Percentage Medium
Calculate profit percentage when cost price is 800 INR and selling price is 920 INR.

Step 1: Identify cost price (CP) and selling price (SP).

CP = 800 INR, SP = 920 INR.

Step 2: Calculate profit:

\[ \text{Profit} = SP - CP = 920 - 800 = 120 \]

Step 3: Calculate profit percentage:

\[ \text{Profit \%} = \frac{\text{Profit}}{CP} \times 100 = \frac{120}{800} \times 100 = 15\% \]

Answer: Profit percentage is 15%.

Example 5: Converting Between Percentage and Fraction Easy
Convert 0.375 to percentage and fraction forms.

Step 1: Convert decimal to percentage:

\[ 0.375 \times 100 = 37.5\% \]

Step 2: Convert decimal to fraction:

0.375 = \(\frac{375}{1000}\)

Simplify the fraction by dividing numerator and denominator by 125:

\[ \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \]

Answer: 0.375 = 37.5% = \(\frac{3}{8}\)

Formula Bank

Formula Bank

Percentage of a Number
\[ \text{Percentage of a number} = \frac{\text{Percentage}}{100} \times \text{Number} \]
where: Percentage = given percent value, Number = the base number
Percentage Increase
\[ \text{New Value} = \text{Original Value} \times \left(1 + \frac{\text{Percentage Increase}}{100}\right) \]
where: Original Value = initial amount, Percentage Increase = percent increase
Percentage Decrease
\[ \text{New Value} = \text{Original Value} \times \left(1 - \frac{\text{Percentage Decrease}}{100}\right) \]
where: Original Value = initial amount, Percentage Decrease = percent decrease
Profit Percentage
\[ \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 \]
where: Profit = Selling Price - Cost Price, Cost Price = original price
Loss Percentage
\[ \text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100 \]
where: Loss = Cost Price - Selling Price, Cost Price = original price
Successive Percentage Changes
\[ \text{Net Change} = \left(1 + \frac{p}{100}\right) \times \left(1 + \frac{q}{100}\right) - 1 \]
where: p, q = percentage changes (use negative for decrease)

Tips & Tricks

Tip: Remember that "percent" means "per hundred", so always divide by 100 when converting to decimal.

When to use: Whenever converting percentage to decimal or fraction.

Tip: For successive percentage changes, multiply the factors \(\left(1 + \frac{p}{100}\right)\) and \(\left(1 + \frac{q}{100}\right)\) instead of adding percentages directly.

When to use: When dealing with multiple percentage increases or decreases.

Tip: Use the shortcut: 10% of a number is just dividing by 10, 1% is dividing by 100, to quickly estimate percentages.

When to use: Quick mental calculations during exams.

Tip: In profit and loss, always remember: Profit = Selling Price - Cost Price, Loss = Cost Price - Selling Price.

When to use: Solving profit and loss percentage problems.

Tip: Convert all percentages to decimals before performing multiplication or division to avoid confusion.

When to use: Complex calculations involving percentages.

Common Mistakes to Avoid

❌ Adding percentage increases directly instead of multiplying for successive changes.
✓ Use the formula for successive percentage changes: multiply \(\left(1 + \frac{p}{100}\right)\) and \(\left(1 + \frac{q}{100}\right)\).
Why: Students assume percentages add linearly, ignoring compound effect.
❌ Confusing percentage increase with percentage of a number.
✓ Clarify that percentage increase is applied on the original value to find new value, not just a percentage part.
Why: Misunderstanding of what percentage increase means.
❌ Using selling price instead of cost price in profit/loss percentage formula.
✓ Always use cost price as denominator in profit/loss percentage calculations.
Why: Misremembering formula leads to incorrect results.
❌ Not converting percentage to decimal before multiplying in calculations.
✓ Convert percentage to decimal by dividing by 100 before multiplication.
Why: Leads to answers off by factor of 100.
❌ Forgetting to subtract discount percentage from 100% when calculating final price after discount.
✓ Calculate final price as Original Price x \(\left(1 - \frac{\text{Discount \%}}{100}\right)\).
Why: Students sometimes multiply directly by discount percentage instead of remaining percentage.
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Percentage · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.