Login
OR
Don't worry. We'll fix this for you!
Have you ever wondered how shops calculate discounts during sales or how banks compute interest on your savings? The answer lies in understanding percentages. The term percentage means "per hundred," and it is a way to express any number as a part of 100. For example, if you score 80 out of 100 marks in an exam, your score is 80%, which means 80 per hundred.
Percentages are everywhere-in price changes, profit and loss, interest rates, and even in statistics like population growth. In competitive exams, questions on percentages test your ability to quickly and accurately work with these concepts. This chapter will guide you step-by-step to master percentage calculations, starting from the basics and moving towards more complex applications.
Let's begin by understanding what a percentage is and how it relates to fractions and decimals.
Percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is %.
For example, 25% means 25 out of 100, or \(\frac{25}{100}\).
Since percentages are closely related to fractions and decimals, it is important to know how to convert between these forms:
| Fraction | Decimal | Percentage |
|---|---|---|
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{3}{5}\) | 0.6 | 60% |
| \(\frac{7}{10}\) | 0.7 | 70% |
| \(\frac{9}{20}\) | 0.45 | 45% |
To convert between these forms, use the following rules:
In daily life, prices of goods often change. Sometimes they increase, and sometimes they decrease. Understanding how to calculate the percentage increase or decrease helps you analyze these changes clearly.
Percentage Increase tells us by what percent a value has grown compared to its original value.
Percentage Decrease tells us by what percent a value has reduced compared to its original value.
The formulas are:
Percentage Increase
\[ \text{Percentage Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]Percentage Decrease
\[ \text{Percentage Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \]Note the importance of the original value as the base for the calculation. This is a common point of confusion, so always remember: percentage change is always relative to the original value.
graph TD A[Start] --> B[Identify Original Value and New Value] B --> C[Calculate Difference] C --> D{Is New Value > Original?} D -->|Yes| E[Calculate Percentage Increase] D -->|No| F[Calculate Percentage Decrease] E --> G[End] F --> GStep 1: Identify the original and new values.
Original Price = INR 500, New Price = INR 600
Step 2: Calculate the increase in price.
Increase = New Price - Original Price = 600 - 500 = INR 100
Step 3: Use the percentage increase formula:
\[ \text{Percentage Increase} = \frac{100}{500} \times 100 = 20\% \]
Answer: The price increased by 20%.
Step 1: Identify the original price and selling price.
Original Price = INR 1200, Selling Price = INR 900
Step 2: Calculate the discount amount.
Discount = Original Price - Selling Price = 1200 - 900 = INR 300
Step 3: Calculate discount percentage:
\[ \text{Discount \%} = \frac{300}{1200} \times 100 = 25\% \]
Answer: The discount percentage is 25%.
Step 1: Identify cost price (CP) and selling price (SP).
CP = INR 1500, SP = INR 1800
Step 2: Calculate profit.
Profit = SP - CP = 1800 - 1500 = INR 300
Step 3: Calculate profit percentage:
\[ \text{Profit \%} = \frac{300}{1500} \times 100 = 20\% \]
Answer: The profit percentage is 20%.
Step 1: Identify principal (P), rate (R), and time (T).
P = INR 10,000, R = 5%, T = 3 years
Step 2: Use the simple interest formula:
\[ SI = \frac{P \times R \times T}{100} = \frac{10,000 \times 5 \times 3}{100} = 1500 \]
Answer: The simple interest is INR 1500.
Step 1: Understand that the selling price is 80% of the original price (because 100% - 20% = 80%).
Step 2: Let the original price be \(x\).
Then, \(80\%\) of \(x = 800\), or
\[ \frac{80}{100} \times x = 800 \]
Step 3: Solve for \(x\):
\[ x = \frac{800 \times 100}{80} = 1000 \]
Answer: The original price was INR 1000.
When to use: Whenever converting percentages to decimals for calculations.
When to use: When calculating percentage change between two values.
When to use: When dealing with discount problems to save time.
When to use: In reverse percentage problems where original value is unknown.
When to use: When applying percentage rates to quantities, like interest or population growth.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|