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Have you ever wondered how to easily compare parts of a whole? For example, if you scored 45 marks out of 50 in an exam, how do you express this in a simple way to understand your performance? This is where percentage comes in. The word "percentage" literally means "per hundred", and it is a way to express any quantity as a fraction of 100.
Percentage is part of our everyday life - when we talk about discounts during shopping, bank interest rates, or population statistics, percentages help us understand proportions quickly and clearly.
In simple terms, percentage expresses how many parts out of 100 a number represents. For example, saying 25% (read as 25 percent) means 25 parts out of 100.
One of the reasons percentage is so useful is its close relationship with fractions and decimals. Because it is always relative to 100, it acts like a bridge between these forms of numbers, making calculations and comparisons easier.
Let us now clearly define percentage and how it connects to fractions and decimals.
A percentage is defined as:
This means, if we know a fraction, say \(\frac{3}{4}\), we can convert it to percentage by multiplying it with 100:
\[\frac{3}{4} \times 100 = 75\\%\]
Similarly, we can convert between percentages and decimals easily:
| 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 |
| 10% | \(\frac{10}{100} = \frac{1}{10}\) | 0.1 |
| 5% | \(\frac{5}{100} = \frac{1}{20}\) | 0.05 |
Understanding these conversions helps in simplifying calculations and interpreting data given in different formats.
One of the most common tasks you will encounter is calculating what a certain percentage of a given quantity is. For example, what is 20% of 500 INR?
The key idea is that "x% of a quantity" means \(x\) parts out of every 100 of that quantity.
The calculation process is straightforward and involves these steps:
graph TD A[Start: Given x% and Quantity Q] --> B[Convert x% to fraction: \(\frac{x}{100}\)] B --> C[Multiply \(\frac{x}{100} \times Q\)] C --> D[Result: \(x\\%\) of \(Q\)]Expressing this as a formula:
Step 1: Write 15% as a fraction: \(\frac{15}{100} = 0.15\)
Step 2: Multiply by 200 meters:
\(0.15 \times 200 = 30\)
Answer: 15% of 200 meters is 30 meters.
In many situations, values change over time, either increasing or decreasing. Expressing these changes using percentages helps us understand the magnitude of change clearly.
Let's define these terms:
The key formulas for these are:
| Type of Change | Formula | Description |
|---|---|---|
| Percentage Increase | \[ \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100 \] | Increase = New Value - Original Value |
| Percentage Decrease | \[ \text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100 \] | Decrease = Original Value - New Value |
By using these, we can understand how much change occurred relative to the initial amount.
Step 1: Calculate the increase amount:
\(20\\% \text{ of } 1500 = \frac{20}{100} \times 1500 = 300\)
Step 2: Add the increase to the original price:
\(1500 + 300 = 1800\)
Answer: The new price after 20% increase is INR 1800.
Step 1: Let the original price be \(x\).
Step 2: After 10% discount, the selling price is 90% of original price:
\(0.9 \times x = 900\)
Step 3: Calculate \(x\):
\[ x = \frac{900}{0.9} = 1000 \]
Answer: The original price is INR 1000.
Sometimes, a quantity undergoes more than one percentage change, one after another. For instance, a salary may be increased by 10% one year, and then further increased by 20% the next year.
It is important to note that successive percentage changes do not simply add up. Instead, each change applies to the value after the previous change.
To calculate the final value after successive percentage changes, use the formula:
graph TD A[Start: Initial Value V] --> B[Apply first change: \(V \times (1 \pm \frac{p_1}{100})\)] B --> C[Apply second change: Multiply by \(1 \pm \frac{p_2}{100}\)] C --> D[Continue for further changes if any] D --> E[Obtain Final Value after successive changes]Step 1: Calculate length after 10% increase:
\(500 \times \\left(1 + \frac{10}{100}\\right) = 500 \times 1.1 = 550\) meters
Step 2: Calculate length after 20% decrease on 550 meters:
\(550 \times \\left(1 - \frac{20}{100}\\right) = 550 \times 0.8 = 440\) meters
Answer: The final length after successive changes is 440 meters.
Step 1: Calculate the increase amount:
\(1500 - 1200 = 300\)
Step 2: Use formula for percentage increase:
\[ \frac{300}{1200} \times 100 = 25\\% \]
Answer: The percentage increase is 25%.
Step 1: Calculate profit amount:
\(10\% \text{ of } 800 = \frac{10}{100} \times 800 = 80\)
Step 2: Selling price is cost price plus profit:
\(800 + 80 = 880\)
Answer: Selling price = INR 880.
Step 1: Let original price be \(x\).
Step 2: After 20% discount, selling price is 80% of original price:
\(0.8 \times x = 960\)
Step 3: Solve for \(x\):
\[ x = \frac{960}{0.8} = 1200 \]
Answer: Original price is INR 1200.
Step 1: Calculate length after first increase:
\(500 \times 1.10 = 550\) meters
Step 2: Calculate length after second increase:
\(550 \times 1.20 = 660\) meters
Answer: Final length after successive increases is 660 meters.
When to use: When calculating multiple percentage increases or decreases in sequence.
When to use: To rapidly convert percentages to decimals during calculations.
When to use: In problems requiring quick rough answers or during initial steps of problem-solving.
When to use: In discount and loss-related problems to avoid common errors.
When to use: In all word problems involving measurements and currency.
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