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Percentages and percentage calculations in real life

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Learning objective
Calculate percentages and apply percentage concepts to real-life problems.

Introduction to Percentages

Have you ever heard someone say, "You got 75% on the test," or seen a sign that says "50% off"? These are examples of percentages in daily life. But what exactly is a percentage?

A percentage is a way to express a number as a part of 100. The word "percent" literally means "per hundred." So, when we say 50%, it means 50 out of 100 parts.

Percentages help us compare quantities easily, no matter the size of the whole. Whether it's calculating discounts while shopping, understanding population growth, or figuring out profits, percentages are everywhere!

To understand percentages fully, we will connect them to fractions and decimals, which you might already know. This connection will make it easier to work with percentages in various problems.

Understanding Percentage

A percentage is a fraction with denominator 100. For example, 25% means 25 parts out of 100 parts, or \(\frac{25}{100}\).

We can express percentages as decimals or fractions, and convert between these forms easily.

Conversion between Fraction, Decimal, and Percentage
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%
\(\frac{2}{5}\) 0.4 40%

How to convert:

  • From fraction to percentage: Multiply the fraction by 100. For example, \(\frac{1}{2} \times 100 = 50\%\).
  • From decimal to percentage: Multiply the decimal by 100. For example, \(0.75 \times 100 = 75\%\).
  • From percentage to decimal: Divide the percentage by 100. For example, \(25\% = \frac{25}{100} = 0.25\).
  • From percentage to fraction: Write the percentage over 100 and simplify. For example, \(40\% = \frac{40}{100} = \frac{2}{5}\).
Key Concept

Percentage

A way to express a number as parts per hundred.

Calculating Percentage of a Number

Often, you need to find what a certain percentage of a number is. For example, what is 20% of 500 INR?

The formula to find the percentage of a number is:

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

This means you convert the percentage to a decimal by dividing by 100, then multiply by the number.

100 units 20% (76 units)

In this diagram, the full bar represents 100 units. The blue section shows 20% of the total, which is 20 out of 100 units.

Percentage Increase and Decrease

Percentages are also used to describe how much a quantity has increased or decreased compared to its original value. This is very common in price changes, population growth, or depreciation of items.

Percentage Increase tells us how much a value has grown relative to its original amount.

Percentage Decrease tells us how much a value has reduced relative to its original amount.

graph TD    A[Start with Original Value] --> B[Find Difference: New Value - Original Value]    B --> C{Is Difference Positive?}    C -->|Yes| D[Calculate Percentage Increase]    C -->|No| E[Calculate Percentage Decrease]    D --> F[Divide Difference by Original Value]    E --> F    F --> G[Multiply by 100]    G --> H[Interpret Result as % Increase or Decrease]

Formulas:

\[ \text{Percentage Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100 \]
\[ \text{Percentage Decrease} = \left( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \right) \times 100 \]

Remember, the original value is always the starting point or the value before the change.

Worked Examples

Example 1: Calculating 20% of 500 INR Easy
Find 20% of 500 INR.

Step 1: Convert 20% to decimal by dividing by 100: \( \frac{20}{100} = 0.20 \).

Step 2: Multiply the decimal by 500: \( 0.20 \times 500 = 100 \).

Answer: 20% of 500 INR is 100 INR.

Example 2: Finding the New Price After 15% Discount on 1200 INR Medium
A product costs 1200 INR. It is sold at a 15% discount. What is the selling price after the discount?

Step 1: Calculate the discount amount: \( \frac{15}{100} \times 1200 = 180 \) INR.

Step 2: Subtract the discount from the original price: \( 1200 - 180 = 1020 \) INR.

Answer: The selling price after 15% discount is 1020 INR.

Example 3: Calculating Percentage Increase in Population from 50,000 to 55,000 Medium
The population of a town increased from 50,000 to 55,000. Find the percentage increase.

Step 1: Find the increase: \( 55,000 - 50,000 = 5,000 \).

Step 2: Use the percentage increase formula:

\[ \text{Percentage Increase} = \left( \frac{5,000}{50,000} \right) \times 100 = 10\% \]

Answer: The population increased by 10%.

Example 4: Profit Percentage When Cost Price is 800 INR and Selling Price is 920 INR Medium
A shopkeeper buys an item for 800 INR and sells it for 920 INR. Find the profit percentage.

Step 1: Calculate profit: \( 920 - 800 = 120 \) INR.

Step 2: Use the profit percentage formula:

\[ \text{Profit \%} = \left( \frac{120}{800} \right) \times 100 = 15\% \]

Answer: The profit percentage is 15%.

Example 5: Finding Original Price Before 10% Increase Resulting in 1100 INR Hard
After a 10% increase, the price of an item is 1100 INR. What was the original price?

Step 1: Let the original price be \( x \) INR.

Step 2: After 10% increase, new price = \( x + \frac{10}{100} \times x = 1.10x \).

Step 3: Given new price is 1100, so \( 1.10x = 1100 \).

Step 4: Solve for \( x \): \( x = \frac{1100}{1.10} = 1000 \) INR.

Answer: The original price was 1000 INR.

Formula Bank

Percentage Formula
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]
where: Part = portion of the whole, Whole = total amount
Finding Percentage of a Number
\[ \text{Percentage of a number} = \frac{\text{Percentage}}{100} \times \text{Number} \]
where: Percentage = given percent value, Number = the quantity to find percentage of
Percentage Increase
\[ \text{Percentage Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100 \]
where: New Value = increased amount, Original Value = initial amount
Percentage Decrease
\[ \text{Percentage Decrease} = \left( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \right) \times 100 \]
where: Original Value = initial amount, New Value = decreased amount
Profit Percentage
\[ \text{Profit \%} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100 \]
where: Profit = Selling Price - Cost Price, Cost Price = original price
Loss Percentage
\[ \text{Loss \%} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100 \]
where: Loss = Cost Price - Selling Price, Cost Price = original price

Tips & Tricks

Tip: Convert percentages to decimals by dividing by 100 before calculations.

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

Tip: For quick percentage increase or decrease, use: New Value = Original Value x (1 ± Percentage/100).

When to use: When calculating final amounts after increase or decrease.

Tip: Use 10% as a base to find other percentages quickly (e.g., 5% is half of 10%).

When to use: For mental math and quick estimations.

Tip: Always check if your percentage result makes sense by comparing it to the original and new values.

When to use: After solving percentage problems to avoid calculation errors.

Tip: In profit and loss problems, clearly identify cost price and selling price before calculations.

When to use: To avoid confusion and mistakes in commercial math problems.

Common Mistakes to Avoid

❌ Confusing percentage increase with percentage decrease formulas.
✓ Use the correct formula based on whether the new value is greater or smaller than the original.
Why: Students often forget to check if the value increased or decreased, leading to wrong sign usage.
❌ Calculating percentage of a number by dividing instead of multiplying.
✓ Multiply the number by the percentage expressed as a decimal (percentage/100).
Why: Misunderstanding the operation needed to find part of a whole.
❌ Using the new value as the denominator instead of the original value in percentage change calculations.
✓ Always divide the difference by the original value, not the new value.
Why: Incorrect denominator leads to wrong percentage calculation.
❌ Not converting percentages to decimals before calculations.
✓ Always convert percentage to decimal form by dividing by 100 before multiplication or division.
Why: Leads to incorrect magnitude of results.
❌ Mixing up profit and loss formulas.
✓ Identify if selling price is greater or less than cost price to apply correct formula.
Why: Confusion between profit and loss scenarios causes wrong answers.
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