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Percentage calculations

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Introduction to Percentage Calculations

Have you ever wondered how a shopkeeper calculates the discount on your favourite shirt or how banks calculate interest on your savings? The answer lies in understanding percentages. A percentage is a way to express a number as a part of 100. The word "percent" literally means "per hundred." This concept helps us compare quantities easily, understand changes in values, and solve many real-life problems.

For example, if you score 80 marks out of 100 in an exam, your score is 80%. Similarly, if a product is sold at a 20% discount, it means the price is reduced by 20 out of every 100 rupees.

Percentages are closely related to fractions and decimals. Understanding these connections will make percentage calculations easier and faster, especially in competitive exams where time is limited.

Definition and Conversion

A percentage represents a fraction with denominator 100. For example, 25% means 25 parts out of 100 parts.

Mathematically,

Percentage Formula

\[\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\]

To find what percent a part is of the whole

Part = portion of the quantity
Whole = total quantity

Percentages can be converted to decimals and fractions, and vice versa. This flexibility helps solve problems in different formats.

Conversion between Fractions, Decimals, and Percentages
Fraction Decimal Percentage
1/2 0.5 50%
1/4 0.25 25%
3/4 0.75 75%
1/5 0.2 20%
2/5 0.4 40%
7/10 0.7 70%

How to convert:

  • Fraction to Percentage: Divide numerator by denominator, multiply by 100.
  • Decimal to Percentage: Multiply decimal by 100.
  • Percentage to Decimal: Divide by 100.
  • Percentage to Fraction: Write percentage over 100 and simplify.

Percentage Increase and Decrease

Often, quantities change over time or due to some event. We use percentage increase and percentage decrease to measure how much a value has grown or shrunk relative to its original amount.

Percentage Increase

\[\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100\]

To calculate percentage increase from original value

Increase = New Value - Original Value

Percentage Decrease

\[\text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100\]

To calculate percentage decrease from original value

Decrease = Original Value - New Value 
graph TD    A[Start with Original Value] --> B[Find New Value]    B --> C{Is New Value > Original?}    C -- Yes --> D[Calculate Increase = New - Original]    C -- No --> E[Calculate Decrease = Original - New]    D --> F[Percentage Increase = (Increase / Original) x 100]    E --> G[Percentage Decrease = (Decrease / Original) x 100]    F --> H[Result]    G --> H[Result]

Worked Example 1: Calculating Percentage Increase

Example 1: Percentage Increase in Price Easy
The price of a pen increases from INR 500 to INR 600. Calculate the percentage increase.

Step 1: Identify the original and new values.

Original Price = INR 500

New Price = INR 600

Step 2: Calculate the increase.

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%.

Worked Example 2: Finding Discount Percentage

Example 2: Discount Percentage Calculation Medium
A shirt marked at INR 1200 is sold for INR 900. Find the discount percentage.

Step 1: Identify the marked price and selling price.

Marked Price (Original Value) = INR 1200

Selling Price (New Value) = INR 900

Step 2: Calculate the discount amount.

Discount = Marked Price - Selling Price = 1200 - 900 = INR 300

Step 3: Use the percentage decrease formula (since price decreased).

\[ \text{Discount Percentage} = \frac{300}{1200} \times 100 = 25\% \]

Answer: The discount given is 25%.

Worked Example 3: Profit Percentage Calculation

Example 3: Profit Percentage Medium
A shopkeeper buys a watch for INR 1500 and sells it for INR 1800. Calculate the profit percentage.

Step 1: Identify cost price and selling price.

Cost Price (CP) = INR 1500

Selling Price (SP) = INR 1800

Step 2: Calculate profit.

Profit = SP - CP = 1800 - 1500 = INR 300

Step 3: Use the profit percentage formula.

\[ \text{Profit \%} = \frac{300}{1500} \times 100 = 20\% \]

Answer: The profit percentage is 20%.

Worked Example 4: Percentage Problem Involving Population Growth

Example 4: Population Growth Percentage Hard
The population of a town increases from 1,20,000 to 1,26,000 in one year. Find the percentage increase.

Step 1: Identify original and new populations.

Original Population = 1,20,000

New Population = 1,26,000

Step 2: Calculate the increase.

Increase = 1,26,000 - 1,20,000 = 6,000

Step 3: Calculate percentage increase.

\[ \text{Percentage Increase} = \frac{6000}{120000} \times 100 = 5\% \]

Answer: The population increased by 5% in one year.

Worked Example 5: Complex Percentage Problem with Successive Discounts

Example 5: Successive Discounts Hard
An item priced at INR 2000 is sold with two successive discounts of 10% and 5%. Find the final price.

Step 1: Identify original price and discounts.

Original Price (P) = INR 2000

First Discount (d₁) = 10%

Second Discount (d₂) = 5%

Step 2: Use the successive discount formula:

Successive Discount Formula

\[P \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right)\]

Calculate price after two successive discounts

P = Original Price
\(d_1\) = First discount %
\(d_2\) = Second discount %

Step 3: Calculate the net price.

\[ \text{Net Price} = 2000 \times \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{5}{100}\right) = 2000 \times 0.9 \times 0.95 = 2000 \times 0.855 = 1710 \]

Answer: The final price after successive discounts is INR 1710.

Formula Bank

Percentage
\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]
where: Part = portion of the quantity, Whole = total quantity
Percentage Increase
\[ \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100 \]
where: Increase = New Value - Original Value
Percentage Decrease
\[ \text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100 \]
where: Decrease = Original Value - New Value
Profit Percentage
\[ \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 \]
where: Profit = Selling Price - Cost Price
Loss Percentage
\[ \text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100 \]
where: Loss = Cost Price - Selling Price
Successive Discount
\[ \text{Net Price} = P \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \]
where: P = Original Price, d₁ = First discount %, d₂ = Second discount %

Tips & Tricks

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

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

Tip: Use the formula Percentage = (Part/Whole) x 100 as a universal approach.

When to use: For any percentage calculation problem.

Tip: For successive discounts, multiply the complements of discount rates instead of adding percentages.

When to use: When calculating final price after multiple discounts.

Tip: Remember that percentage increase and decrease are always relative to the original value.

When to use: To avoid confusion in increase/decrease problems.

Tip: Estimate percentages by rounding numbers for quick approximation.

When to use: During time-limited exams when exact calculation is not required.

Common Mistakes to Avoid

❌ Calculating percentage increase/decrease using the new value as the base instead of the original value.
✓ Always use the original value as the denominator when calculating percentage change.
Why: Students confuse which value to consider as base, leading to incorrect results.
❌ Adding discount percentages directly instead of calculating successive discounts correctly.
✓ Calculate successive discounts by multiplying the complements, not by adding percentages.
Why: Misunderstanding of how successive discounts compound rather than add up.
❌ Confusing profit percentage with profit amount.
✓ Profit percentage is profit divided by cost price times 100, not just the profit amount.
Why: Students often forget to relate profit to cost price, leading to wrong percentage calculation.
❌ Not converting fractions or decimals properly before calculating percentage.
✓ Always convert fractions and decimals to a consistent form before percentage calculations.
Why: Mixing forms causes errors in calculations.
❌ Ignoring units (like INR or metric units) in word problems leading to confusion.
✓ Always note and maintain consistent units throughout the problem.
Why: Unit inconsistency can cause misinterpretation of the problem.
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