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Percentage Calculations

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Introduction to Percentage

Percentage is a way of expressing a number as a part of 100. The word "percent" literally means "per hundred." When we say 25%, it means 25 parts out of 100.

This concept is used widely in daily life: calculating discounts at a store, understanding exam scores, measuring statistics, and more. Understanding percentage helps us compare quantities easily and intuitively.

Think of a pie chart divided into 100 equal slices. Each slice represents 1%. If you eat 25 slices, you've eaten 25% of the pie.

25% Total = 100%

Just like fractions and decimals, percentages provide a way to express proportions, but always with respect to 100. This makes it easier to understand parts of a whole.

Percentage Definition and Conversion

To work fluently with percentages, it's essential to convert between fractions, decimals, and percentages. Remember, all three represent parts of a whole, just in different formats.

Fraction Decimal Percentage
1/2 0.5 50%
1/4 0.25 25%
3/5 0.6 60%
7/10 0.7 70%
2/5 0.4 40%

Converting Fractions to Percentage: Multiply the fraction by 100.

Example: \( \frac{1}{2} = 0.5 \times 100 = 50\% \)

Converting Decimals to Percentage: Multiply the decimal by 100.

Example: \( 0.75 \times 100 = 75\% \)

Converting Percentage to Decimal: Divide by 100.

Example: \( 85\% = \frac{85}{100} = 0.85 \)

Converting Percentage to Fraction: Write the percentage over 100 and simplify.

Example: \( 25\% = \frac{25}{100} = \frac{1}{4} \)

Calculating Percentage of a Number

To find what is a certain percentage of any number, we use a simple formula:

Finding Part from Percentage

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

Calculate the part value when given percentage and whole

Percentage = Rate in %
Whole = Total quantity or number

This means: multiply the whole number by the fraction form of the percentage.

graph TD    A[Start] --> B[Identify the Whole Number]    B --> C[Identify the Percentage Rate]    C --> D[Convert Percentage to Decimal\\n(by dividing by 100)]    D --> E[Multiply Whole Number by Decimal Equivalent]    E --> F[Get the Resulting Part]
Example 1: Calculating 15% of 2500 INR Easy
Find 15% of Rs.2500.

Step 1: Convert 15% into decimal form: \( \frac{15}{100} = 0.15 \).

Step 2: Multiply the whole amount by this decimal: \( 2500 \times 0.15 = 375 \).

Answer: 15% of Rs.2500 is Rs.375.

Percentage Increase and Decrease

Percentage change helps us understand how a quantity grows or reduces relative to its original value.

Percentage Increase occurs when the new value is more than the original. The formula is:

Percentage Increase

\[\%\text{ Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\]

Calculate how much percent the value has increased

New Value = Increased amount
Original Value = Initial amount

Percentage Decrease occurs when the new value is less than the original. The formula is:

Percentage Decrease

\[\%\text{ Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100\]

Calculate how much percent the value has decreased

Original Value = Initial amount
New Value = Decreased amount
Scenario Original Value (Rs.) New Value (Rs.) Percentage Change
Price Increase 1200 1380 \(\frac{1380-1200}{1200} \times 100 = 15\%\) Increase
Discount given 1500 1200 \(\frac{1500-1200}{1500} \times 100 = 20\%\) Decrease
Example 2: Finding the % Increase when Price goes from 1200 INR to 1380 INR Medium
A shopkeeper raises the price of an item from Rs.1200 to Rs.1380. Find the percentage increase.

Step 1: Calculate the increase: \( 1380 - 1200 = 180 \).

Step 2: Use the percentage increase formula:

\[ \% \text{ Increase} = \frac{180}{1200} \times 100 = 15\% \]

Answer: The price increased by 15%.

Example 3: Discount Calculation: 20% Discount on a 1500 INR Item Medium
A product priced at Rs.1500 is sold with a 20% discount. Find the discount amount and final price.

Step 1: Calculate discount amount:

\( \text{Discount} = \frac{20}{100} \times 1500 = 300 \) INR

Step 2: Find the sale price:

\( \text{Final Price} = 1500 - 300 = 1200 \) INR

Answer: Discount amount is Rs.300; final price after discount is Rs.1200.

Example 4: Finding Original Price Given Sale Price & Discount Percentage Hard
An item is sold for Rs.960 after giving a 20% discount. Find the original price.

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

Step 2: Since the discount is 20%, the sale price is 80% of the original price:

\( 960 = \frac{80}{100} \times x \)

Step 3: Solve for \( x \):

\[ x = \frac{960 \times 100}{80} = 1200 \]

Answer: The original price was Rs.1200.

Formula Bank

Formula Bank

Percentage Formula
\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]
where: Part = portion of interest, Whole = total quantity
Finding Part from Percentage
\[ \text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole} \]
where: Percentage = rate in %, Whole = total quantity
Percentage Increase
\[ \% \text{ Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]
where: New Value = increased amount, Original Value = initial amount
Percentage Decrease
\[ \% \text{ Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \]
where: Original Value = initial amount, New Value = decreased amount
Original Price from Discount
\[ \text{Original Price} = \frac{\text{Sale Price} \times 100}{100 - \text{Discount Percentage}} \]
where: Sale Price = price after discount, Discount Percentage = discount given in %
Example 5: Successive Percentage Changes Hard
A price increases by 10% one day and then decreases by 20% the next day. What is the net percentage change over the two days?

Step 1: Assume original price is Rs.100.

Step 2: After 10% increase:

New price = \( 100 + \frac{10}{100} \times 100 = 110 \) INR

Step 3: Next, price decreases by 20%:

New price = \( 110 - \frac{20}{100} \times 110 = 110 \times 0.8 = 88 \) INR

Step 4: Calculate net change:

Difference = \( 88 - 100 = -12 \)

Percentage change = \( \frac{-12}{100} \times 100 = -12\% \)

Answer: Net percentage change is a 12% decrease.

Tips & Tricks

Tip: Use 10% as a stepping stone for other percentages.

When to use: Calculate 5%, 15%, 25% by halving or adding multiples of 10%; e.g., 15% = 10% + 5%

Tip: Remember successive percentage changes multiply, not add.

When to use: To find overall effect of multiple increases/decreases, multiply growth factors instead of simply adding percentages.

Tip: For discount problems, convert percentage to decimal and subtract from 1 to get the multiplier.

When to use: Quickly finding final price after discount as \( \text{Price} \times (1 - \frac{\text{Discount}}{100}) \).

Tip: Reverse percentage calculations can be done by dividing by (1 - discount%) or (1 + increase%).

When to use: Finding original price when only final price and percentage change are known.

Tip: Always identify the correct whole in word problems before calculating.

When to use: To avoid errors in denominator selection, carefully analyze problem context.

Common Mistakes to Avoid

❌ Using percentage increase formula instead of decrease formula when price falls
✓ Apply the percentage decrease formula properly using \(\frac{\text{Original} - \text{New}}{\text{Original}} \times 100\)
Why: Increase and decrease formulas have similar structure, so confusion arises easily
❌ Adding percentages directly in successive changes
✓ Multiply the successive change factors instead of adding percentages
Why: Successive changes compound multiplicatively, not additively
❌ Mistaking percentage value for actual value
✓ Always convert percentage to fraction or decimal before performing operations
Why: Treating percentages as whole numbers causes incorrect calculations
❌ Not converting percentage discount to decimal multiplier correctly
✓ Calculate final price as Original x (1 - Discount/100)
Why: Neglecting multiplier leads to wrong final price
❌ Incorrectly identifying whole in percentage problems
✓ Analyze problem context to identify the base quantity (whole)
Why: Confusing the base leads to denominator errors and incorrect answers
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