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Percentages

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Introduction to Percentages

Percentages are a way to express a number as a part of 100. The word "percent" comes from Latin, meaning "per hundred". For example, 45% means 45 out of 100. Understanding percentages is essential because they are everywhere-in exams, shopping discounts, bank interest rates, and statistics. They help us compare quantities easily and understand proportions in real life.

To understand percentages deeply, it's important to see their connection with fractions and decimals. For example, 50% is the same as the fraction \(\frac{50}{100}\) and the decimal 0.5. This connection lets us convert a percentage into a number we can use in calculations.

Definition and Conversion

A percentage represents a part per hundred of any quantity. The general way to write a percentage is:

"Percentage means 'per hundred' or 'out of 100'."

For example:

  • 25% means 25 out of 100
  • 100% means the whole or the complete quantity
  • 5% means 5 parts out of 100

We can convert between three forms-fractions, decimals, and percentages-using simple steps:

Fraction Decimal Percentage
\(\frac{1}{2}\) 0.5 50%
\(\frac{3}{4}\) 0.75 75%
\(\frac{1}{5}\) 0.2 20%
\(\frac{7}{10}\) 0.7 70%
\(\frac{9}{100}\) 0.09 9%

How to convert:

  • Fraction to Percentage: Multiply by 100 (e.g., \(\frac{1}{2} \times 100 = 50\%\))
  • Percentage to Fraction: Divide by 100 and simplify (e.g., \(45\% = \frac{45}{100} = \frac{9}{20}\))
  • Decimal to Percentage: Multiply by 100 (e.g., 0.75 x 100 = 75%)
  • Percentage to Decimal: Divide by 100 (e.g., 20% = \(\frac{20}{100} = 0.2\))

Calculating Percentage of a Number

Finding a percentage of a number means calculating how much that percent is out of the given quantity. For example, what is 10% of 200?

There are two common methods:

graph TD    Start[Start]    A[Convert % to decimal or fraction]    B[Multiply converted number by given number]    C[Obtain result]    Start --> A --> B --> C

Step-by-step:

  1. Convert percentage to decimal (divide by 100), e.g. 10% -> 0.10
  2. Multiply this decimal by the number, e.g., 0.10 x 200 = 20
  3. The result is the required percentage part of the number.
Example 1: Finding 15% of INR 500 Easy
Find 15% of INR 500 using decimal conversion.

Step 1: Convert 15% to decimal by dividing by 100:

15% = \(\frac{15}{100} = 0.15\)

Step 2: Multiply decimal by the number:

0.15 x 500 = 75

Answer: 15% of INR 500 is INR 75.

Percentage Increase and Decrease

When a quantity changes, we often want to find the percentage of that change relative to the original amount. This is called percentage increase or percentage decrease.

Percentage Increase means the quantity has grown more than the original, and Percentage Decrease means it has gone down.

Steps to solve such problems are:

graph TD    Start[Start]    A[Find difference between new and original]    B[Divide difference by original value]    C[Multiply result by 100]    D[Obtain percentage increase/decrease]    Start --> A --> B --> C --> D

Formulas:

  • Percentage Increase = \(\frac{\text{New} - \text{Original}}{\text{Original}} \times 100\)
  • Percentage Decrease = \(\frac{\text{Original} - \text{New}}{\text{Original}} \times 100\)
Example 2: Price increase from INR 400 to INR 460 Medium
Calculate the percentage increase in price.

Step 1: Find the difference:

460 - 400 = 60

Step 2: Divide the difference by the original price:

\(\frac{60}{400} = 0.15\)

Step 3: Multiply by 100 to get percentage:

0.15 x 100 = 15%

Answer: The price increased by 15%.

Applications: Discounts

A discount reduces the price of an item by a certain percentage on the marked price. Calculating discount and the final selling price are common problems.

Steps to find the selling price after discount:

graph TD    MP[Marked Price]    DA[Calculate Discount Amount]    SP[Subtract Discount from Marked Price]    MP --> DA    DA --> SP

Formulas:

  • Discount = \(\frac{\text{Discount %}}{100} \times \text{Marked Price}\)
  • Selling Price = Marked Price - Discount
Example 3: 20% discount on an INR 1500 smartphone Easy
Calculate the discount amount and the final price after discount.

Step 1: Calculate discount amount:

Discount = \(\frac{20}{100} \times 1500 = 0.20 \times 1500 = 300\)

Step 2: Calculate selling price:

Selling Price = 1500 - 300 = 1200

Answer: Discount is INR 300, and final price is INR 1200.

Applications: Simple Interest

Simple Interest is the percentage of the principal amount earned or paid over a certain period at a fixed rate of interest.

It depends on three factors:

Variable Description Impact
Principal (P) Initial invested amount More principal means more interest
Rate (R) Interest per annum (as percent) Higher rate means more interest
Time (T) Duration in years Longer time means more interest

The simple interest formula is:

\[ \text{SI} = \frac{P \times R \times T}{100} \]
Example 4: Simple interest on INR 10,000 at 5% per annum for 3 years Medium
Calculate the total simple interest earned and the final amount.

Step 1: Identify variables:

Principal \(P = 10,000\), Rate \(R = 5\%\), Time \(T = 3\) years

Step 2: Apply simple interest formula:

\(\text{SI} = \frac{10,000 \times 5 \times 3}{100} = \frac{150,000}{100} = 1,500\)

Step 3: Calculate total amount after interest:

Total amount = Principal + Interest = 10,000 + 1,500 = 11,500

Answer: Total interest is INR 1,500 and final amount is INR 11,500.

Formula Application and Problem Solving

Competitive exams often present problems requiring rearranging formulas or combining percentage calculations with other concepts such as ratios or mixtures. The key is to identify what is given, what to find, and use the related formula carefully.

Reverse Percentage Problems are common - where the final amount after a percentage reduction or increase is known, and we must find the original value.

Example formula for reverse percentage when a discount is applied:

\[ \text{Original Price} = \frac{\text{Final Price} \times 100}{100 - \text{Discount \%}} \]
Example 5: Reverse Percentage Problem - Final price INR 1200 after 20% discount Hard
Find the original marked price before the discount.

Step 1: Identify given values:

Final price after discount = INR 1200, Discount = 20%

Step 2: Use reverse percentage formula:

\[ \text{Original Price} = \frac{1200 \times 100}{100 - 20} = \frac{1200 \times 100}{80} = 1500 \]

Answer: The original marked price was INR 1500.

Formula Bank

Percentage
\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]
where: Part = portion of the whole, Whole = total quantity
Finding Percentage of a Number
\[ \text{Percentage Value} = \frac{\text{Percentage}}{100} \times \text{Number} \]
where: Percentage = % to find, Number = total quantity
Percentage Increase
\[ \text{Percentage Increase} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100 \]
where: Original = initial value, New = increased value
Percentage Decrease
\[ \text{Percentage Decrease} = \frac{\text{Original} - \text{New}}{\text{Original}} \times 100 \]
where: Original = initial value, New = decreased value
Discount Amount
\[ \text{Discount} = \frac{\text{Discount \%}}{100} \times \text{Marked Price} \]
where: Discount % = percentage discount, Marked Price = original price
Selling Price after Discount
\[ \text{Selling Price} = \text{Marked Price} - \text{Discount} \]
where: Marked Price = original price, Discount = discount amount
Simple Interest
\[ \text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \]
where: Principal = amount invested, Rate = annual interest rate %, Time = years
Compound Percentage (Reverse Percentage)
\[ \text{Original Price} = \frac{\text{Final Price} \times 100}{100 - \text{Discount \%}} \]
where: Final Price = reduced price, Discount % = percentage discount

Tips & Tricks

Tip: Convert percentages into decimals before calculations to simplify multiplication.

When to use: Any problem requiring finding percent of a number or percentage increase/decrease.

Tip: For quick discount calculations, use the shortcut: price after discount = marked price x \(\frac{100 - \text{discount %}}{100}\).

When to use: Calculating selling price after discount in shopping or financial problems.

Tip: When calculating percentage increase or decrease, always divide by the original value, not the new value.

When to use: Calculating percent change between two quantities.

Tip: In reverse percentage problems, convert the final value back to the original by dividing by \((1 - \text{discount}/100)\).

When to use: Working backwards from discounted or increased amounts.

Tip: Memorize the simple interest formula as it applies directly to many percentage-based financial problems.

When to use: Problems involving principal, rate, and time with interest calculations.

Common Mistakes to Avoid

❌ Dividing by final amount instead of original amount when calculating percentage increase.
✓ Always divide the difference by the original value to get the correct percent increase or decrease.
Why: Students often confuse the reference point, leading to incorrect percentage values.
❌ Treating percentages as whole numbers instead of converting to decimals during calculations.
✓ Convert percentage to decimal by dividing by 100 before multiplication.
Why: Avoids answers that are 100 times too large.
❌ Subtracting discount % directly from price instead of calculating percentage of price.
✓ Calculate discount amount as percentage of marked price, not subtract directly.
Why: Percentages represent parts of a whole, not a simple subtraction.
❌ Misapplying simple interest formula with incorrect time or rate units.
✓ Ensure time is in years and rate is annual interest rate (%) before calculation.
Why: Unit inconsistency causes wrong answers in financial problems.
❌ Forgetting to add interest to principal after calculating simple interest.
✓ Total amount = Principal + Interest, not just interest alone.
Why: Students sometimes mistake interest earned for total amount payable or receivable.
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