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Percentages

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Introduction to Percentages

Have you ever noticed how often we hear about things like "50% off," "75% attendance," or "20% tax"? These numbers are all examples of percentages. But what exactly is a percentage?

A percentage is a way to express a part of a whole as a fraction of 100. The word "percent" literally means "per hundred." So, when we say 25%, it means 25 parts out of 100 parts.

Percentages are everywhere-in shopping discounts, exam scores, interest rates on loans, and even in interpreting data in graphs and tables. Understanding percentages is essential not only for daily life but also for competitive exams.

Percentages are closely related to fractions and decimals. For example, 50% is the same as the fraction \(\frac{50}{100}\) or the decimal 0.5. In this chapter, we will explore how to work with percentages, convert between forms, calculate increases and decreases, and apply these concepts to real-world problems.

Definition and Conversion

A percentage is a number or ratio expressed as a fraction of 100. It is denoted using the symbol "%". For example, 40% means 40 out of 100.

To understand percentages better, let's see how they relate to fractions and decimals.

Conversion between Fraction, Decimal, and Percentage
Fraction Decimal Percentage
\(\frac{1}{2}\) 0.5 50%
\(\frac{1}{4}\) 0.25 25%
\(\frac{3}{5}\) 0.6 60%
\(\frac{7}{10}\) 0.7 70%
\(\frac{9}{20}\) 0.45 45%

How to convert:

  • Fraction to Percentage: Multiply the fraction by 100. For example, \(\frac{3}{5} = \frac{3}{5} \times 100 = 60\%\).
  • Percentage to Decimal: Divide the percentage by 100. For example, \(60\% = \frac{60}{100} = 0.6\).
  • Decimal to Percentage: Multiply the decimal by 100. For example, \(0.25 \times 100 = 25\%\).
Key Concept

Percentage

A way to express a number as parts per hundred.

Calculating Percentage of a Number

One of the most common tasks with percentages is finding what a certain percentage of a number is. For example, what is 15% of 2500 INR?

The general method is:

graph TD    A[Start: Given percentage p% and number N] --> B[Convert p% to decimal by dividing by 100]    B --> C[Multiply decimal by N]    C --> D[Result is p% of N]

Mathematically, this is:

Percentage of a Number

\[\text{Percentage of } N = \frac{p}{100} \times N\]

Calculate p% of a number N

p = percentage
N = number
Example 1: Calculate 15% of 2500 INR Easy
Find 15% of 2500 INR.

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

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

Step 2: Multiply 0.15 by 2500.

\(0.15 \times 2500 = 375\)

Answer: 15% of 2500 INR is 375 INR.

Percentage Increase and Decrease

Percentages are often used to describe how much a quantity has increased or decreased compared to its original value.

Percentage Increase tells us how much a value has gone up, expressed as a percentage of the original value.

Percentage Decrease tells us how much a value has gone down, expressed as a percentage of the original value.

The formulas are:

Percentage Increase

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

Calculate percentage increase from original to new value

Original = initial value
New = increased value

Percentage Decrease

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

Calculate percentage decrease from original to new value

Original = initial value
New = decreased value
Original 100 Increased 120 (+20%) Decreased 80 (-20%)
Example 2: Price Increase from 1200 INR to 1380 INR Medium
Calculate the percentage increase when the price rises from 1200 INR to 1380 INR.

Step 1: Find the increase in price.

\(1380 - 1200 = 180\) INR

Step 2: Use the percentage increase formula.

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

Answer: The price increased by 15%.

Successive Percentage Changes

Sometimes, a quantity undergoes more than one percentage change one after another. For example, a price may increase by 10% one month and then decrease by 20% the next month.

It is important to understand that successive percentage changes are not simply added or subtracted. Instead, we multiply the factors corresponding to each change.

The formula for successive percentage changes \(p_1\%\) and \(p_2\%\) is:

Successive Percentage Change

\[\text{Overall Change} = \left(1 + \frac{p_1}{100}\right) \times \left(1 + \frac{p_2}{100}\right) - 1\]

Calculate overall percentage change after successive changes

\(p_1, p_2\) = percentage changes (use negative for decrease)
Example 3: Successive Percentage Changes Hard
Calculate the overall percentage change if a price of 5000 INR increases by 10% and then decreases by 20%.

Step 1: Convert percentage changes to factors.

Increase by 10%: \(1 + \frac{10}{100} = 1.10\)

Decrease by 20%: \(1 - \frac{20}{100} = 0.80\)

Step 2: Multiply the factors.

\(1.10 \times 0.80 = 0.88\)

Step 3: Find overall change.

\(0.88 - 1 = -0.12\), which means a 12% decrease overall.

Answer: The overall change is a 12% decrease.

Formula Bank

Percentage to Decimal
\[ p\% = \frac{p}{100} \]
where: \(p\) = percentage value
Percentage of a Number
\[ \text{Percentage of } N = \frac{p}{100} \times N \]
where: \(p\) = percentage, \(N\) = number
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
Successive Percentage Change
\[ \text{Overall Change} = \left(1 + \frac{p_1}{100}\right) \times \left(1 + \frac{p_2}{100}\right) - 1 \]
where: \(p_1, p_2\) = percentage changes (use negative for decrease)
Profit
\[ \text{Profit} = \frac{\text{Profit \%} \times \text{Cost Price}}{100} \]
where: Profit % = profit percentage, Cost Price = initial price
Simple Interest
\[ SI = \frac{P \times R \times T}{100} \]
where: \(P\) = principal, \(R\) = rate of interest per annum, \(T\) = time in years

Profit Calculation Using Percentage

In commerce, profit is often expressed as a percentage of the cost price. If you buy an item and sell it for more than you paid, the extra money is your profit.

The formula to calculate profit amount when profit percentage and cost price are known is:

Profit Calculation

\[\text{Profit} = \frac{\text{Profit \%} \times \text{Cost Price}}{100}\]

Calculate profit amount from cost price and profit percentage

Profit % = profit percentage
Cost Price = initial price
Example 4: Profit Calculation Using Percentage Medium
Calculate the profit if the cost price is 1500 INR and the profit percentage is 12%.

Step 1: Use the profit formula.

\[ \text{Profit} = \frac{12 \times 1500}{100} = 180 \text{ INR} \]

Answer: The profit is 180 INR.

Simple Interest Using Percentage

Simple Interest (SI) is the interest calculated on the original principal amount for a certain period at a given rate.

The formula for Simple Interest is:

Simple Interest

\[SI = \frac{P \times R \times T}{100}\]

Calculate simple interest

P = principal
R = rate of interest per annum
T = time in years
Example 5: Simple Interest Calculation Medium
Calculate the simple interest on 10,000 INR at 8% per annum for 3 years.

Step 1: Use the simple interest formula.

\[ SI = \frac{10000 \times 8 \times 3}{100} = 2400 \text{ INR} \]

Answer: The simple interest earned is 2400 INR.

Tips & Tricks

Tip: Convert percentages to decimals quickly by moving the decimal point two places to the left.

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

Tip: For successive percentage changes, multiply the factors \((1 + p/100)\) instead of adding percentages.

When to use: When dealing with multiple percentage increases or decreases.

Tip: Remember that a percentage increase followed by the same percentage decrease does not return to the original value.

When to use: To avoid common misconceptions in percentage problems.

Tip: Use approximation for quick estimation by rounding percentages to the nearest 10%.

When to use: During time-limited exams to check answers quickly.

Tip: Convert fractions to percentages by multiplying the numerator by 100 and dividing by the denominator.

When to use: When given fractions and asked to find equivalent percentages.

Common Mistakes to Avoid

❌ Adding percentage increases directly instead of multiplying factors in successive changes.
✓ Multiply \((1 + \frac{p_1}{100})\) and \((1 + \frac{p_2}{100})\) and subtract 1 to find overall change.
Why: Students assume percentages are additive rather than multiplicative.
❌ Confusing percentage increase with percentage of increase.
✓ Always divide the increase by the original value before multiplying by 100.
Why: Students sometimes divide by the new value or use incorrect base.
❌ Using percentage as a whole number instead of decimal in calculations.
✓ Convert percentage to decimal by dividing by 100 before multiplying.
Why: Misunderstanding of percentage representation.
❌ Assuming profit percentage is profit amount, not percentage of cost price.
✓ Use formula Profit = \(\frac{\text{Profit %} \times \text{Cost Price}}{100}\).
Why: Confusion between absolute amounts and percentages.
❌ Ignoring units or currency in word problems.
✓ Always include units (INR, kg, meters) to avoid misinterpretation.
Why: Leads to incorrect answers or confusion.
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