Percentage

 

Complete Percentage Guide

 

                Percentage is a mathematical concept that means "per hundred" or "out of every 100." The word percentage comes from the Latin words per (for every) and centum (hundred). It was introduced to make comparisons easier because fractions with different denominators (such as 3/5 and 7/10) are difficult to compare directly. By converting all values to a common base of 100, comparisons become simple and clear. The symbol % represents "out of 100." For example, 25% means 25 out of 100, and 100% means the whole quantity. Today, percentages are widely used in examinations, banking, business, discounts, taxes, statistics, and everyday calculations because they provide a standard and easy way to express parts of a whole.

   

25%=25\100​=0.25

Fractions, decimals, and percentages 

                Fractions, decimals, and percentages are three different ways of representing the same numerical value or part of a whole. In mathematics, it is often necessary to convert one form into another to make calculations easier and to solve problems efficiently. Fractions express a part of a whole, decimals represent the same value in decimal form, and percentages express the value as "per hundred" (out of 100).

 

1. Fraction → Percentage

A fraction represents a part of a whole. To express this part out of 100, we convert it into a percentage. This is done by multiplying the fraction by 100 and adding the % symbol.

Formula:

Percentage = Fraction × 100%

A percentage always means "per hundred" or "out of 100." Fractions have different denominators (2, 3, 4, 5, etc.), so multiplying by 100 converts them into an equivalent value out of 100, making comparison easier.

Examples:

• 3/4 × 100 = 75%
• Three-fourths of a whole equals 75 out of 100.
• 2/5 × 100 = 40%
• Two-fifths of a whole equals 40 out of 100.

2. Decimal → Percentage

A decimal is another way of expressing a fraction. To convert a decimal into a percentage, multiply it by 100 and add the % symbol.

Formula:

Percentage = Decimal × 100%

Why do we multiply by 100?

A decimal shows part of a whole. Multiplying by 100 changes that part into a value out of 100, which is exactly what a percentage represents.

Examples:

• 0.75 × 100 = 75%
• 0.75 means 75 hundredths, so it is 75%.
• 1.20 × 100 = 120%
• This means the value is greater than the whole (100%).

3. Percentage → Fraction

A percentage is already a fraction with a denominator of 100. Therefore, to convert a percentage into a fraction, write it over 100 and simplify it to its lowest terms.

Formula:

Fraction = Percentage/100

Then simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF).

Examples:

• 80% = 80/100 = 4/5
• 12.5% = 12.5/100 = 1/8

Three rules:

Fraction → Percentage: × 100 Decimal → Percentage: × 100 Percentage → Fraction: ÷ 100, then simplify.

 

 

Percentage Increase and Decrease

Percentage increase and percentage decrease are used to measure how much a quantity has increased or decreased in comparison with its original value. These concepts are widely used in everyday life to calculate changes in prices, salaries, population, marks, profits, losses, and interest rates.

Percentage Increase

When the new value is greater than the original value, it is called a percentage increase.

Formula:

Example:

Price increases from ₹400 to ₹500.

Increase = ₹100

Percentage Increase = (100 ÷ 400) × 100 = 25%

Percentage Decrease

When the new value is less than the original value, it is called a percentage decrease.

Formula:

Example:

Price decreases from ₹800 to ₹600.

Decrease = ₹200

Percentage Decrease = (200 ÷ 800) × 100 = 25%

 

Successive Percentage

Successive Percentage = Two percentage changes applied one after another.

For example:

• First, a salary increases by 20%.
• Then, it increases again by 10%.

The second 10% is not calculated on the original salary. It is calculated on the new salary after the first increase.

Formula: If a value changes by a% and then b%, then Net Percentage Change = a + b + (ab/100) Use + for increase and − for decrease.

 

1. Increase → Increase ( + , + )

Formula:

Net % = a + b + (ab/100)

Example:

20% ↑ then 10% ↑

= 20 + 10 + 2

= 32% Increase

2. Decrease → Decrease ( − , − )

Formula:

Net % = -(a + b) + (ab/100)

Example:

20% ↓ then 10% ↓

= -(20 + 10) + 2

= -28%

= 28% Decrease

3. Increase → Decrease ( + , − )

Formula:

Net % = a - b - (ab/100)

Example:

20% ↑ then 10% ↓

= 20 - 10 - 2

= 8% Increase

4. Decrease → Increase ( − , + )

Formula:

Net % = -a + b - (ab/100)

Example:

25% ↓ then 20% ↑

= -25 + 20 - 5

= -10%

= 10% Decrease

·       Same signs (++, --) → Add ab/100 ·       Different signs (+-, -+) → Subtract ab/100

 

Real-Life Uses of Percentage

Percentages are an essential part of mathematics and are widely used in our daily lives to compare quantities, measure changes, and express values in a simple and standardized form. Since percentages represent values out of every 100, they make comparisons easy regardless of the size of the numbers. Some of the most common applications of percentages are given below.

1. Education

Percentages are used to calculate students' examination marks, grades, and overall performance. Schools, colleges, and competitive examinations use percentages to compare the achievements of students.

Example:

If a student scores 450 marks out of 500, the percentage is:

(450 ÷ 500) × 100 = 90%

 

2. Business and Shopping

In business, percentages are used to calculate profit, loss, discounts, commission, GST, and taxes. During shopping, discounts are usually offered as percentages.

Example:

A shirt costs ₹2,000 and is offered at a 20% discount.

Discount = ₹2,000 × 20% = ₹400

Selling Price = ₹2,000 − ₹400 = ₹1,600

 

3. Banking and Finance

Banks use percentages to calculate interest on savings accounts, fixed deposits, and loans. Investment returns are also expressed as percentages.

Example:

If a bank offers 7% annual interest, a customer earns interest equal to 7% of the deposited amount every year.

 

4. Population

Governments use percentages to measure population growth or decline over a period of time. It helps in planning schools, hospitals, roads, and other public services.

Example:

If a city's population increases by 5%, it means the population has grown by 5 people for every 100 people.

 

5. Salary

Companies often increase employees' salaries by a certain percentage based on performance or annual increments.

Example:

If an employee earns ₹40,000 per month and receives a 10% salary increase, the new salary becomes ₹44,000.

 

6. Statistics and Data Analysis

Percentages help represent survey results, election results, literacy rates, unemployment rates, and many other statistical data in an easy-to-understand format.

Example:

If 75% of people support a proposal, it means 75 out of every 100 people are in favour of it.

 

Common Mistakes to Avoid

Students often make mistakes while solving percentage problems. Avoiding these common errors can improve both accuracy and speed.

1. Using the Wrong Base Value

Percentage increase or decrease should always be calculated using the original value, not the new value.

Incorrect: Dividing by the new value.

Correct: Divide by the original value.

 

2. Forgetting to Multiply by 100

After dividing the required values, students sometimes forget to multiply the answer by 100 to convert it into a percentage.

Example:

25 ÷ 100 = 0.25

Percentage = 25%, not 0.25%.

 

3. Confusing Increase with Decrease

Always check whether the new value is greater or smaller than the original value before deciding whether it is an increase or a decrease.

 

4. Adding Successive Percentages Directly

Two percentage changes applied one after another cannot simply be added because the second change is calculated on the new value.

Example:

Increase by 20%, then 10%.

Correct Net Increase = 32%, not 30%.

 

5. Forgetting to Simplify Fractions

When converting percentages into fractions, always simplify the fraction to its lowest terms.

Example:

50% = 50/100 = 1/2

Not simply 50/100.

 

Frequently Asked Questions (FAQs)

1. What is a percentage?

A percentage is a mathematical term meaning "per hundred" or "out of every 100." It is represented by the symbol %.

2. Why do we use percentages?

Percentages make it easier to compare different quantities because they convert all values into a common base of 100.

3. What does the symbol (%) mean?

The symbol % means "per hundred" or "out of every 100."

4. How do you convert a fraction into a percentage?

Multiply the fraction by 100 and add the % symbol.

Example:

3/5 × 100 = 60%

5. How do you find the percentage increase?

Subtract the original value from the new value, divide the increase by the original value, and multiply by 100.

Formula:

Percentage Increase = (Increase ÷ Original Value) × 100

6. How do you find the percentage decrease?

Subtract the new value from the original value, divide the decrease by the original value, and multiply by 100.

Formula:

Percentage Decrease = (Decrease ÷ Original Value) × 100

7. Why is percentage important for SSC CGL?

Percentage is one of the most important topics in SSC CGL Quantitative Aptitude. It forms the foundation for topics such as Profit and Loss, Simple and Compound Interest, Ratio and Proportion, Data Interpretation, and Average. A strong understanding of percentages helps solve many exam questions quickly and accurately.

 

            Percentage increase and decrease provide a simple way to measure changes in quantities such as population, salary, and prices. By applying the appropriate formula, we can easily determine the new value after an increase or decrease.