Concept of Percentage

When we say a number is expressed as a percentage, we are indicating how much out of 100 it represents. For example, if you have 25 apples out of 100 total apples, you can say that you have $25\%$ of the apples.

Mathematically, the percentage can be calculated using the following formula:

$$ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 $$

• Part: The portion of the whole that you are interested in.
• Whole: The total amount that is considered.

To find the percentage of a number, you can follow these steps:

1. Identify the part and the whole.
2. Use the percentage formula to calculate the percentage.

Example: What percentage of 50 is 15?

$$ \text{Percentage} = \left( \frac{15}{50} \right) \times 100 = 30\% $$

Percentages are often used to describe increases and decreases in values.

3.1 Increase by Percentage

To increase a salary of $2000$ by $10\%$, the formula used is:

$$ \text{New Salary} = \text{Original Salary} + \left( \text{Original Salary} \times \frac{\text{Percentage}}{100} \right) $$

Substituting the values:

$$ \text{New Salary} = 2000 + \left( 2000 \times \frac{10}{100} \right) $$

Calculating the increase:

$$ \text{New Salary} = 2000 + (2000 \times 0.1) = 2000 + 200 = 2200 $$

Thus, after a $10\%$ increase, the new salary is $2200$.

3.2 Decrease by Percentage

To decrease a number by a percentage, the formula is:

$$ \text{New Value} = \text{Original Value} – \left( \text{Original Value} \times \frac{\text{Percentage}}{100} \right) $$

Example: If a product originally costs $\$150$ and is discounted by $20\%$, the new price would be calculated as follows:

$$ \text{New Price} = 150 – \left( 150 \times \frac{20}{100} \right) = 150 – 30 = 120 $$

Percentages can also be converted to fractions and decimals:

4.1 To Fraction

To convert a percentage to a fraction, simply place the percentage number over 100 and simplify:

$$ 25\% = \frac{25}{100} = \frac{1}{4} $$

4.2 To Decimal

To convert a percentage to a decimal, divide by 100:

$$ 25\% = \frac{25}{100} = 0.25 $$

Percentages are widely used in various fields:

• Finance: Calculating interest rates, discounts, and profit margins.
• Statistics: Analyzing data distributions and population samples.
• Education: Grading systems often use percentages to evaluate student performance.

Understanding percentages is crucial for making informed decisions in everyday life. Mastering the calculation and application of percentages enables better financial management, data analysis, and interpretation of statistical results.