Understanding Percentages

What is a Percentage?

A percentage is a way to express a number as a fraction of 100. It shows how much there is of something in relation to the whole, where the whole is always represented as 100. The word “percent” actually means “per hundred” in Latin.

We use the “%” symbol to represent percentages. For example, 35% means 35 out of 100, which can also be written as the fraction 35/100 or the decimal 0.35.

How Percentages Work

Percentages are calculated by taking a ratio and multiplying it by 100. For instance, if 25 out of 50 students in a class are male, the ratio is 25/50 = 0.5. Multiplying this by 100 gives us 50%. This means 50% of the students are male.

The Percentage Formula

The basic percentage formula is:

P × V₁ = V₂

Where:

• P is the percentage
• V₁ is the initial value
• V₂ is the result after applying the percentage

For example, if we want to know what percentage (P) of 30 equals 1.5:
P × 30 = 1.5
P = 1.5 ÷ 30 = 0.05
To convert this to a percentage, we multiply by 100: 0.05 × 100 = 5%

Percentage Difference

The percentage difference between two values tells us how much they differ relative to their average value. The formula is:

Percentage Difference = (|V₁ – V₂| ÷ ((V₁ + V₂)/2)) × 100

For example, to find the percentage difference between 10 and 6:
|10 – 6| = 4
(10 + 6)/2 = 8
4 ÷ 8 = 0.5
0.5 × 100 = 50%

So, the percentage difference between 10 and 6 is 50%.

Percentage Change

Percentage change measures how much a value has increased or decreased compared to its original value. The formula is:

Percentage Change = ((New Value – Original Value) ÷ Original Value) × 100

For percentage increases, we can also calculate new values directly:

• To increase a number by a percentage: Original × (1 + Percentage/100)
• To decrease a number by a percentage: Original × (1 – Percentage/100)

Examples:

• 500 increased by 10%: 500 × (1 + 0.10) = 500 × 1.1 = 550
• 500 decreased by 10%: 500 × (1 – 0.10) = 500 × 0.9 = 450

Why Percentages Are Useful

Percentages are widely used because they allow us to easily compare different quantities, regardless of their actual sizes. For example, saying “20% of students passed the test” gives clearer information than saying “50 out of 250 students passed the test,” especially when comparing with another class where “30 out of 150 students passed” (which is also 20%).

They’re used in many everyday situations including:

• Calculating discounts during sales
• Understanding interest rates on loans or savings
• Analyzing statistics and data
• Comparing performance metrics
• Calculating tips at restaurants

Understanding percentages helps us make better financial decisions, interpret data accurately, and compare different options effectively.