The word “percentage” originates from the Latin term “per centum,” meaning “by the hundred.” Percentages reflect fractions with 100 as their denominator, capturing the relationship between a part and a whole, with the whole always considered as 100.

Imagine a student scoring 25 out of 80 on a math test. To derive the corresponding percentage, express “marks earned” as a fraction of the “total marks” and multiply the result by 100. In simpler terms, the percentage of marks would be 25 / 80 × 100 = 31.25%. Explore more about percentages and their conversion to fractions and decimals.

### Understanding Percentages:

A “percentage” is a fraction or ratio where the whole (denominator) is consistently set at 100. For instance, if Lisa achieves 40% in a science test, it means she scored 40 marks out of 100. This is represented as 40/100 in fraction form and 40:100 as a ratio. The symbol “%” denotes a percentage and is pronounced as “percent” or “percentage.” This symbol can always be substituted with “divided by 100” to transform it into an equivalent fraction or decimal.

**Examples of Percentages:**

15% = 15/100 (equals 3/20 or 0.15)

35% = 35/100 (equals 7/20 or 0.35)

8.5% = 8.5/100 (equals 17/200 or 0.085)

60% = 60/100 (equals 3/5 or 0.6)

### Calculating Percentages:

Calculating a percentage involves determining the portion in relation to the whole, based on 100. There are two approaches to calculating percentages:

- Adjusting the denominator of the fraction to 100: In this approach, find the equivalent fraction of the given fraction so that the resulting denominator is 100. The numerator itself becomes the percentage. For instance:

6/45 = 6/45 × 2/2 = 12/90 = 13.33% - Utilizing the unitary method: In this method, the fraction is directly multiplied by 100 to get the percentage. For instance, the percentage corresponding to the fraction 6/45 is:

6/45 × 100 = 600/45 = 13.33%

It’s important to note that the first method is suitable only when the denominator is a factor of 100. Otherwise, the unitary method is preferred. Let’s explore the detailed procedure for finding percentages using these two methods.

### Finding Percentages When the Total is 100:

When values add up to 100, the percentage of individual values concerning the total value is the value itself. For example, John buys candies of three flavors. The purchase details are provided in the table below.

Flavor | Number of Candies | Percentage |
---|---|---|

Mint | 25 | 25% |

Fruit | 45 | 45% |

Chocolate | 30 | 30% |

### Finding Percentages When the Total is NOT 100:

When the total number of items doesn’t sum up to 100, percentages can be calculated by adjusting the denominators to 100. For instance, Kate has a necklace with 15 turquoise beads and 20 pearl beads. The total beads are 15 + 20 = 35. In this case:

Percentage of turquoise beads = 15/35 × 5/5 = 75/100 = 75%

Percentage of pearl beads = 20/35 × 5/5 = 100/100 = 100%

Here’s another example highlighting the unitary method’s advantage.

Example: Calculate the percentage of a student’s 48 out of 55 history marks.

Solution: Given marks = 48/55. As the denominator isn’t a factor of 100, the unitary method is useful here.

Percentage of marks = 48/55 × 100 = 87.27%.

### Percentage Formula:

The percentage formula gauges a portion in terms of 100. All three methods for calculating percentages, as discussed earlier, can be conveniently computed using this formula:

Percentage = (Value/Total Value) × 100

Example: In a classroom with 36 students, 12 are boys. What percentage of the class are boys?

Solution: Number of boys = 12.

Total students = 36.

By the percentage formula:

Percentage of boys = 12/36 × 100 = 33.33%.

Converting Between Percentages and Decimals:

Replacing “%” with “/100” transforms percentages into decimals. Converting decimals into percentages involves multiplication by 100.

Example: 0.18 can be expressed as 0.18 × 100 = 18%.

Percentage Change Between Two Numbers:

Percentage change signifies the shift in value as a percentage over time. It’s useful for tracking increases or decreases. The formula for calculating percentage change varies based on whether it’s an increase or decrease:

Percentage Increase = (Increased Value – Original Value) / Original Value × 100

Example: The cost of a book rises from $10 to $18. The percentage increase is:

Solution: Percentage increase = (18 – 10) / 10 × 100 = 80%.

Percentage Decrease = (Original Value – Decreased Value) / Original Value × 100

Example: Temperature drops from 30°C to 24°C. The corresponding percentage decrease is:

Solution: Percentage decrease = (30 – 24) / 30 × 100 = 20%.

### Key Points on Percentages:

- To find the percentage of a number from the total, use the formula number / total number × 100.
- Percentages represent changes, whether increases or decreases.
- Conversion between fractions and percentages is achievable. Multiply by 100 for fractions to percentages and divide by 100 for percentages to fractions.
- Percentages are reciprocal. For instance, 25% of 80 is the same as 80% of 25.