A percentage is a number or ratio expressed as a fraction of 100. It represents a part of a whole, where the whole is considered to be 100. Think of it as a way to standardize comparisons and understand proportions easily. The word “percent” itself comes from the Latin “per centum,” meaning “by the hundred.” This explains why a percentage is essentially a fraction with 100 as the denominator. For example, 45% is equivalent to 45/100, 45:100 (or simplified to 45:55), or 0.45.
Table of Contents
Expressing Percentages
Percentages are denoted by several symbols: %, pct., pct, or pc. The most common symbol is %. For instance, 50% of students means 50 out of every 100 students.
Uses of Percentages
Percentages are widely used to express proportions, represent changes (increases or decreases), calculate discounts, understand interest rates, and present data in various fields like finance, statistics, science, and everyday life. They make it easy to compare different quantities relative to a common base of 100. For example, a $0.15 increase on $2.50 represents a 6% increase, making it clear how significant the change is relative to the original price. If 60% of students are female and 5% of all students are Computer Science (CS) majors, then 3% of all students are female CS majors. Further, if 10% of all students are CS majors, and 3% of all students are female CS majors, then 30% of CS majors are female.
Can Percentages be Greater than 100% or Less than 0%?
Yes. A percentage greater than 100% indicates a value more than the whole. For example, a 150% increase in sales means sales have increased by one and a half times the original amount. A percentage less than 0% is typically not used, but mathematically, it could represent a negative change relative to a baseline.
Calculating Percentages
The basic calculation involves multiplying the numeric value of the ratio by 100. For example, to express 15/50 as a percentage, multiply 15/50 by 100 to get 30%. Similarly, 4/25 as a percentage is (4/25) * 100 = 16%. The unitary method simplifies this by finding the value of one unit and then multiplying by 100. For example, if 4/25 represents a percentage, then 1/25 represents (1/4) of that percentage, which is 4%. Therefore, 4/25 represents 4 * 4 = 16%.
The percentage equivalent of 35/40 is 87.5%. If there are 10 girls in a class of 40, the percentage of girls is (10/40) * 100 = 25%.
Calculating Percentage Increase and Decrease
Percentage increase is calculated as: [(New Value – Original Value) / Original Value] * 100. If the price of a jacket increases from $100 to $150, the percentage increase is [(150-100)/100] * 100 = 50%. A 100% increase results in a doubling of the original quantity.
Percentage decrease is calculated similarly: [(Original Value – Decreased Value) / Original Value] * 100. If rainfall decreases from 127mm to 103mm, the percentage decrease is approximately [(127-103)/127] * 100 = 18.9%. A 100% decrease results in a final value of zero.
Compounding Percentages
Compounding percentages occur when a percentage change is applied to an already changed value. For example, if a stock increases by 10% in the first year and then another 10% in the second year, the total increase is not 20% but slightly more because the second 10% increase is applied to the larger value after the first year’s growth. Calculating percentages of percentages involves converting them to fractions or decimals and multiplying. For example, 50% of 40% is (0.50 * 0.40) * 100 = 20%.
Percentage Change vs. Percentage Point Change
Percentage change expresses the relative change between two values, while percentage point change describes the arithmetic difference between two percentages. For example, if unemployment rises from 5% to 6%, the percentage point change is 1%, but the percentage change is [(6-5)/5] * 100 = 20%.
Common Errors in Percentage Calculations
Common errors include confusing percentage change with percentage point change, incorrectly applying compounding percentages, and failing to convert percentages to decimals or fractions before multiplication.
Writing Percentages According to Style Guides
Most style guides recommend writing the number and the percent sign (%) without a space. For instance, 25%. However, ISO 31-0 requires a space between the number and the percent sign (25 %). In humanistic texts, writing “percent” is often preferred, while scientific texts often use the symbol (%).
Historical Development of Percentages
Ancient Rome utilized fractions in multiples of 1/100 for calculations. During the Middle Ages, computations with a denominator of 100 became increasingly common. By the 17th century, the use of interest rates expressed in hundredths had become standardized.
Percentages in Sports and Road Steepness
Sports statistics often use decimal proportions instead of percentages. Road or railway steepness is described by percent grade, calculated as 100 multiplied by the rise divided by the run.
More on Definitions and Calculations
A percentage can be defined simply as a part per hundred. 10% is equivalent to 10/100, 1/10, or 0.1. The percentage equivalent of 2/5 is 40%. 10% of 80 is 8.
Converting between percentages and decimals involves simple multiplication or division by 100. To convert a percentage to a decimal, replace the percent sign (%) with “/100” (e.g., 50% = 50/100 = 0.50). To convert a decimal to a percentage, multiply by 100 (e.g., 0.25 = 0.25 * 100 = 25%).
The percentage formula can be expressed as (Value / Total Value) × 100. Calculating the percentage of a number involves the formula P% of Number = X, or (P/100) * Number = X.
The percentage difference between two numbers is calculated as the absolute difference between the two numbers divided by their average, multiplied by 100. For example, the percentage difference between 20 and 30 is |20-30|/((20+30)/2) * 100 ≈ 22.22%.
Percentages are reversible. For example, 50% of 60 is equal to 60% of 50 (both equal 30). This is a useful “percentage trick.”
Solving Percentage Problems
If 16% of 40% of a number is 8, then the number is 125. This can be solved by setting up the equation (0.16 * 0.40 * x) = 8 and solving for x.
1/35 is 10% of 2/7. This is calculated by dividing (1/35) by (2/7) and multiplying by 100.
40% less than 90 is 54. This is calculated as 90 – (0.40 * 90) = 54.
The sum of (16% of 24.2) and (10% of 2.42) is 4.114.
If a fruit seller sells 40% of their apples and has 420 left, they originally had 700 apples. This can be solved by recognizing that 420 represents 60% (100% – 40%) of the original amount, so the original amount is (420/0.60) = 700.
If 40% of a larger number equals 60% of a smaller number, and their sum is 150, the larger number is 90. This can be solved using a system of equations. Let ‘L’ represent the larger number and ‘S’ the smaller number. The equations are 0.40L = 0.60S and L + S = 150. Solving these simultaneously yields L = 90 and S = 60.
45 out of 150 is 30%.
40% of 120 is 48.
This comprehensive guide provides a detailed understanding of percentages, their calculation, applications, and historical development. It aims to clarify common misconceptions and equip readers with the tools to confidently tackle percentage problems.