Understanding Percentages

Understanding Percentages: Fundamental Concepts

Percentages are a mathematical representation of a ratio or fraction with a standard denominator of 100, originating from the Latin phrase 'per centum' which translates to 'by the hundred'. This concept is denoted by the percentage symbol (%). For instance, if 62 out of 100 students in a class have brown hair, it is said that 62% of the class possesses brown hair. Percentages are invaluable for comparing different data sets as they normalize values to a consistent scale. This allows for straightforward comparisons, such as evaluating a student's performance across different subjects, like 51% in mathematics versus 63% in English, regardless of the varying exam formats and subject matter.

Calculating Percentages from Raw Data

The calculation of a percentage involves dividing the part or individual score by the total amount and then multiplying the result by 100. This is encapsulated in the formula: percentage = (part/total) × 100%. For example, a student who correctly answers 32 out of 48 questions on a test has a percentage score of (32/48) × 100%, which simplifies to 66.67%. This percentage clearly reflects the student's performance relative to the total possible points. If the threshold for passing is 70%, the student can easily determine that they need to improve their performance by a modest amount to achieve a passing grade.

Finding the Percentage of an Amount

Determining a specific percentage of a given amount is a common requirement in various practical situations, such as computing discounts or figuring out the necessary score to pass an examination. To find standard percentages of amounts, one can use simple methods: halve the amount for 50%, divide by four for 25%, by ten for 10%, and by one hundred for 1%. For more complex percentages, these basic calculations can be combined. For instance, to find 28% of an amount, one could calculate 20% (2 times 10%), add 5%, and then add 3% (3 times 1%).

Converting Between Fractions, Decimals, and Percentages

Percentages, fractions, and decimals are interrelated methods of expressing proportions. To convert a decimal to a percentage, multiply the decimal by 100. To change a percentage into a decimal, divide it by 100. Fractions can be transformed into percentages by multiplying the fraction by 100 and percentages into fractions by placing the percentage over 100 and reducing it to its simplest form if possible. For example, 34% as a fraction is 34/100, which simplifies to 17/50, and as a decimal, it is 0.34. Conversely, the fraction 3/10 is equivalent to 30% as a percentage, and the decimal 0.07 translates to 7% when expressed as a percentage.

Understanding Percentage Change

Percentage change quantifies the extent of increase or decrease in a value relative to its original amount. The formula for this calculation is: percentage change = (difference/original amount) × 100%, where 'difference' is the absolute change between the initial and subsequent value. This metric is especially useful in economic contexts, such as monitoring the fluctuation in prices of commodities or investments. For instance, if a property's price rises from £296,000 to £400,000, the percentage increase is determined by the formula: (£400,000 - £296,000)/£296,000 × 100%, which yields a 35.1% rise.

Applying Percentage Increase and Decrease

Percentage increase and decrease are commonly applied concepts in daily life, for instance, when calculating sales discounts or adjusting prices. To compute a percentage increase, determine the percentage of the original amount and add it to the original figure. For example, if a property's value ascends by 10% from £250,000, the increase is £25,000 (10% of £250,000), resulting in a new value of £275,000. In contrast, a percentage decrease is calculated by finding the percentage of the original amount and subtracting it from that amount. For example, a £30 t-shirt with a 30% discount is reduced to £21, which is the result of subtracting the £9 discount (30% of £30).