Comparing Quantities - Percentage as a way of Comparing Quantities

Meaning of Percentage: The word 'percent' is derived from the Latin 'per centum' meaning 'per hundred'. It is represented by the symbol $\%$ and denotes a fraction with a denominator of 100. Visualize a large square grid divided into $10 \times 10$ smaller equal squares (100 total); if 20 squares are shaded, the shaded portion represents $20\%$.

Converting Fractions to Percentages: To convert any fraction to a percentage, multiply the fraction by $100$ and attach the $\%$ sign. For example, if you have a circular pie cut into 4 equal slices and 1 slice is eaten, the visual fraction is $\frac{1}{4}$. Numerically, $\frac{1}{4} \times 100\% = 25\%$.

Converting Decimals to Percentages: To convert a decimal into a percentage, multiply the decimal by 100 (which effectively shifts the decimal point two places to the right). On a number line scaled from 0 to 1, a point at $0.75$ corresponds exactly to $75\%$.

Converting Percentages to Fractions or Decimals: To convert a percentage back to a fraction, remove the $\%$ sign and divide by 100, then simplify. To convert to a decimal, move the decimal point two places to the left. For instance, $50\%$ represents half of a whole bar, simplified as $\frac{50}{100} = \frac{1}{2}$ or $0.5$.

Ratios to Percentages: When parts are given as a ratio, such as $2:3$, first find the total parts ($2 + 3 = 5$). Each part can then be written as a fraction of the total and converted to a percentage. In a bar divided into 5 segments where 2 are blue and 3 are red, the blue part is $\frac{2}{5} \times 100 = 40\%$ and the red part is $\frac{3}{5} \times 100 = 60\%$.

Percentage Increase or Decrease: Percentages are often used to express how much a quantity has changed relative to its original value. Visualize a bar graph where the height of a bar grows from 100 units to 120 units; this represents a $20\%$ increase over the original base height.

Using Percentages for Comparison: Percentages provide a uniform base (100) to compare quantities that have different totals. For example, if Student A scores 20 out of 25 and Student B scores 30 out of 40, comparing their percentages ($80\%$ vs $75\%$) makes it clear who performed better.