8.8 Summary
Key Points
- A percentage is a fraction with denominator 100: x\% = \dfrac{x}{100}. The trio fraction \leftrightarrow decimal \leftrightarrow percentage are three costumes for one proportion, e.g. \tfrac{4}{10} = 0.4 = 40\%.
- To express a fraction as a percentage, multiply by 100. To find y\% of a quantity, multiply by \dfrac{y}{100}. The 10% anchor makes many percentages doable in your head.
- A ratio can be turned into a fraction and then a percentage; percentages let us compare proportions fairly even when the wholes differ.
- Percentage change = \dfrac{\text{amount of change}}{\text{original amount}}\times 100, used for increase/decrease, profit, loss, discount and tax (all measured against the right base, profit and loss against the cost price).
- Growth: without compounding, A = p(1 + rt); with compounding, A = p(1 + r)^{t}. Decline multiplies by (1 - r) each period.
- Percentages compare proportions, not absolute amounts, and successive changes multiply rather than add, so “40% then 20%” compounds to a 52% cut, a 50% markup undone by a 50% discount is a 25% loss, and “x\% of y” always equals “y\% of x.” Estimate before you calculate.