Percentages

Percentages represent ratios out of $100$, expressed using the symbol $\%$. Calculating with percentages involves converting between fractions, decimals, and relative amounts to solve for totals, changes, or rates.

Core Percentage Calculations

A percentage expresses a fraction of $100$: $p\% = \frac{p}{100}$.

• Finding the Part: Multiply the total value by the percentage written as a decimal or fraction.

  Example: Find $15\%$ of $80 \implies 0.15 \times 80 = 12$.

• Finding the Percentage (Rate): Divide the part by the whole and multiply by $100\%$.

  Example: $18$ out of $24$ is what percentage? $\implies \frac{18}{24} = 0.75 \implies 75\%$.

• Finding the Whole (Total): Divide the known part by the percentage written as a decimal.

  Example: If $30$ is $40\%$ of a number $x \implies 0.40x = 30 \implies x = \frac{30}{0.40} = 75$.

Percentage Increase, Decrease, and Change

Percentage change measures the relative growth or decay of a value relative to its original starting amount.

• Percent Change Formula: Calculate the change as a fraction of the original value:

  $$\text{Percent Change} = \dfrac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$$

  Example: A price rises from $\$50$ to $\$65$. Change $= \frac{65 - 50}{50} \times 100\% = \frac{15}{50} \times 100\% = 30\% \text{ increase}$.

• Multiplier for Increase ($1 + r$): An increase of $r\%$ scales the original amount by $(1 + r)$.

  Example: A $\$200$ item increased by $8\%$ costs $200 \times (1 + 0.08) = 200 \times 1.08 = \$216$.

• Multiplier for Decrease ($1 - r$): A discount or decrease of $r\%$ scales the original amount by $(1 - r)$.

  Example: A $\$120$ jacket discounted by $25\%$ costs $120 \times (1 - 0.25) = 120 \times 0.75 = \$90$.

Successive Percentage Changes

When multiple percentage changes occur in sequence, apply multipliers consecutively to the updated values rather than simply adding the percentages together.

• Sequential Discount/Growth: Multiply the original value by each successive multiplier.

  Example: An item priced at $\$100$ is discounted by $20\%$, then the new price is discounted by an additional $10\%$:

  $$\text{Final Price} = 100 \times (1 - 0.20) \times (1 - 0.10) = 100 \times 0.80 \times 0.90 = \$72$$

  (Note: A $20\%$ discount followed by a $10\%$ discount equals a total $28\%$ discount, not $30\%$).

1. Use single decimal multipliers directly for rapid percent increase and decrease calculations.

   Example: Increase $\$85$ by $12\%$.

   • Growth multiplier: $1 + 0.12 = 1.12$

   • Final amount: $85 \times 1.12 = \$95.20$

2. Use single decimal multipliers directly for rapid percent increase and decrease calculations. 

   Example: A population drops from $500$ to $400$. What is the percent decrease?

   • Change: $500 - 400 = 100$

   • Original baseline: $500$

   • Calculation: $\frac{100}{500} \times 100\% = 20\%$

3. Work backward from final amounts by dividing by the multiplier. 

   Example: A shirt costs $\$36$ after a $20\%$ discount. What was the original price?

   • Discount multiplier: $1 - 0.20 = 0.80$

   • Equation: $0.80 \times \text{Original} = 36$

   • Original price: $\frac{36}{0.80} = \$45$

4. Do not add successive percentages together; multiply their corresponding scaling factors. 

   Example: A stock goes up by $50\%$ on day one and down by $50\%$ on day two. What is the net change?

   • Combined multiplier: $(1 + 0.50) \times (1 - 0.50) = 1.50 \times 0.50 = 0.75$

   • Net effect: $0.75 - 1 = -0.25 \implies 25\% \text{ decrease overall}$.

5. Use $100$ as a convenient starting value for problems with unknown initial totals.

   Example: A price increases by $20\%$, then decreases by $15\%$. What is the net percent change?

   • Start with hypothetical base: $\$100$

   • After $20\%$ increase: $100 \times 1.20 = \$120$

   • After $15\%$ decrease: $120 \times 0.85 = \$102$

   • Net change: $\$102 - \$100 = 2\% \text{ overall increase}$.