1. Percent Basics

• Definition: Percent means “per hundred.”
  Example: $$25% = 25$$ out of $$100 = 25/100 = 0.25$$.

• Conversions:

  • Fraction → Percent: multiply by 100.
    Example: $$\frac{3}{4} = 0.75 = 75%$$.

  • Decimal → Percent: move decimal two places right.
    Example: $$0.32 = 32%$$.

  • Percent → Decimal: move decimal two places left.
    Example: $$18% = 0.18$$.

2. Percent Word Problems

These are very common on the ACT:

• Finding a percent of a number:
  Formula:
  Part = Percent × Whole
  Example: What is $$30%$$ of $$80$$? → $$0.30 × 80 = 24$$.

• Finding the whole given a part and percent:
  $$\text{Whole}=\frac{\text{Part}}{\text{Percent}}$$​
  Example: 12 is 20% of what number? → $$\frac{12}{0.20}=60$$

• Finding the percent given part and whole:
  Percent=PartWhole×100\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100Percent=WholePart​×100
  Example: What percent of 50 is 20? → $$(\frac{20}{50})\times100=40$$

3. Percent Change

• Formula:
  $$\text{Percent Change}=\frac{\text{New Value}-\text{Original Value}}{\text{Original Value}}\times100$$

• Percent Increase: when the new value is larger than the original.
  Example: A shirt costs $40 and increases to $50.
  Percent increase = $$\frac{50-40}{40}\times100=25\%$$

• Percent Decrease: when the new value is smaller than the original.
  Example: Price drops from $60 to $45.
  Percent decrease = $$\frac{60-45}{60}\times100=25%$$.

4. Successive Percent Changes

Be careful: a $$20%$$ increase followed by a $$20%$$ decrease does NOT return to the original value.

• Example: Starting at 100:

  • Increase 20% → 120.

  • Decrease 20% → 120 – 24 = 96.
    Final is 4% less than the original.

5. Percent Applications

• Simple Interest:
  $$I=P\times r\times t$$
  (P = principal, r = annual rate, t = time in years).

• Discounts & Markups:
  Sale Price = Original Price – (Discount % × Original Price).
  Markup Price = Cost + (Markup % × Cost).

• Tax & Tips:
  Total = Original Price × (1 + Tax/Tip %).

ACT Strategy Tip: Always convert percents into decimals before calculations. Also, when working with percent change, carefully identify the original value (denominator), a common ACT trap is to flip it.