Ratios and Percentages

Definition: A ratio is a comparison of two quantities using a quotient.
Notation: There are three main ways to write a ratio of a number 'a' to a number 'b' (where 'b' is not equal to zero):
- $a ext{ to } b$
- $a:b$
- $\frac{a}{b}$
Writing Ratios from Phrases:
- When comparing quantities, ensure they are in the same units (e.g., convert hours to minutes).
- Example: Three weeks to thirty minutes is not directly comparable. If it were thirty minutes to two hours, you'd convert two hours to $120$ minutes, making the ratio $30 ext{ minutes}:120 ext{ minutes}$ . This simplifies to $1:4$ .

Proportion: An equation stating that two ratios are equal.
Cross Products: A method used to solve proportions.
- If you have a proportion $\frac{a}{b} = \frac{c}{d}$ , then the cross products are equal: $a imes d = b imes c$ .
- This equality must hold true for the proportion to be valid.
Example 1: Solving for an Unknown Variable
- Given the equation: $\frac{x}{6} = \frac{35}{42}$
- Apply cross products: $x imes 42 = 6 imes 35$
- $42x = 210$
- Divide by $42$ : $x = \frac{210}{42}$
- Solve: $x = 5$
Example 2: Solving for an Unknown Variable with Expressions
- Given the equation: $\frac{x+6}{2} = \frac{2}{5}$
- Apply cross products: $(x+6) imes 5 = 2 imes 2$
- $5(x+6) = 4$
- Distribute the $5$ : $5x + 30 = 4$
- Subtract $30$ from both sides: $5x = 4 - 30$
- $5x = -26$
- Divide by $5$ : $x = -\frac{26}{5}$
- Checking the Solution:
  - Substitute $x = -\frac{26}{5}$ back into the original equation:
    $\frac{-\frac{26}{5} + 6}{2} = \frac{2}{5}$
  - Find a common denominator for the numerator: $-\frac{26}{5} + \frac{30}{5} = \frac{4}{5}$
  - The left side becomes: $\frac{\frac{4}{5}}{2}$
  - To divide by $2$ ,