Ratios and Percentages

Ratios and Proportions

Understanding Ratios

  • 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):
    • aexttoba ext{ to } b
    • a:ba:b
    • ab\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 120120 minutes, making the ratio 30extminutes:120extminutes30 ext{ minutes}:120 ext{ minutes}. This simplifies to 1:41:4.

Cross Products and Proportions

  • Proportion: An equation stating that two ratios are equal.
  • Cross Products: A method used to solve proportions.
    • If you have a proportion ab=cd\frac{a}{b} = \frac{c}{d}, then the cross products are equal: aimesd=bimesca 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: x6=3542\frac{x}{6} = \frac{35}{42}
    • Apply cross products: ximes42=6imes35x imes 42 = 6 imes 35
    • 42x=21042x = 210
    • Divide by 4242: x=21042x = \frac{210}{42}
    • Solve: x=5x = 5
  • Example 2: Solving for an Unknown Variable with Expressions
    • Given the equation: x+62=25\frac{x+6}{2} = \frac{2}{5}
    • Apply cross products: (x+6)imes5=2imes2(x+6) imes 5 = 2 imes 2
    • 5(x+6)=45(x+6) = 4
    • Distribute the 55: 5x+30=45x + 30 = 4
    • Subtract 3030 from both sides: 5x=4−305x = 4 - 30
    • 5x=−265x = -26
    • Divide by 55: x=−265x = -\frac{26}{5}
    • Checking the Solution:
      • Substitute x=−265x = -\frac{26}{5} back into the original equation:
        −265+62=25\frac{-\frac{26}{5} + 6}{2} = \frac{2}{5}
      • Find a common denominator for the numerator: −265+305=45-\frac{26}{5} + \frac{30}{5} = \frac{4}{5}
      • The left side becomes: 452\frac{\frac{4}{5}}{2}
      • To divide by 22,