{"id":1834,"date":"2021-09-10T18:20:41","date_gmt":"2021-09-10T18:20:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&p=1834"},"modified":"2022-02-07T18:47:28","modified_gmt":"2022-02-07T18:47:28","slug":"solving-proportions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/solving-proportions\/","title":{"raw":"Solving Proportions","rendered":"Solving Proportions"},"content":{"raw":"
\r\n

Learning Outcomes<\/h3>\r\n
\r\n
    \r\n \t
  • Solve proportional equations<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nPreviously when solving equations involving fractions, we cleared fractions by multiplying both sides of the equation by the lowest common denominator. For example, to solve the equation\r\n\r\n[latex]\\frac{x}{2} + \\frac{x}{3} = 5[\/latex]\r\n\r\nwe multiply both sides of the equation by the lowest common denominator, which is the least common multiple of the denominators in our equation. Since the least common multiple of 2 and 3 is 6, we\u2019ll multiply both sides of the equation by 6.\r\n

    [latex]6(\\frac{x}{2})=6 \\cdot 5[\/latex]\r\n[latex]\\frac{6x}{2} + \\frac{6x}{3} = 30[\/latex]\r\n[latex]3x+2x=30[\/latex]\r\n[latex]5x=30[\/latex]\r\n[latex]\\frac{5x}{5} = \\frac{30}{5}[\/latex]\r\n[latex]x=6[\/latex]<\/span><\/p>\r\nThis approach can be applied to equations in which a variable appears in a denominator.\r\n

    \r\n

    Example<\/h3>\r\nSolve [latex]\\frac{12}{n}=3[\/latex].\r\n\r\n[reveal-answer q=\"111304\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"111304\"]\r\n\r\nSince our only fraction involves the denominator [latex]n[\/latex], our LCD is [latex]n[\/latex].\r\n\r\nMultiply both sides of the equation by the LCD.\r\n

    [latex]n (\\frac{12}{n}) = 3n[\/latex]\r\n[latex]\\frac{12n}{n} = 3n[\/latex]<\/p>\r\nCancel common factors.\r\n

    [latex]12=3n[\/latex]<\/p>\r\nDivide each side by [latex]3[\/latex].\r\n

    [latex]\\frac{12}{3} = \\frac{3n}{3}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]4=n[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Proportion<\/h3>\r\nA proportion<\/strong> is an equation of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex].\r\nThe proportion states two ratios or rates are equal. The proportion is read \"[latex]a[\/latex] is to [latex]b[\/latex], as [latex]c[\/latex] is to [latex]d[\/latex].\"\r\n\r\n<\/div>\r\nIn the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.\r\n
    \r\n

    Example<\/h3>\r\nSolve [latex]\\frac{x}{6} = \\frac{2}{3}[\/latex].\r\n\r\n[reveal-answer q=\"289139\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"289139\"]\r\n\r\nTo isolate x<\/em>, multiply both sides of the equation by the LCD, 6.\r\n

    [latex]6(\\frac{x}{6}) = 6(\\frac{2}{3})[\/latex]\r\n[latex]\\frac{6x}{6} = \\frac{6 \\cdot 2}{3}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]x=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    try it<\/h3>\r\n[ohm_question]146811[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show another example of how to solve a proportion equation using the LCD.\r\n\r\nhttps:\/\/youtu.be\/pXvzpSU4DyU\r\n\r\nAnother approach to solving a proportion involves finding the cross products of the proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign).\r\n\r\nFor any proportion of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex], its cross products are equal.\r\n\r\n\"a\r\n
    \r\n

    Example<\/h3>\r\nSolve [latex]\\frac{x}{15}=\\frac{2}{5}[\/latex].\r\n\r\n[reveal-answer q=\"667845\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"667845\"]\r\n\r\nSet the cross products equal to each other.\r\n

    [latex]5x=15 \\cdot 2[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]5x=30[\/latex]<\/p>\r\nDivide each side by [latex]5[\/latex] to isolate [latex]x[\/latex].\r\n

    [latex]\\frac{5x}{5} = \\frac{30}{5}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]x=6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLet\u2019s look at an example of a proportion involving a variable in the denominator.\r\n

    \r\n

    Example<\/h3>\r\nSolve [latex]\\frac{48}{n}=\\frac{4}{3}[\/latex].\r\n\r\n[reveal-answer q=\"623605\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"623605\"]\r\n\r\nSet the cross products equal to each other.\r\n

    [latex]48 \\cdot 3 = 4n[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]144=4n[\/latex]<\/p>\r\nDivide each side by [latex]4[\/latex] to isolate [latex]n[\/latex].\r\n

    [latex]\\frac{144}{4} = \\frac{4n}{4}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]36=n[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    try it<\/h3>\r\n[ohm_question]146813[\/ohm_question]\r\n\r\n<\/div>","rendered":"
    \n

    Learning Outcomes<\/h3>\n
    \n
      \n
    • Solve proportional equations<\/li>\n<\/ul>\n<\/section>\n<\/div>\n

      Previously when solving equations involving fractions, we cleared fractions by multiplying both sides of the equation by the lowest common denominator. For example, to solve the equation<\/p>\n

      [latex]\\frac{x}{2} + \\frac{x}{3} = 5[\/latex]<\/p>\n

      we multiply both sides of the equation by the lowest common denominator, which is the least common multiple of the denominators in our equation. Since the least common multiple of 2 and 3 is 6, we\u2019ll multiply both sides of the equation by 6.<\/p>\n

      [latex]6(\\frac{x}{2})=6 \\cdot 5[\/latex]
      \n[latex]\\frac{6x}{2} + \\frac{6x}{3} = 30[\/latex]
      \n[latex]3x+2x=30[\/latex]
      \n[latex]5x=30[\/latex]
      \n[latex]\\frac{5x}{5} = \\frac{30}{5}[\/latex]
      \n[latex]x=6[\/latex]<\/span><\/p>\n

      This approach can be applied to equations in which a variable appears in a denominator.<\/p>\n

      \n

      Example<\/h3>\n

      Solve [latex]\\frac{12}{n}=3[\/latex].<\/p>\n

      Show Answer<\/span><\/p>\n
      \n

      Since our only fraction involves the denominator [latex]n[\/latex], our LCD is [latex]n[\/latex].<\/p>\n

      Multiply both sides of the equation by the LCD.<\/p>\n

      [latex]n (\\frac{12}{n}) = 3n[\/latex]
      \n[latex]\\frac{12n}{n} = 3n[\/latex]<\/p>\n

      Cancel common factors.<\/p>\n

      [latex]12=3n[\/latex]<\/p>\n

      Divide each side by [latex]3[\/latex].<\/p>\n

      [latex]\\frac{12}{3} = \\frac{3n}{3}[\/latex]<\/p>\n

      Simplify.<\/p>\n

      [latex]4=n[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Proportion<\/h3>\n

      A proportion<\/strong> is an equation of the form [latex]{\\Large\\frac{a}{b}}={\\Large\\frac{c}{d}}[\/latex], where [latex]b\\ne 0,d\\ne 0[\/latex].
      \nThe proportion states two ratios or rates are equal. The proportion is read “[latex]a[\/latex] is to [latex]b[\/latex], as [latex]c[\/latex] is to [latex]d[\/latex].”<\/p>\n<\/div>\n

      In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.<\/p>\n

      \n

      Example<\/h3>\n

      Solve [latex]\\frac{x}{6} = \\frac{2}{3}[\/latex].<\/p>\n

      Show Answer<\/span><\/p>\n
      \n

      To isolate x<\/em>, multiply both sides of the equation by the LCD, 6.<\/p>\n

      [latex]6(\\frac{x}{6}) = 6(\\frac{2}{3})[\/latex]
      \n[latex]\\frac{6x}{6} = \\frac{6 \\cdot 2}{3}[\/latex]<\/p>\n

      Simplify.<\/p>\n

      [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      try it<\/h3>\n