{"id":1223,"date":"2019-06-12T14:49:37","date_gmt":"2019-06-12T18:49:37","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/ohsmath\/?post_type=chapter&p=1223"},"modified":"2020-09-15T08:54:11","modified_gmt":"2020-09-15T12:54:11","slug":"unit-9-ratio-proportion-and-percent","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/ohsmath\/chapter\/unit-9-ratio-proportion-and-percent\/","title":{"raw":"1.4. Ratios, Rates, and Percent","rendered":"1.4. Ratios, Rates, and Percent"},"content":{"raw":"[Latexpage]\r\n

Ratios and Rates<\/h1>\r\nRatio<\/strong>: a relationship between two numbers, expressed as a quotient with the same unit<\/em><\/strong> in the denominator and the numerator. There are three ways to write a ratio.\r\n

The ratio of a<\/em> and b<\/em> is:\u00a0\u00a0\u00a0 \u00a0a<\/em> to b<\/em>\u00a0\u00a0 \u00a0\u00a0\u00a0or\u00a0 \u00a0\u00a0\u00a0\u00a0a <\/em>: b<\/em> \u00a0\u00a0 \u00a0\u00a0or\u00a0 \u00a0 \u00a0 $\\frac{a}{b}$<\/p>\r\n\r\n

\r\n\r\nExample: <\/strong>\u00a0Write the ratio of 5 cents to 9 cents.\r\n

5 to 9\u00a0 \u00a0\u00a0\u00a0\u00a0or \u00a0\u00a0\u00a0\u00a0\u00a05 : 9\u00a0 \u00a0\u00a0\u00a0\u00a0or\u00a0 \u00a0 \u00a0 $\\frac{5}{9}$<\/p>\r\n\r\n<\/div>\r\n \r\n

    \r\n \t
  • Write a ratio in lowest terms (simplify):<\/li>\r\n<\/ul>\r\n

    - Write the ratio in a fractional form.<\/p>\r\n

    - Simplify and drop the units if given (as they cancel each other out).<\/p>\r\n\r\n

    \r\n

    Example:\u00a0 \u00a0 \u00a0 <\/strong>\u00a0$4 : 28 = \\frac{\\bcancel{4}}{\\bcancel{28}}=\\frac{1}{7}$<\/p>\r\n

    \u00a0 $\\div4$<\/p>\r\n

    \u00a0 \u00a0\u00a0<\/span><\/p>\r\n

    Example<\/strong>:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.75 metres to 0.25 metres\u00a0 \u00a0 \u00a0 \u00a0$\\frac{0.75m}{0.25m}=\\frac{\\bcancel{75}}{\\bcancel{25}}=\\frac{3}{1}=3$<\/p>\r\n

    \u00a0 \u00a0 \u00a0$\\times100$\u00a0 \u00a0 $\\div25$<\/span><\/p>\r\n\r\n<\/div>\r\n \r\n\r\nRate:<\/strong>\u00a0a ratio of two quantities with different units.\r\n

    \r\n\r\nExample:<\/strong>\u00a0 teachers to students;\u00a0\u00a0 money to time;\u00a0\u00a0 distance to time, etc.\r\n

    $\\frac{2\\;teachers}{83\\;students}$\u00a0 \u00a0,\u00a0 \u00a0$\\frac{24\\;dollars}{3\\;hours}$\u00a0 \u00a0,\u00a0 \u00a0$\\frac{85\\;miles}{2\\;hours}$<\/p>\r\n\r\n<\/div>\r\n

      \r\n \t
    • Write a rate in lowest terms (simplify the rate):<\/li>\r\n<\/ul>\r\n
      \r\n\r\nExample: <\/strong>\u00a080 kilometres per 320 minutes:\u00a0 \u00a0 \u00a0 \u00a0$\\frac{\\bcancel{80}km}{\\bcancel{320}min}=\\frac{1km}{4min}$\r\n

      \u00f780<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nUnit rate:<\/strong> a rate in which the number in the denominator<\/em><\/strong>\u00a0is 1.\r\n

      Example:\u00a0 <\/strong>15 dollars per hour: \u00a0\u00a0\u00a0\u00a0<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong> $\\frac{\\\\$15}{1h}$ = \\$15\/h<\/div>\r\n
        \r\n \t
      • Some unit rates:<\/li>\r\n<\/ul>\r\n

        - Miles (or kilometres) per hour (or minute).<\/p>\r\n

        - Cost (dollars\/cents) per item or quantity.<\/p>\r\n

        - Earnings (dollars) per hour (or week).<\/p>\r\n\r\n

        \r\n

        <\/h1>\r\n<\/div>\r\nProportion<\/strong>: an equation with a ratio (or rate) on two sides ($\\frac{a}{b}=\\frac{c}{d}$), in which the two ratios are equal.\r\n
        \r\n\r\n\u00a0Example: <\/strong>\u00a0Write the following sentence as a proportion.\r\n\r\n3 printers is to 18 computers as 2 printers is to 12 computers.\r\n\r\n$\\frac{3\\;printers}{18\\;computers}=\\frac{2\\;printers}{12\\;computers}$\r\n\r\n<\/div>\r\n \r\n\r\nSolving a propor<\/strong>tion: <\/strong>\r\n
          \r\n \t
        • Cross multiply: multiply along two diagonals.\u00a0 \u00a0 \u00a0$\\frac{a}{b}=\\frac{c}{d}$<\/li>\r\n \t
        • Solve for the unknown.<\/li>\r\n<\/ul>\r\n \r\n
          \r\n

          Example 1.4.1<\/p>\r\n\r\n<\/header>\r\n

          \r\n\r\n4 litres of milk cost \\$4.38. What is the cost of 2 litres?\r\n\r\n\r\n\r\n
          \r\n
            \r\n \t
          • Facts and unknown:<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
            4 L milk<\/td>\r\n2 L milk<\/td>\r\n<\/tr>\r\n
            \\$4.38<\/td>\r\n\\$ x = <\/em>?<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            \r\n
              \r\n \t
            • Proportion:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
            		$\\frac{4\\;L}{\\\\$4.38}=\\frac{2\\;L}{\\\\$x}$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
            \r\n
              \r\n \t
            • Cross multiply:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
            		$(4)(x)=(2)(4.38)$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
            \r\n
              \r\n \t
            • Solve for x<\/em>:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
            		$\\frac{\\bcancel{4}x}{\\bcancel{4}}=\\frac{2(4.38)}{4}$<\/td>\r\nDivide both sides by 4.<\/td>\r\n<\/tr>\r\n
            <\/td>\r\n$x=\\frac{(2)(4.38)}{4}=2.19$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
            <\/td>\r\n2 litres of milk cost \\$2.19.<\/td>\r\n<\/td>\r\n<\/tr>\r\n
            \r\n
              \r\n \t
            • Check:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
            		$\\frac{4\\;L}{\\\\$4.38}=\\frac{2\\;L}{\\\\$2.19}$<\/td>\r\nReplace x<\/em> with 2.19.<\/td>\r\n<\/tr>\r\n
            <\/td>\r\n(4) (2.19) = (2) (4.38)<\/td>\r\n<\/td>\r\n<\/tr>\r\n
            <\/td>\r\n8.76 = 8.76<\/td>\r\nCorrect!<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n
            \r\n

            Example 1.4.2<\/p>\r\n\r\n<\/header>\r\n

            \r\n\r\nTom\u2019s height is 1.75 metres, and his shadow is 1.09 metres long. A building\u2019s shadow is 10 metres long at the same time. How high is the building?\r\n\r\n\r\n\r\n
            \r\n
              \r\n \t
            • Facts and unknown:<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
              \u00a0 \u00a0 Tom\u2019s height = 1.75 m<\/td>\r\n\u00a0 Building\u2019s height (x<\/em>) = ?<\/td>\r\n<\/tr>\r\n
              \u00a0 \u00a0 Tom\u2019s shadow = 1.09 m<\/td>\r\n\u00a0 Building\u2019s shadow = 10m<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              \r\n
                \r\n \t
              • Proportion:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$\\frac{1.75m}{1.09m}=\\frac{xm}{10m}$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              \r\n
                \r\n \t
              • Cross multiply:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$(1.75)(10)=(1.09)(x)$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              \r\n
                \r\n \t
              • Solve for x<\/em>:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$x=\\frac{(1.75)(10)}{1.09}=\\frac{(\\bcancel{1.09})x}{\\bcancel{1.09}}$<\/td>\r\nDivide both sides by 1.09.<\/td>\r\n<\/tr>\r\n
              <\/td>\r\n$x=\\frac{(1.75)(10)}{1.09}\\approx16.055$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              <\/td>\r\nThe building\u2019s height is 16.055m.<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              \r\n
                \r\n \t
              • Check:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$\\frac{1.75m}{1.09m}=\\frac{16.055m}{10m}$<\/td>\r\nReplace x<\/em> with 16.055.<\/td>\r\n<\/tr>\r\n
              <\/td>\r\n$(1.75) (10) = (16.055) (1.09)$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              <\/td>\r\n$17.5 = 17.5$<\/td>\r\nCorrect!<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n
              \r\n

              Example 1.4.3<\/p>\r\n\r\n<\/header>\r\n

              \r\n\r\nIf 15 mL of medicine must be mixed with 180 mL of water, how many mL of medicine must be mixed in 230 mL of water?\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              \r\n
                \r\n \t
              • Proportion:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$\\frac{15\\;mL}{180\\;mL}=\\frac{x\\;mL}{230\\;mL}$<\/td>\r\n$\\frac{15\\;mL\\;medicine}{180\\;mL\\;water}=\\frac{x\\;mL\\;medicine}{230\\;mL\\;water}$<\/td>\r\n<\/tr>\r\n
              \r\n
                \r\n \t
              • Cross multiply:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$(15)(230)=(180)(x)$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              \r\n
                \r\n \t
              • Solve for x<\/em>:<\/li>\r\n<\/ul>\r\n<\/td>\r\n
              		$x=\\frac{(15\\;mL)(230\\;mL)}{180\\;mL}\\approx19.17\\;mL$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              <\/td>\r\n19.17 mL of medicine must be mixed in 230 mL of water.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n
              \r\n

              Percent<\/strong><\/h1>\r\n<\/div>\r\nPercent (%):<\/strong> one part per hundred.\r\n\r\nConverting between percent, decimals and fractions:<\/strong>\r\n\r\n\r\n\r\n
              Conversion<\/strong><\/td>\r\nSteps<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              Percent \u21d2 Decimal<\/strong><\/td>\r\nMove the decimal point two places to the left, then remove %.<\/td>\r\n31%\u00a0 =\u00a0 31.% = 0.31<\/td>\r\n<\/tr>\r\n
              Decimal \u21d2 Percent<\/strong><\/td>\r\nMove the decimal point two places to the right, then insert %.<\/td>\r\n0.317 = 0. 317 = 31.7 %<\/td>\r\n<\/tr>\r\n
              Percent \u21d2 Fraction<\/strong><\/td>\r\nRemove %, divide by 100, then simplify.<\/td>\r\n\u00a015% = $\\frac{15}{100}=\\frac{3}{20}$<\/td>\r\n<\/tr>\r\n
              Fraction \u21d2 Percent<\/strong><\/td>\r\nDivide, move the decimal point two places to the right, then insert %.<\/td>\r\n$\\frac{1}{4}$ = 1\u00f74 = 0.25 = 25 %<\/td>\r\n<\/tr>\r\n
              Decimal \u21d2 Fraction<\/strong><\/td>\r\nConvert the decimal to a percent, then convert the percent to a fraction.<\/td>\r\n0.35 = 35 % = $\\frac{35}{100}=\\frac{7}{20}$\r\n\r\n% = per one hundred<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nThere are two methods to solve percent problems:<\/strong>\r\n
                \r\n \t
              • Percent proportion method<\/li>\r\n \t
              • Translation (translate the words into mathematical symbols.)<\/li>\r\n<\/ul>\r\n \r\n\r\nPercent proportion method: <\/strong>\r\n\r\n\r\n\r\n
                $\\frac{Part}{Whole}=\\frac{Percent}{100}$<\/td>\r\nor<\/td>\r\n$\\frac{\"is\"\\;number}{\"of\"\\;number}=\\frac{\\%}{100}$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\r\n\r\n
                Step<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                <\/td>\r\n8<\/em><\/strong> percent of<\/em> <\/strong>what number is<\/strong> 4<\/em> <\/strong>?\r\n<\/em><\/strong><\/td>\r\n<\/tr>\r\n
                \r\n
                  \r\n \t
                • Identify the part, whole, and percent.<\/li>\r\n<\/ul>\r\n<\/td>\r\n
                		\u21d1\r\n\r\nPercent<\/td>\r\n\u21d1\r\n\r\nWhole (x<\/em>)<\/td>\r\n\u21d1\r\n\r\nPart<\/td>\r\n<\/tr>\r\n
                \r\n
                  \r\n \t
                • Set up the proportion equation.<\/li>\r\n<\/ul>\r\n<\/td>\r\n
                		$\\frac{4}{x}=\\frac{8}{100}$<\/td>\r\n$\\frac{Part}{Whole}=\\frac{\\%}{100}$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
                \r\n
                  \r\n \t
                • Solve for unknown (x<\/em>).<\/li>\r\n<\/ul>\r\n<\/td>\r\n
                		$x=\\frac{(4)(100)}{8}=50$<\/td>\r\n$x=50$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
                \r\n

                <\/h1>\r\n<\/div>\r\nTranslation method: <\/strong>translate the words into mathematical symbols.\r\n
                  \r\n \t
                • What ____\u00a0 x <\/em>:\u00a0\u00a0\u00a0 the word \u201cwhat\u201d represents an unknown quantity x<\/em>.<\/li>\r\n \t
                • Is\u00a0 \u00a0 \u00a0 ____ \u00a0= : \u00a0\u00a0\u00a0the word \u201cis\u201d represents an equal sign.<\/li>\r\n \t
                • of\u00a0 \u00a0 \u00a0____\u00a0 \u00d7 :\u00a0\u00a0\u00a0 the word \u201cof\u201d represents a multiplication sign.<\/li>\r\n \t
                • % \u00a0\u00a0\u00a0to\u00a0\u00a0\u00a0\u00a0 decimal:\u00a0 \u00a0\u00a0always change the percent to a decimal.<\/li>\r\n<\/ul>\r\n \r\n
                  \r\n

                  Example 1.4.4<\/p>\r\n\r\n<\/header>\r\n

                  \r\n\r\n1) <\/strong>\u00a0\u00a0What \u00a0is \u00a015% \u00a0of \u00a080?\r\n\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0x<\/em>\u00a0 =\u00a0 0.15 \u2022 80\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0\u00a0<\/strong>x<\/em> = (0.15)(80) = 12\u00a0\u00a0 <\/strong>\r\n\r\n\u00a0<\/strong>2) \u00a0<\/strong>\u00a0What \u00a0percent \u00a0of \u00a090 \u00a0is \u00a045?\r\n\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong>x<\/em> \u2022 90 = 45\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/strong>\u00a0\u00a0x<\/em> = $\\frac{45}{90}$ = 0.5 = 50%\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/strong>\r\n

                  \u00a0<\/strong>Divide both sides by 90.<\/p>\r\n\u00a0<\/strong>3) \u00a0<\/strong>\u00a012 \u00a0is \u00a08% \u00a0of \u00a0what number?\r\n\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>12 = 0.08 \u2022 x<\/em> \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0\u00a0x<\/em> = $\\frac{12}{0.08}$ =\u00a0<\/strong>150 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong>\r\n

                  Divide both sides by 0.08.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nPercent increase or decrease:<\/strong>\r\n\r\n\r\n\r\n
                  Application<\/strong><\/td>\r\nFormula<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
                  Percent increase<\/strong><\/td>\r\nPercent increase = $\\frac{New\\;value-Original\\;value}{Original\\;value}$<\/td>\r\n$x=\\frac{N-O}{O}$<\/td>\r\n<\/tr>\r\n
                  Percent decrease<\/strong><\/td>\r\nPercent decrease = $\\frac{Original\\;value-New\\;value}{Original\\;value}$<\/td>\r\n$x=\\frac{O-N}{O}$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n
                  \r\n

                  Example 1.4.5<\/p>\r\n\r\n<\/header>\r\n

                  \r\n\r\nA product increased production from 1500 last month<\/em><\/strong> to 1650 this month<\/em><\/strong>. \u00a0Find the percent increase<\/em><\/strong>.<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n
                  New value (N):<\/td>\r\n1650<\/td>\r\nThis month.<\/td>\r\n<\/tr>\r\n
                  Original value (O):<\/td>\r\n1500<\/td>\r\nLast month.<\/td>\r\n<\/tr>\r\n
                  Percent increase:<\/td>\r\n$x=\\frac{N-O}{O}=\\frac{1650-1500}{1500}=0.1=10\\%}$<\/td>\r\nA 10% increase.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nA product was reduced<\/em><\/strong> from $\\bf{\\\\$}$33<\/em><\/strong> to $\\bf{\\\\$}$29<\/em>.<\/strong> \u00a0What percent reduction <\/em><\/strong>is this?\r\n\r\n\r\n\r\n
                  Percent decrease:<\/td>\r\n$x=\\frac{O-N}{O}=\\frac{33-29}{33}\\approx0.12=12\\%$<\/td>\r\nA 12 % decrease.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n
                  \r\n

                  Practice questions<\/strong><\/h1>\r\n<\/div>\r\n1.<\/strong> Write the following as a ratio or rate in lowest terms:\r\n

                  a.<\/strong> 4 nickels to 16 nickels.<\/p>\r\n

                  b.<\/strong> 350 people for 1500 tickets.<\/p>\r\n

                  c.<\/strong> 160 kilometres per 740 minutes.<\/p>\r\n2.<\/strong> A train travelled 459 km in 6 hours. What is the unit rate?\r\n\r\n3.<\/strong> Write the following sentence as a proportion: 24 hours is to 1,940 kilometres as 12 hours is to 985 kilometres. \u00a0\u00a0<\/strong>\r\n\r\n4.<\/strong> 4 litres of juice cost \\$7.38, how much do 2 litres cost?\r\n\r\n5.<\/strong> Sarah earns \\$4,500 in 30 days. How much does she earn in 120 days?\r\n\r\n6.<\/strong> A product increased production from 2,800 last year to 3,920 this year. Find the percent increase.\r\n\r\n7.<\/strong> The table below gives survey data regarding Ryerson Occupational and Public Health students and summer jobs, but some data are missing:\r\n\r\n\r\n\r\n
                  Year<\/td>\r\n2015<\/td>\r\n2016<\/td>\r\n2017<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
                  Unemployed students<\/td>\r\n350<\/td>\r\n?<\/td>\r\n396<\/td>\r\n<\/tr>\r\n
                  Total number of students<\/td>\r\n1250<\/td>\r\n1100<\/td>\r\n?<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                  a.<\/strong> What % of students were unemployed in 2015?<\/p>\r\n

                  b.<\/strong> If the % of unemployed students remained the same in 2016, find the number of unemployed students in 2016.<\/p>\r\n

                  c.<\/strong> If the % of unemployed students increased by an additional 5% in 2017, find the total number of students in 2017.<\/p>\r\n ","rendered":"

                  Ratios and Rates<\/h1>\n

                  Ratio<\/strong>: a relationship between two numbers, expressed as a quotient with the same unit<\/em><\/strong> in the denominator and the numerator. There are three ways to write a ratio.<\/p>\n

                  The ratio of a<\/em> and b<\/em> is:\u00a0\u00a0\u00a0 \u00a0a<\/em> to b<\/em>\u00a0\u00a0 \u00a0\u00a0\u00a0or\u00a0 \u00a0\u00a0\u00a0\u00a0a <\/em>: b<\/em> \u00a0\u00a0 \u00a0\u00a0or\u00a0 \u00a0 \u00a0 \"\frac{a}{b}\"<\/p>\n

                  \n

                  Example: <\/strong>\u00a0Write the ratio of 5 cents to 9 cents.<\/p>\n

                  5 to 9\u00a0 \u00a0\u00a0\u00a0\u00a0or \u00a0\u00a0\u00a0\u00a0\u00a05 : 9\u00a0 \u00a0\u00a0\u00a0\u00a0or\u00a0 \u00a0 \u00a0 \"\frac{5}{9}\"<\/p>\n<\/div>\n

                   <\/p>\n

                    \n
                  • Write a ratio in lowest terms (simplify):<\/li>\n<\/ul>\n

                    – Write the ratio in a fractional form.<\/p>\n

                    – Simplify and drop the units if given (as they cancel each other out).<\/p>\n

                    \n

                    Example:\u00a0 \u00a0 \u00a0 <\/strong>\u00a0\"4 : 28 = \frac{\bcancel{4}}{\bcancel{28}}=\frac{1}{7}\"<\/p>\n

                    \u00a0 \"\div4\"<\/p>\n

                    \u00a0 \u00a0\u00a0<\/span><\/p>\n

                    Example<\/strong>:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.75 metres to 0.25 metres\u00a0 \u00a0 \u00a0 \u00a0\"\frac{0.75m}{0.25m}=\frac{\bcancel{75}}{\bcancel{25}}=\frac{3}{1}=3\"<\/p>\n

                    \u00a0 \u00a0 \u00a0\"\times100\"\u00a0 \u00a0 \"\div25\"<\/span><\/p>\n<\/div>\n

                     <\/p>\n

                    Rate:<\/strong>\u00a0a ratio of two quantities with different units.<\/p>\n

                    \n

                    Example:<\/strong>\u00a0 teachers to students;\u00a0\u00a0 money to time;\u00a0\u00a0 distance to time, etc.<\/p>\n

                    \"\frac{2\;teachers}{83\;students}\"\u00a0 \u00a0,\u00a0 \u00a0\"\frac{24\;dollars}{3\;hours}\"\u00a0 \u00a0,\u00a0 \u00a0\"\frac{85\;miles}{2\;hours}\"<\/p>\n<\/div>\n

                      \n
                    • Write a rate in lowest terms (simplify the rate):<\/li>\n<\/ul>\n
                      \n

                      Example: <\/strong>\u00a080 kilometres per 320 minutes:\u00a0 \u00a0 \u00a0 \u00a0\"\frac{\bcancel{80}km}{\bcancel{320}min}=\frac{1km}{4min}\"<\/p>\n

                      \u00f780<\/p>\n<\/div>\n

                       <\/p>\n

                      Unit rate:<\/strong> a rate in which the number in the denominator<\/em><\/strong>\u00a0is 1.<\/p>\n

                      Example:\u00a0 <\/strong>15 dollars per hour: \u00a0\u00a0\u00a0\u00a0<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong> \"\frac{\$15}{1h}\" = $15\/h<\/div>\n
                        \n
                      • Some unit rates:<\/li>\n<\/ul>\n

                        – Miles (or kilometres) per hour (or minute).<\/p>\n

                        – Cost (dollars\/cents) per item or quantity.<\/p>\n

                        – Earnings (dollars) per hour (or week).<\/p>\n

                        \n

                        <\/h1>\n<\/div>\n

                        Proportion<\/strong>: an equation with a ratio (or rate) on two sides (\"\frac{a}{b}=\frac{c}{d}\"), in which the two ratios are equal.<\/p>\n

                        \n

                        \u00a0Example: <\/strong>\u00a0Write the following sentence as a proportion.<\/p>\n

                        3 printers is to 18 computers as 2 printers is to 12 computers.<\/p>\n

                        \"\frac{3\;printers}{18\;computers}=\frac{2\;printers}{12\;computers}\"<\/p>\n<\/div>\n

                         <\/p>\n

                        Solving a propor<\/strong>tion: <\/strong><\/p>\n

                          \n
                        • Cross multiply: multiply along two diagonals.\u00a0 \u00a0 \u00a0\"\frac{a}{b}=\frac{c}{d}\"<\/li>\n
                        • Solve for the unknown.<\/li>\n<\/ul>\n

                           <\/p>\n

                          \n
                          \n

                          Example 1.4.1<\/p>\n<\/header>\n

                          \n

                          4 litres of milk cost $4.38. What is the cost of 2 litres?<\/p>\n\n\n\n
                          \n
                            \n
                          • Facts and unknown:<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                            4 L milk<\/td>\n2 L milk<\/td>\n<\/tr>\n
                            $4.38<\/td>\n$ x = <\/em>?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
                            \n
                              \n
                            • Proportion:<\/li>\n<\/ul>\n<\/td>\n
                            		\"\frac{4\;L}{\$4.38}=\frac{2\;L}{\$x}\"<\/td>\n<\/td>\n<\/tr>\n
                            \n
                              \n
                            • Cross multiply:<\/li>\n<\/ul>\n<\/td>\n
                            		\"(4)(x)=(2)(4.38)\"<\/td>\n<\/td>\n<\/tr>\n
                            \n
                              \n
                            • Solve for x<\/em>:<\/li>\n<\/ul>\n<\/td>\n
                            		\"\frac{\bcancel{4}x}{\bcancel{4}}=\frac{2(4.38)}{4}\"<\/td>\nDivide both sides by 4.<\/td>\n<\/tr>\n
                            <\/td>\n\"x=\frac{(2)(4.38)}{4}=2.19\"<\/td>\n<\/td>\n<\/tr>\n
                            <\/td>\n2 litres of milk cost $2.19.<\/td>\n<\/td>\n<\/tr>\n
                            \n
                              \n
                            • Check:<\/li>\n<\/ul>\n<\/td>\n
                            		\"\frac{4\;L}{\$4.38}=\frac{2\;L}{\$2.19}\"<\/td>\nReplace x<\/em> with 2.19.<\/td>\n<\/tr>\n
                            <\/td>\n(4) (2.19) = (2) (4.38)<\/td>\n<\/td>\n<\/tr>\n
                            <\/td>\n8.76 = 8.76<\/td>\nCorrect!<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                             <\/p>\n

                            \n
                            \n

                            Example 1.4.2<\/p>\n<\/header>\n

                            \n

                            Tom\u2019s height is 1.75 metres, and his shadow is 1.09 metres long. A building\u2019s shadow is 10 metres long at the same time. How high is the building?<\/p>\n\n\n\n
                            \n
                              \n
                            • Facts and unknown:<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                              \u00a0 \u00a0 Tom\u2019s height = 1.75 m<\/td>\n\u00a0 Building\u2019s height (x<\/em>) = ?<\/td>\n<\/tr>\n
                              \u00a0 \u00a0 Tom\u2019s shadow = 1.09 m<\/td>\n\u00a0 Building\u2019s shadow = 10m<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
                              \n
                                \n
                              • Proportion:<\/li>\n<\/ul>\n<\/td>\n
                              		\"\frac{1.75m}{1.09m}=\frac{xm}{10m}\"<\/td>\n<\/td>\n<\/tr>\n
                              \n
                                \n
                              • Cross multiply:<\/li>\n<\/ul>\n<\/td>\n
                              		\"(1.75)(10)=(1.09)(x)\"<\/td>\n<\/td>\n<\/tr>\n
                              \n
                                \n
                              • Solve for x<\/em>:<\/li>\n<\/ul>\n<\/td>\n
                              		\"x=\frac{(1.75)(10)}{1.09}=\frac{(\bcancel{1.09})x}{\bcancel{1.09}}\"<\/td>\nDivide both sides by 1.09.<\/td>\n<\/tr>\n
                              <\/td>\n\"x=\frac{(1.75)(10)}{1.09}\approx16.055\"<\/td>\n<\/td>\n<\/tr>\n
                              <\/td>\nThe building\u2019s height is 16.055m.<\/td>\n<\/td>\n<\/tr>\n
                              \n
                                \n
                              • Check:<\/li>\n<\/ul>\n<\/td>\n
                              		\"\frac{1.75m}{1.09m}=\frac{16.055m}{10m}\"<\/td>\nReplace x<\/em> with 16.055.<\/td>\n<\/tr>\n
                              <\/td>\n\"(1.75) (10) = (16.055) (1.09)\"<\/td>\n<\/td>\n<\/tr>\n
                              <\/td>\n\"17.5 = 17.5\"<\/td>\nCorrect!<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                               <\/p>\n

                              \n
                              \n

                              Example 1.4.3<\/p>\n<\/header>\n

                              \n

                              If 15 mL of medicine must be mixed with 180 mL of water, how many mL of medicine must be mixed in 230 mL of water?<\/p>\n\n\n\n\n\n\n
                              \n
                                \n
                              • Proportion:<\/li>\n<\/ul>\n<\/td>\n
                              		\"\frac{15\;mL}{180\;mL}=\frac{x\;mL}{230\;mL}\"<\/td>\n\"\frac{15\;mL\;medicine}{180\;mL\;water}=\frac{x\;mL\;medicine}{230\;mL\;water}\"<\/td>\n<\/tr>\n
                              \n
                                \n
                              • Cross multiply:<\/li>\n<\/ul>\n<\/td>\n
                              		\"(15)(230)=(180)(x)\"<\/td>\n<\/td>\n<\/tr>\n
                              \n
                                \n
                              • Solve for x<\/em>:<\/li>\n<\/ul>\n<\/td>\n
                              		\"x=\frac{(15\;mL)(230\;mL)}{180\;mL}\approx19.17\;mL\"<\/td>\n<\/td>\n<\/tr>\n
                              <\/td>\n19.17 mL of medicine must be mixed in 230 mL of water.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                               <\/p>\n

                              \n

                              Percent<\/strong><\/h1>\n<\/div>\n

                              Percent (%):<\/strong> one part per hundred.<\/p>\n

                              Converting between percent, decimals and fractions:<\/strong><\/p>\n\n\n\n
                              Conversion<\/strong><\/td>\nSteps<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n\n
                              Percent \u21d2 Decimal<\/strong><\/td>\nMove the decimal point two places to the left, then remove %.<\/td>\n31%\u00a0 =\u00a0 31.% = 0.31<\/td>\n<\/tr>\n
                              Decimal \u21d2 Percent<\/strong><\/td>\nMove the decimal point two places to the right, then insert %.<\/td>\n0.317 = 0. 317 = 31.7 %<\/td>\n<\/tr>\n
                              Percent \u21d2 Fraction<\/strong><\/td>\nRemove %, divide by 100, then simplify.<\/td>\n\u00a015% = \"\frac{15}{100}=\frac{3}{20}\"<\/td>\n<\/tr>\n
                              Fraction \u21d2 Percent<\/strong><\/td>\nDivide, move the decimal point two places to the right, then insert %.<\/td>\n\"\frac{1}{4}\" = 1\u00f74 = 0.25 = 25 %<\/td>\n<\/tr>\n
                              Decimal \u21d2 Fraction<\/strong><\/td>\nConvert the decimal to a percent, then convert the percent to a fraction.<\/td>\n0.35 = 35 % = \"\frac{35}{100}=\frac{7}{20}\"<\/p>\n

                              % = per one hundred<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                               <\/p>\n

                              There are two methods to solve percent problems:<\/strong><\/p>\n

                                \n
                              • Percent proportion method<\/li>\n
                              • Translation (translate the words into mathematical symbols.)<\/li>\n<\/ul>\n

                                 <\/p>\n

                                Percent proportion method: <\/strong><\/p>\n\n\n\n
                                \"\frac{Part}{Whole}=\frac{Percent}{100}\"<\/td>\nor<\/td>\n\"\frac{"is"\;number}{"of"\;number}=\frac{\%}{100}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                 <\/p>\n\n\n\n
                                Step<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n
                                <\/td>\n8<\/em><\/strong> percent of<\/em> <\/strong>what number is<\/strong> 4<\/em> <\/strong>?
                                \n<\/em><\/strong><\/td>\n<\/tr>\n
                                \n
                                  \n
                                • Identify the part, whole, and percent.<\/li>\n<\/ul>\n<\/td>\n
                                		\u21d1<\/p>\n

                                Percent<\/td>\n

                                		\u21d1<\/p>\n

                                Whole (x<\/em>)<\/td>\n

                                		\u21d1<\/p>\n

                                Part<\/td>\n<\/tr>\n

                                \n
                                  \n
                                • Set up the proportion equation.<\/li>\n<\/ul>\n<\/td>\n
                                		\"\frac{4}{x}=\frac{8}{100}\"<\/td>\n\"\frac{Part}{Whole}=\frac{\%}{100}\"<\/td>\n<\/td>\n<\/tr>\n
                                \n
                                  \n
                                • Solve for unknown (x<\/em>).<\/li>\n<\/ul>\n<\/td>\n
                                		\"x=\frac{(4)(100)}{8}=50\"<\/td>\n\"x=50\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                \n

                                <\/h1>\n<\/div>\n

                                Translation method: <\/strong>translate the words into mathematical symbols.<\/p>\n

                                  \n
                                • What ____\u00a0 x <\/em>:\u00a0\u00a0\u00a0 the word \u201cwhat\u201d represents an unknown quantity x<\/em>.<\/li>\n
                                • Is\u00a0 \u00a0 \u00a0 ____ \u00a0= : \u00a0\u00a0\u00a0the word \u201cis\u201d represents an equal sign.<\/li>\n
                                • of\u00a0 \u00a0 \u00a0____\u00a0 \u00d7 :\u00a0\u00a0\u00a0 the word \u201cof\u201d represents a multiplication sign.<\/li>\n
                                • % \u00a0\u00a0\u00a0to\u00a0\u00a0\u00a0\u00a0 decimal:\u00a0 \u00a0\u00a0always change the percent to a decimal.<\/li>\n<\/ul>\n

                                   <\/p>\n

                                  \n
                                  \n

                                  Example 1.4.4<\/p>\n<\/header>\n

                                  \n

                                  1) <\/strong>\u00a0\u00a0What \u00a0is \u00a015% \u00a0of \u00a080?<\/p>\n

                                  \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0x<\/em>\u00a0 =\u00a0 0.15 \u2022 80\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0\u00a0<\/strong>x<\/em> = (0.15)(80) = 12\u00a0\u00a0 <\/strong><\/p>\n

                                  \u00a0<\/strong>2) \u00a0<\/strong>\u00a0What \u00a0percent \u00a0of \u00a090 \u00a0is \u00a045?<\/p>\n

                                  \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong>x<\/em> \u2022 90 = 45\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/strong>\u00a0\u00a0x<\/em> = \"\frac{45}{90}\" = 0.5 = 50%\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/strong><\/p>\n

                                  \u00a0<\/strong>Divide both sides by 90.<\/p>\n

                                  \u00a0<\/strong>3) \u00a0<\/strong>\u00a012 \u00a0is \u00a08% \u00a0of \u00a0what number?<\/p>\n

                                  \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>12 = 0.08 \u2022 x<\/em> \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0\u00a0x<\/em> = \"\frac{12}{0.08}\" =\u00a0<\/strong>150 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n

                                  Divide both sides by 0.08.<\/p>\n<\/div>\n<\/div>\n

                                   <\/p>\n

                                  Percent increase or decrease:<\/strong><\/p>\n\n\n\n
                                  Application<\/strong><\/td>\nFormula<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                                  Percent increase<\/strong><\/td>\nPercent increase = \"\frac{New\;value-Original\;value}{Original\;value}\"<\/td>\n\"x=\frac{N-O}{O}\"<\/td>\n<\/tr>\n
                                  Percent decrease<\/strong><\/td>\nPercent decrease = \"\frac{Original\;value-New\;value}{Original\;value}\"<\/td>\n\"x=\frac{O-N}{O}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  \n
                                  \n

                                  Example 1.4.5<\/p>\n<\/header>\n

                                  \n

                                  A product increased production from 1500 last month<\/em><\/strong> to 1650 this month<\/em><\/strong>. \u00a0Find the percent increase<\/em><\/strong>.<\/strong><\/p>\n\n\n\n\n\n
                                  New value (N):<\/td>\n1650<\/td>\nThis month.<\/td>\n<\/tr>\n
                                  Original value (O):<\/td>\n1500<\/td>\nLast month.<\/td>\n<\/tr>\n
                                  Percent increase:<\/td>\n\"x=\frac{N-O}{O}=\frac{1650-1500}{1500}=0.1=10\%}\"<\/td>\nA 10% increase.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                   <\/p>\n

                                  A product was reduced<\/em><\/strong> from \"\bf{\$}\"33<\/em><\/strong> to \"\bf{\$}\"29<\/em>.<\/strong> \u00a0What percent reduction <\/em><\/strong>is this?<\/p>\n\n\n\n
                                  Percent decrease:<\/td>\n\"x=\frac{O-N}{O}=\frac{33-29}{33}\approx0.12=12\%\"<\/td>\nA 12 % decrease.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                                   <\/p>\n

                                  \n

                                  Practice questions<\/strong><\/h1>\n<\/div>\n

                                  1.<\/strong> Write the following as a ratio or rate in lowest terms:<\/p>\n

                                  a.<\/strong> 4 nickels to 16 nickels.<\/p>\n

                                  b.<\/strong> 350 people for 1500 tickets.<\/p>\n

                                  c.<\/strong> 160 kilometres per 740 minutes.<\/p>\n

                                  2.<\/strong> A train travelled 459 km in 6 hours. What is the unit rate?<\/p>\n

                                  3.<\/strong> Write the following sentence as a proportion: 24 hours is to 1,940 kilometres as 12 hours is to 985 kilometres. \u00a0\u00a0<\/strong><\/p>\n

                                  4.<\/strong> 4 litres of juice cost $7.38, how much do 2 litres cost?<\/p>\n

                                  5.<\/strong> Sarah earns $4,500 in 30 days. How much does she earn in 120 days?<\/p>\n

                                  6.<\/strong> A product increased production from 2,800 last year to 3,920 this year. Find the percent increase.<\/p>\n

                                  7.<\/strong> The table below gives survey data regarding Ryerson Occupational and Public Health students and summer jobs, but some data are missing:<\/p>\n\n\n\n
                                  Year<\/td>\n2015<\/td>\n2016<\/td>\n2017<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                                  Unemployed students<\/td>\n350<\/td>\n?<\/td>\n396<\/td>\n<\/tr>\n
                                  Total number of students<\/td>\n1250<\/td>\n1100<\/td>\n?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                  a.<\/strong> What % of students were unemployed in 2015?<\/p>\n

                                  b.<\/strong> If the % of unemployed students remained the same in 2016, find the number of unemployed students in 2016.<\/p>\n

                                  c.<\/strong> If the % of unemployed students increased by an additional 5% in 2017, find the total number of students in 2017.<\/p>\n

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