Answers and Solutions


    1. \(\dfrac{6}{10} = \dfrac{60}{100} = 60 \%\)
    2. \(\dfrac{13}{20} = \dfrac{65}{100}=65\%\)
    3. \(\dfrac{13}{25}=\dfrac{52}{100}=52\%\)
    4. \(\dfrac{3}{4}=\dfrac{75}{100} = 75\%\)
    5. \(\dfrac{6}{5} = \dfrac{120}{100} = 120\%\)
    1. \(\dfrac{5}{8}\) is half way between \(\dfrac{4}{8} = 0.5\) and \(\dfrac{6}{8} = \dfrac{3}{4} = 0.75\) which is \(0.625\). Therefore, \(\dfrac{5}{8}\) is \(62.5\%\).
    2. \(\dfrac{14}{16} = \dfrac{7}{8}\) is half way between \(\dfrac{6}{8} = 0.75\) and \(\dfrac{8}{8} =1\) which is \(0.875\). Therefore, \(\dfrac{14}{16}\) is \(87.5\%\).
    3. \(\dfrac{11}{40}\) is half way between \(\dfrac{10}{40} = 0.25\) and \(\dfrac{12}{40} = \dfrac{3}{10} = 0.3\) which is \(0.275\). Therefore, \(\dfrac{11}{40}\) is \(27.5\%\).
    4. \(\dfrac{12}{32}=\dfrac{3}{8}\) which is solved in the question. Therefore, \(\dfrac{12}{32}\) is \(37.5\%\).
    5. \(\dfrac{15}{24} = \dfrac{5}{8}=0.625\) from part a). Therefore, \(\dfrac{15}{24}\) is \(62.5\%\).
  3. Since \(\dfrac{2}{3}=0.\bar{6} = 0.66\bar{6}\), this fraction can be expressed as a percent as \(66.\bar{6}\%\). Since the decimal representation of \(\dfrac{2}{3}\) is a repeating decimal, this is also true of the percent. Since the decimal representation of \(\dfrac{2}{7}\) starts with \(0.2857...\), which is approximately \(0.286\), we might write an approximate percent value of \(28.6\%\). Could you write the exact percent? 
  4.   Starting Fraction \(\dfrac{\boxed{\phantom\square}}{100}\) Decimal Value Percent
    a. \(\dfrac{1}{20}\) \(\dfrac{5}{100}\) \(0.05\) \(5\%\)
    b. \(\dfrac{3}{50}\) \(\dfrac{6}{100}\) \(0.06\) \(6\%\)
    c. \(\dfrac{2}{25}\) \(\dfrac{8}{100}\) \(0.08\) \(8\%\)
    d. \(\dfrac{3}{40}\) \(\dfrac{7.5}{100}\) \(0.075\) \(7.5\%\)
    e. \(\dfrac{6}{200}\) \(\dfrac{3}{100}\) \(0.03\) \(3\%\)
    f. \(\dfrac{36}{400}\) \(\dfrac{9}{100}\) \(0.09\) \(9\%\)
    g. \(\dfrac{9}{360}\) \(\dfrac{2.5}{100}\) \(0.025\) \(2.5\%\)
      1. \(\dfrac{2}{5}\) or \(40\%\)
      2. \(\dfrac{3}{5}\) or \(60\%\)
      3. \(40\% + 60\% = 100\%\)
        Since all of the shapes are either small or tall, the sum of the two percents should equal \(1\) or \(100\%\).
      1. \(\dfrac{3}{5}\) or \(60\%\)
      2. \(\dfrac{2}{5}\) or \(40\%\)
      3. \(60\% + 40\% = 100\%\)
        Since all of the shapes are either shaded or not shaded, the sum of the two percents should equal \(1\) or \(100\%\).
      1. \(\dfrac{1}{5}\) or \(20\%\)
      2. \(\dfrac{1}{5}\) or \(20\%\)
      3. \(20\% + 20\% = 40\%\).
        Since there are shapes that are neither small and shaded nor tall and not shaded, the sum of the two percents should be less than \(1\) or \(100\%\). 
    1. Ahmad received \(85\%\) in science, \(80\%\) in geography, and \(84\%\) in French.
    2. Let's call these two percents Percent 1 and Percent 2, in order. Percent 2 should be higher than Percent 1. In Percent 1, we are giving the same weight to each of the assignments, regardless of how many marks were available in each assignment. In Percent 2, we are giving a larger weight to any assignment that is out of a larger number of marks. For example, an assignment out of \(80\) would carry more weight than an assignment out of \(21\) or \(40\). Since Ahmad's lowest mark corresponds to the assignment out of \(80\), this will bring his mark down in Percent 2. We see in our calculations that, in fact, Percent 2 is lower than Percent 1:
      • Since \(\dfrac{85+80+84}{3} = 83\), Percent 1 is \(83\%\).
      • Since \(\dfrac{34+64+21}{40+80+25} =\dfrac{119}{145}\), which is approximately \(0.821\), Percent 2 is \(82\%\).
    1. Since the perimeter of the hexagon is \(2 \times 6 = 12\) cm, the side length is \(\dfrac{2}{12} = \dfrac{1}{6}\) of the perimeter. This fraction is \(16.\bar{6}\% \approx 16.7\%\).
    2. Since the side length of the hexagon is \(2\) cm, and a hexagon is made up of \(6\) equilateral triangles as shown, the length of the line connecting diametrically opposite vertices is \(4\) cm.
      The perimeter of the trapezoid is then \(3 \times 2 + 4 = 10\) cm. The base is \(\dfrac{4}{10} = \dfrac{2}{5}\) of the perimeter of the trapezoid. This fraction is \(40\%\).
    1. Since \(15 \times 1.3 = 19.5\) and \(16\times 1.3 = 20.8\), and the floor requires at least \(20\) m\(^2\), the minimum number of boxes that Rod can purchase is \(16\). 
    2. Since \(10\%\) of \(20\) is \(0.1 \times 20 = 2\), Rod should buy enough boxes to get at least \(22\) m\(^{2}\) of flooring in total. Since \(17 \times 1.3 = 22.1\), Rod should purchase at least \(17\) boxes.
  9. Let's assume that the jug contains \(1\) L of water at the start. The following table shows the quantity of water poured into each glass and the quantity of water remaining after each pour.  We stop when the quantity of water remaining in the jug is less than \(0.5\) L.
    Number of glasses Number of litres poured Number of litres remaining
    1 \(10\% \text{ of } 1= 0.1\) \(1-0.1 = 0.9\)
    2 \(10 \% \text{ of } 0.9 = 0.09\) \(0.9-0.09=0.81\)
    3 \(10\% \text{ of } 0.81 = 0.081\) \(0.81 - 0.081 = 0.729\)
    4 \(10\% \text{ of } 0.729 = 0.0729\) \(0.729 - 0.0729 = 0.6561\)
    5 \(10 \% \text{ of } 0.6561 = 0.06561\) \(0.6561-0.06561 = 0.59049\)
    6 \(10 \% \text{ of } 0.59049 = 0.059049\) \(0.59049 - 0.059049 = 0.531441\)
    7 \(10\% \text{ of } 0.531441 = 0.0531441\) \(0.531441 - 0.0531441 = 0.4782969\)

    We can see from the table that the minimum number of glasses that Kim must pour so that less than half of the water remains in the jug is \(7\).

  10. Let's suppose that there are \(100\) songs on the updated playlist. Since \(40\%\) of the songs on the updated playlist are Country, the remaining \(100\% - 40\%\) or \(60\%\) must be either Hip Hop or Pop songs. This means there are \(40\) Country songs and \(60\) songs that are either Hip Hop or Pop. Since the ratio of Hip Hop songs to Pop songs does not change, then \(65\%\) of these \(60\) songs must be Hip Hop songs. Since \(65\%\) of \(60\) is \(0.65 \times 60 = 39\), the updated playlist must have \(39\) Hip Hop songs. Since the total number of songs is \(100\) this means that \(39\%\) of the total number of songs on the playlist are Hip Hop songs. (Note that we could also find this final percentage directly by multiplying \(65\% \times 60\% = 0.65 \times 0.6 = 0.39 = 39\%\).)