Answers and Solutions


  1. In a tennis court, the ratio of length to width is \(24:8\), which simplifies to \(3:1\). In a soccer field, this ratio is \(100:75\), which simplifies to \(4:3\). Since the ratios are not equivalent, the fields are not in the same proportion.
  2. The ratio of length to width in the original image is \(12:8\) or \(3:2\). 
    1. Since the ratio of length to width in the enlarged image is \(30:15\) or \(2:1\), this is not equivalent to the original ratio. Therefore, the enlarged image is not in proportion.
    2. The width should be \(20\) cm to get a ratio of \(3:2\).
    3. The length should be \(22.5\) cm to get a ratio of \(3:2\). Since \(2 \times 7.5 = 15\), we need a length of \(3 \times 7.5 = 22.5\) to get the needed ratio of \(3:2\).
  3. The ratios of shaded area to total area are as follows:
    Figure A: \(1:4\)
    A triangle is divided into 4 equal parts, 1 of the parts is shaded.
    Figure B: \(2:6\) or \(1:3\)
    A hexagon is divided into 6 equal parts, 2 of the parts are shaded.
    Figure C: \(4:16\) or \(1:4\)
    A square is divided into 16 equal parts, 4 of the parts are shaded.
    Figure D: \(3:9\) or \(1:3\)
    A rectangle is divided into 9 equal parts, 3 of the parts are shaded.
    Figure A and Figure C have the same proportion of shaded area to total area, and Figure B and Figure D have the same proportion of shaded area to total area.
    1. We want equivalent rates of the form \(\dfrac{$25}{12 \text{ bracelets}}\) and \(\dfrac{\boxed{\phantom\square}}{60 \text{ bracelets}}\). Notice that \(12 \times 5 = 60\). Therefore, the missing numerator is \(25 \times 5 = 125\). You would be paid \($ 125\) for \(60\) bracelets.
    2. We want equivalent rates of the form \(\dfrac{$25}{12 \text{ bracelets}}\) and \(\dfrac{$175}{\boxed{\phantom\square} \text{ bracelets}}\). Notice that \(25 \times 7 = 175\). Therefore, the missing denominator is \(12 \times 7 = 84\).  Your payment of \($ 175\) must have been for \(84\) bracelets.
    1. You made \(12\) bracelets in \(36\) minutes, and \(24\) bracelets in \(60\) minutes. Since \(12 \times 2 = 24\) but \(36 \times 2 = 72\), the rate for making \(12\) bracelets is not equivalent to the rate for making \(24\) bracelets. Therefore, this situation is not proportional.
    2. If you are making new bracelets, then the first batch will likely take the most time to make. This might be because you are getting used to the pattern for the bracelet. As you become more comfortable, you will likely speed up your production. From the table above, each of the four batches of \(12\) takes less time than the previous batch. 
  6. The side lengths of \(\Delta ABC\) are \(3\), \(4\), and \(5\) units.
    1. The ratio of the smallest to the largest side length must be \(3:5\), which is equivalent to \(9:15\). This means the smallest side must have a length of \(9\) cm.
      The ratio of second largest to the largest side length must be \(4:5\) or \(12:15\). This means the second largest side must have a length of \(12\) cm.
    2. Since \(3 \times 0.25 = 0.75\), the other side lengths of the triangle must be \(4 \times 0.25 = 1\) cm and \(5 \times 0.25 = 1.25\) cm in length. 
  7. In this solution, we do not assume that the length of a rectangle must be longer than the width. 
    1. The simplest option is to pick all of the dimensions to be \(5\) mm. Then \(\ell:w\) and \(L:W\) are both \(5:5\) or \(1:1\). Since \(5 : 10\) is equivalent to \(2.5:5\), another possibility is to have \(w = 10\) and \(L = 2.5\). You can generate many more possibilities by trying to choose values for \(w\) and \(L\) that make the following equation true:\[\frac{5}{w}= \frac{L}{5}\]
    2. The area of the first rectangle is \(a = \ell\times w = 2\ell\) since \(w = 2\).
      The area of the second rectangle is \(A = L \times W = 4L\) since \(W=4\).
      So, the ratio of the areas (of the first rectangle to the second rectangle) is \(2\ell : 4L\) or \(\ell : 2L\) when simplified. It turns out that this ratio is always equivalent to the ratio \(1:4\). You can convince yourself of this fact by testing some \(\ell\) and \(L\) values that are in the correct proportions.
      Using algebra, we can explain why we always get \(1:4\):
      Since \(\ell\) and \(L\) must satisfy the equation\[\frac{\ell}{w}=\frac{L}{W} \text{ or } \frac{\ell}{2}=\frac{L}{4}\]it must be the case that \(L\) is twice as large as \(\ell\), or \(2\ell = L\) which means \(4\ell = 2L\). Since \(2 L = 4 \ell \) the ratio \(\ell : 2L\) is the same as the ratio \(\ell : 4 \ell\) which is equivalent to \(1 : 4\). Therefore the ratio of the areas of the rectangles is \(1:4\).