Exercises


  1. Determine the unit rate relating the two proportional quantities given each of the following tables.
    1. Time (s) Distance (m)
      \(0\) \(0\)
      \( 3\) \(6\)
      \(6\) \(12\)
      \(9\) \(18\)
    2. Number of Items Cost \(($)\)
      \(4\) \(6\)
      \(10\) \(15\)
      \(20\) \(30\)
      \(50\) \(75\)
  2. In a local store, almonds are sold at a rate of \($15\) per kg. Write an equation to represent the relationship between the cost, \(C\), in dollars, and the mass, \(m\), in kg.
  3. A company produces new shoes at a rate of \(175\) shoes per week. Assuming that the company works \(7\) days a week, write an equation to represent the relationship between production, \(P\), in number of shoes, and the time, \(d\), in days.
  4. If snow accumulates at a rate of \(1\) mm every \(6\) minutes, which graph most accurately represents the snow fall? Explain your answer.
    1. A graph with Time in minutes along the horizontal axis and Snow Accumulated in millimetres along the vertical axis. Two points are plotted at (3,3) and (6,6).
    2. A graph with Time in minutes along the horizontal axis and Snow Accumulated in millimetres along the vertical axis. Two points are plotted at (2,12) and (6,24).
    3. A graph with Time in minutes along the horizontal axis and Snow Accumulated in millimetres along the vertical axis. Two points are plotted at (3,0.5) and (6,1).
  5. The graph below represents the trips of two bikers, each travelling at a constant speed. Determine the speeds (as unit rates) of Biker 1 and Biker 2.
    A graph with Time in hours along the horizontal axis and Distance Travelled in kilometres along the vertical axis. Biker 1's trip is represented by six points at (0.5,10), (1,20), (1.5,30), (2,40), (2.5,50), and (3,60). Biker 2's trip is represented by three points at (1,15), (2,30), and (4,45).
  6. Track teams participate in a relay race. Each runner runs \(100\) m during the race for a total of \(400\) m. Each \(100\) m piece of the race is called a leg. Below is a table summarizing one team's performance in the race. 
    Time (s) Distance (m)
    \(0\) \(0\)
    \(6\) \(40\)
    \(15\) \(100\)
    \(29\) \(140\)
    \(50\) \(200\)
    \(65\) \(260\)
    \(75\) \(300\)
    \(87\) \(380\)
    \(90\) \(400\)
    1. Determine the four time intervals corresponding to each of the four legs of the race.
    2. Below is a graph of the data from the table above.
      A graph with the Time in seconds along the horizontal axis and Distance in metres along the vertical axis. The points plotted are provided in the table above.
      It is not possible to draw a single straight line that passes through all of the points on the graph, but it is possible to join all of the data points using four line segments. For each leg of the race, draw a line passing through all points during the time interval of that leg. What does the steepness of each line segment represent?
    3. Assuming that the runners each run at a constant speed (perhaps different from one runner to the next), determine the running speeds for each of the four runners.
    4. If you wanted to summarize the team's race using a single equation, then what might that equation be? 
  7. Consider the following graph representing the cost, \(C\), of renting a banquet hall based on the number of guests, \(n\).
    A graph with Number of Guests along the horizontal axis and Cost, in dollars, along the vertical axis. Six points are plotted at (0,20), (10,60), (20,100), (30,140), (40,180), and (50,220).
    1. Make a table containing the data from the graph. Verify that there is no multiplicative relationship between the two columns in the table and so the quantities are not proportional.
    2. Notice that a constant change in \(n\) results in a constant change in \(C\). If \(n\) increases by \(10\), then what is the corresponding increase in \(C\)? Create a unit rate (in dollars per guest) by dividing the corresponding change in \(C\) by the change in \(n\).
    3. Explain why the unit rate from part b) is not enough to describe the situation represented in the graph above. What extra information is needed?
  8. Tanner wants to fill his swimming pool using two hoses, each of which sprays at a constant rate. Hose A fills the pool in \(a\) hours when used by itself, where \(a\) is a positive integer. Hose B fills the pool in \(b\) hours when used by itself, where \(b\) is a positive integer.
    1. Write an equation that represents filling the pool using only Hose A. Do the same for Hose B.
    2. If the two hoses are used together, then the pool will fill faster. Write an equation that represents filling the pool using Hose A along with Hose B.
    3. If, when used together, Hose A and Hose B fill the pool in \(6\) hours, then how many different possibilities are there for the value of \(a\)?