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Chad created a table that shows the ratio of his sports game cards.

Chad’s Cards
Card Type
Outfield
Infield
Catcher
Pitchers
Quantity
3
4
1
2

Next, Chad created the graph below showing possible ratios for Pitchers cards to Infield cards in his deck. Which of the following statements are true?

On a coordinate plane, a graph titled Chad's Cards has Damage on the x-axis and Defense on the y-axis. Points (1, 2), (2, 4), (3, 6), and (4, 8) are plotted.
Chad should not have included the values (1, 2) and (3, 6) because that would mean he only had half a Catcher card.
Chad should not have included the values (1, 2) and (3, 6) because the ratio would be different for these points.
Chad should have changed the axes for the Infield and Pitchers cards so it was the same order as the table.
Chad needs to switch the order of each (x, y) value to show the decreasing values of the Pitchers cards.

1 answer

Let's analyze the information and statements provided regarding Chad's sports game cards and the graph he created.

Chad's card quantities are:

  • Outfield: 3
  • Infield: 4
  • Catcher: 1
  • Pitchers: 2

The specific ratio we are analyzing is the ratio of Pitchers cards to Infield cards. Initially, Chad has:

  • Pitchers: 2
  • Infield: 4

The ratio of Pitchers to Infield cards is \( \frac{2}{4} = \frac{1}{2} \). This means for every 1 Pitcher card, there are 2 Infield cards.

Now let's look at the four plotted points on the graph: (1, 2), (2, 4), (3, 6), and (4, 8). These points represent the ratios of Pitchers (x-axis) to Infield (y-axis). Here’s how we evaluate the statements:

  1. Chad should not have included the values (1, 2) and (3, 6) because that would mean he only had half a Catcher card.

    • False: The points represent the ratio of cards, not whole cards. The value (1, 2) means that for 1 Pitcher card, there are 2 Infield cards. This is a valid representation of the ratio.

  2. Chad should not have included the values (1, 2) and (3, 6) because the ratio would be different for these points.

    • False: The points (1, 2) and (3, 6) are in line with the established ratio of \( \frac{1}{2} \). All points adhere to the same ratio, so including them is justifiable.

  3. Chad should have changed the axes for the Infield and Pitchers cards so it was the same order as the table.

    • True: In the context of the table, Infield cards are more numerous than Pitchers. However, since the graph is comparing Pitchers to Infield, the axes listed can remain as is for interpreting the ratio.

  4. Chad needs to switch the order of each (x, y) value to show the decreasing values of the Pitchers cards.

    • False: The plotted points correctly reflect Pitchers on the x-axis and Infield count on the y-axis, representing the ratio accurately as determined by Chad's quantities. There isn’t any necessity or significance in switching.

In summary, the only statement that can be agreed upon as true is the third one regarding the need to align the axes with the table representation.