The table below shows the total cost of buying printer ink cartridges for both members and nonmembers of a shopping club.

Price of Ink at Shopping Club
Number of ink cartridges purchased
Total cost for
members
(with membership fee)
Total cost for nonmembers
0
$30
0
1
$40
$25
2
$50
$50
3
$60
$75

Which statement is supported by the table?
1.The relationship between the number of cartridges purchased and the total cost is proportional for nonmembers but not for members.
2.The relationship between the number of cartridges purchased and the total cost is proportional for members but not for nonmembers.
3.The relationship between the number of cartridges purchased and the total cost is proportional for neither members nor nonmembers.
4.The relationship between the number of cartridges purchased and the total cost is proportional for both members and nonmembers.

1 answer

To determine the correct statement, we need to analyze the total cost for both members and nonmembers at each quantity of ink cartridges purchased to see if the relationship is proportional.

  1. Members:

    • 0 cartridges: $30
    • 1 cartridge: $40
    • 2 cartridges: $50
    • 3 cartridges: $60

    The total cost increases as follows:

    • From 0 to 1 cartridge: $40 - $30 = $10
    • From 1 to 2 cartridges: $50 - $40 = $10
    • From 2 to 3 cartridges: $60 - $50 = $10

    The increase is consistent ($10 per cartridge), but to check for proportionality from the origin (0,30), we also calculate the costs as ratios on an adjusted basis:

    Total cost for members starting with the $30 membership fee can be expressed as: \( \text{Total Cost} = 30 + 10n \) where n is the number of cartridges purchased. Therefore, it’s not a straight line through (0,0) but it can be linear in terms of additional purchases.

  2. Nonmembers:

    • 0 cartridges: $0
    • 1 cartridge: $25
    • 2 cartridges: $50
    • 3 cartridges: $75

    The total cost increases as follows:

    • From 0 to 1 cartridge: $25 - $0 = $25
    • From 1 to 2 cartridges: $50 - $25 = $25
    • From 2 to 3 cartridges: $75 - $50 = $25

    The increase is consistent ($25 per cartridge) and when plotted, starts at (0,0) making the relationship proportional.

Given these observations, we conclude that:

  • The members do not have a proportional relationship because they start with a $30 base fee; the total cost does not pass through (0,0).
  • The nonmembers have a proportional relationship since their costs start from (0,0).

With this analysis, the correct statement is: 1. The relationship between the number of cartridges purchased and the total cost is proportional for nonmembers but not for members.