Question

The following table represents the total cost, in dollars (y) to join a gym for x number of months. The cost includes a one-time joining fee of 10$. Does the data in the table represent a proportional relationship or a non-proportional relationship? how do you know?

X= 1 2 3 4 5
Y= 25 40 55 70 85

Answer 1

let y' is the monthly cost to join the gym
∵ The cost includes a one-time joining fee 10$
∵ y' = y -10
for x = 1 ⇒ y' = 25 - 10 = 15 ⇒⇒⇒ (y'/x) = 15
for x = 2 ⇒ y' = 40 - 10 = 30 ⇒⇒⇒ (y'/x) = 15
for x = 3 ⇒ y' = 55 - 10 = 45 ⇒⇒⇒ (y'/x) = 15
for x = 4 ⇒ y' = 70 - 10 = 60 ⇒⇒⇒ (y'/x) = 15
for x = 5 ⇒ y' = 85 - 10 = 75 ⇒⇒⇒ (y'/x) = 15

∵ y'/x = constant = 15 ⇒⇒⇒ y' = 15x

∴ y = y' + 10
y = 15x +10

The last equation in the form ⇒⇒⇒ y = m x + c
Which represents a non-proportional relationship
So,
The data in the table represent a non-proportional relationship.
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Another solution:
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A proportional relationship has the form y = m x
Or (y/x) = m = constant
So,
By dividing the values of y by the values of x we will not get a constant value

∴ The data in the table represent a non-proportional relationship.