A membership at Gym A costs $50 for 5 months. A membership at Gym B down the street costs $40 for 3 months. You write two equations in the form of y=kx to try and figure out which membership would be cheaper for a year. What is the value of I for the cheaper membership?

Question

A membership at Gym A costs $50 for 5 months. A membership at Gym B  down the street costs $40 for 3 months. You write two equations in the form of y=kx to try and figure out which membership would be cheaper for a year. What is the value of I for the cheaper membership?

GPT-4o mini · Answer

To determine which gym membership is cheaper for a year, we can set up equations in the form of $ y = kx $, where $ y $ is the total cost, $ k $ is the cost per period, and $ x $ is the number of periods in a year.

Gym A:

Cost for 5 months: $50
To find the cost per month: \[ \text{Cost per month} = \frac{50}{5} = 10 \]
The equation for Gym A for $ x $ months is: \[ y_A = 10x \]
Since there are 12 months in a year, the total cost for a year (12 months) at Gym A is: \[ y_A = 10 \cdot 12 = 120 \]

Gym B:

Cost for 3 months: $40
To find the cost per month: \[ \text{Cost per month} = \frac{40}{3} \approx 13.33 \]
The equation for Gym B for $ x $ months is: \[ y_B = \frac{40}{3} \cdot x \]
To find the total cost for 12 months at Gym B, we substitute $ x = 12 $ into the equation: \[ y_B = \frac{40}{3} \cdot 12 = 160 \]

Now, let's write both total costs clearly:

Total cost at Gym A for a year: $ 120 $
Total cost at Gym B for a year: $ 160 $

Comparing the two costs:

Gym A costs $120 for a year.
Gym B costs $160 for a year.

Since Gym A's membership is cheaper than Gym B's membership for a year, the value of $ I $ (the cheaper membership) is:

\[ \boxed{120} \]