To determine which gym membership is cheaper for a year, we need to find the total cost for each gym when considering their respective membership terms.
For Gym A: The cost is $50 for 5 months. To find the cost per month (k), we can express the cost for one month as follows:
\[ y_A = k_A x \]
where:
- \( k_A = \frac{50}{5} = 10 \) (cost per month for Gym A)
- \( x \) is the number of months
The cost equation for Gym A over 12 months (or 1 year) would be:
\[ y_A = 10x \] where \( x \) would be 12 months.
Now, to calculate the total cost for a year:
Since Gym A has an effective 5-month cycle, we solve:
\[ \text{Total for a year at Gym A} = \left(\frac{12}{5}\right) \times 50 = 2.4 \times 50 = 120 \]
For Gym B: The cost is $40 for 3 months. We express the cost for one month (k) similarly:
\[ y_B = k_B x \]
where:
- \( k_B = \frac{40}{3} \) (approximately $13.33 per month for Gym B)
- \( x \) is the number of months
The cost equation for Gym B over 12 months would be:
\[ y_B = \left(\frac{40}{3}\right)x \]
Now, to calculate the total cost for a year:
\[ \text{Total for a year at Gym B} = \left(\frac{12}{3}\right) \times 40 = 4 \times 40 = 160 \]
Now we compare the yearly totals for both gyms:
- Total for Gym A for one year: $120
- Total for Gym B for one year: $160
Since Gym A is cheaper, the value of \( k \) for the cheaper membership (Gym A) is:
\[ \boxed{10} \]
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