I’m membership at gym A cost $50 for five months . A membership at gym B down the street cause $40 for three months you write two equations in the form of y=kx to try to figure out which membership would be cheaper for a year. What is the value of k for the cheaper membership?
Let's define x as the number of months and y as the total cost in dollars.
For gym A, the equation would be y = 10x since the cost is $50 for 5 months.
For gym B, the equation would be y = (40/3)x since the cost is $40 for 3 months.
To compare both memberships for a year (12 months), we need to find the total cost for each equation when x = 12.
For gym A: y = 10(12) = 120.
For gym B: y = (40/3)(12) = 160.
Comparing the total costs, we can see that gym A is cheaper for a year. Therefore, the value of k for the cheaper membership (gym A) is 10.
Let's define the cost of membership at gym A as y₁ and the number of months as x₁. Similarly, let's define the cost of membership at gym B as y₂ and the number of months as x₂.
Based on the given information, we have:
y₁ = 50 (cost of gym A membership)
x₁ = 5 (number of months for gym A membership)
y₂ = 40 (cost of gym B membership)
x₂ = 3 (number of months for gym B membership)
Now, let's find the value of k for each membership by using the equation y = kx:
For gym A:
y₁ = k₁ * x₁
50 = k₁ * 5
Simplifying this equation, we find:
k₁ = 50 / 5
k₁ = 10
For gym B:
y₂ = k₂ * x₂
40 = k₂ * 3
Simplifying this equation, we find:
k₂ = 40 / 3
k₂ ≈ 13.33
Comparing the values of k₁ (10) and k₂ (13.33), we can see that k₁ is the smaller value. Therefore, the value of k for the cheaper membership is k₁ and the cheaper membership is gym A.
To determine which gym membership would be cheaper for a year, we can compare the total cost of each membership over 12 months.
For gym A:
Cost of membership for 5 months = $50
Cost of membership for 12 months can be calculated using a proportion:
x dollars / 5 months = $50 / 1 month
Simplifying the equation, we have:
x = $50 * 12 / 5 = $120
Therefore, the cost of the gym A membership for 12 months is $120.
For gym B:
Cost of membership for 3 months = $40
Cost of membership for 12 months can be calculated using a proportion:
x dollars / 3 months = $40 / 1 month
Simplifying the equation, we have:
x = $40 * 12 / 3 = $160
Therefore, the cost of the gym B membership for 12 months is $160.
We can represent the cost y of a membership at each gym as a linear function of time x (measured in months):
For gym A: y = kx, where k represents the rate of change of the cost per month.
For gym B: y = kx, where k represents the rate of change of the cost per month.
To find the value of k for the cheaper membership, we compare the slope of the two linear equations. The membership with the smaller slope (k value) will be cheaper for a year.
Comparing the values we obtained:
k for gym A = ($120 - $0) / 12 = $10/month
k for gym B = ($160 - $0) / 12 = $13.33/month
Since gym A has a smaller value of k ($10), the membership at gym A is cheaper for a year compared to gym B.