A fitness club offers two membership options. The first option has an initial fee of $50 and then charges $30 per month. The second option has no initial fee but charges $40 per month. After how many months, x, will the first option be a better value than the second?
This problem works off of the relationship between two equations. In this case, the equations represent the two offered fitness plans. We will start by setting up our two equations:
y = 30x + 50
y = 40x
Our goal is to find when the first option will come out to be better in the long run, meaning that the first plan will cost us less money than the second plan after "x" months. You can use a plug-and-chug method to find this, where you just replace "x" with any number and use that for both equations or.
If you have access to a graphing calculator, you can find the point where the two lines intersect and just assume that "x + 1" will be when the first plan is the better buy.
To determine after how many months the first option will be a better value than the second option, we need to set up an equation and solve for x.
Let's assume that the total cost of the first option after x months is A, and the total cost of the second option after x months is B.
For the first option:
Initial fee = $50
Monthly fee = $30
Total cost after x months = A = 50 + 30x
For the second option:
Initial fee = $0
Monthly fee = $40
Total cost after x months = B = 40x
Now we want to find the value of x when A is less than B. Mathematically, this can be represented as:
A < B
50 + 30x < 40x
To solve this inequality, we can subtract 30x from both sides:
50 < 40x - 30x
50 < 10x
Next, divide both sides by 10:
50/10 < x
5 < x
Therefore, the first option will be a better value than the second option after more than 5 months.