Math problem about the gym

Question

One gym costs $24 per month and has a joining fee of $135. Another gym costs $39 per month but does not have a joining fee. How many months would it take for each gym to cost the same amount of money? 

Answer 1

One way to solve this problem is to consider what the two gyms costs would look like as linear equations then to plot points (or enter data into a graphing calculator) with a cost over time (months) in mind.

Cost equals per month fee times number of months plus joining fee or, in standard form:

Gym one:

f(x) = 24x + 135

Gym two:

g(x) = 39x (+ 0 [since there was no joining fee])

The first month is easy.

Gym one's cost is just the joining fee plus the one month of membership fee. Or...

159 = 24(1) + 135

Gym two's cost is just that first month's membership fee. Or...

39 = 39(1) + 0

I did a second month [f(x) = 183 = 24(2)+135; g(x) = 78 = 39(2)], then a tenth month [f(x) = 375 = 24(10) + 135; g(x) = 390 = 39(10)] and saw that gym two's cost exceeded the cost of gym one, so I knew I'd gone too far. I went back one month and found the answer. By month nine, both gyms had equalized at $351.

If you'd graphed, you'd see the intersection at coordinate (9, 351).

Hope this helps!

Answer 2

This question has two types of cost: fixed cost (joining fee) and variable cost (monthly fee). Gym a has both a fixed cost and a variable cost while gym b has only a variable cost. Whenever you see a question like this it helps to think about the following: total cost = fixed cost + variable cost. Since the question asks you to find how many months it would take for each gym to cost the same, you are looking for the total cost for gym a to equal the total cost for gym b.

Step 1: set up cost equations for gym a gym b

Tca= fixed cost a + variable cost a

Tcb= fixed cost b + variable cost b

*There is no fixed cost for b so this would be zero

* since we don't know how many months, we can use the variable x for months: $24/ month -> $24x

Step 2: since tca= tcb, we can set the two cost equations to equal each other

Tca=Tcb

Fixed cost a + variable cost a = fixed cost b (in this case 0) + variable cost b

Step 3: solve for x to determine how many months