Question

Geno wants to purchase gym membership. He has no more than y dollars to spend. Total Fitness charges an initial fee of $100 plus $30 per month. Gymania charges initial fee of $25 plus $50 per month. Write a system of equations that can be used to determine which company offers the better deal.

Answer 1

Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.

Explanation:

Let the number of months be 'x'.

Given:

Money Geno has = 'y' dollars.

Total Fitness charges:

Monthly fee = $30

Initial fee = $100

Gymania charges:

Monthly fee = $50

Initial fee = $25

Total charges is equal to the sum of initial fee and monthly fee multiplied by number of months.

So, for 'x' months, monthly fee charged by Total Fitness =
`30x`

For 'x' months, monthly fee charged by Gymania =
`50x`

Now, total charge by Total Fitness = Initial fee + Fee for 'x' months

Total charge by Total Fitness =
`100+30x`

Now, total charge by Gymania = Initial fee + Fee for 'x' months

Total charge by Gymania =
`25+50x`

Now, Geno has only 'y' dollars to spend. So, 'y' must be less than or equal to the total charge.

Therefore, the total charge for each membership is:

`y=30x+100\\\\y= 50x+25`

Now, we graph both the equations. The graph is shown below.

From the graph, it is clear that, the total cost for Gymania (blue line) is less than that of Total Fitness (red line) till number of months equals 3.75 or 3 months. After 3.75 months, the graph of Gymania is above Total Fitness. So, if the membership is 4 months or above, then Total Fitness is more efficient.

Therefore, Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.