Brett is joining a new gym. He can join Power Plus or Heavy Hulk. At Power Plus, it will cost $34.58 per month, plus a sign-up fee of $22.00. At Heavy Hulk, it will cost $34.83, plus a sign-up fee of $34.50. Right now, Heavy Hulk is cutting their sign-up fee in half for new members. If Brett signs up at a gym today, after how many months will the cost of the memberships be the same?

Question

Brett is joining a new gym. He can join Power Plus or Heavy Hulk. At Power Plus, it will cost $34.58 per month, plus a sign-up fee of $22.00. At Heavy Hulk, it will cost $34.83, plus a sign-up fee of $34.50. Right now, Heavy Hulk is cutting their sign-up fee in half for new members. If Brett signs up at a gym today, after how many months will the cost of the memberships be the same?
  A. 
The cost will never be the same.
  B. 
50
  C. 
19
  D. 
12.25

GPT-4o mini · Answer

To solve this problem, we will set up an equation for the total cost of each gym based on the number of months Brett is a member.

Let $ m $ be the number of months.

Power Plus:

Monthly cost: $34.58
Sign-up fee: $22.00

Total cost for Power Plus after $ m $ months: \[ \text{Cost}_{PP} = 34.58m + 22 \]

Heavy Hulk:

Monthly cost: $34.83
Original sign-up fee: $34.50 (now cut in half, so it will be $ 34.50 / 2 = 17.25 $)

Total cost for Heavy Hulk after $ m $ months: \[ \text{Cost}_{HH} = 34.83m + 17.25 \]

Now, we want to find the number of months $ m $ when the costs are equal: \[ 34.58m + 22 = 34.83m + 17.25 \]

To solve for $ m $, first, rearrange the equation: \[ 34.58m - 34.83m = 17.25 - 22 \] \[ -0.25m = -4.75 \]

Now, divide both sides by -0.25: \[ m = \frac{-4.75}{-0.25} \] \[ m = 19 \]

Therefore, after 19 months, the cost of the memberships will be the same.

The correct answer is: C. 19