Final answer:
After 43 sessions, both pricing plans offered by Justin will cost the same amount, which is $2445. The mathematical equation for each plan was set equal to each other to find the number of sessions where the cost is equal for both.
Step-by-step explanation:
The student's question involves finding the number of sessions after which two different pricing plans for personal training will cost the same amount. We have two plans: Plan A charges $80 for the initial consultation and then $55 per session, while Plan B charges $37 for the consultation and $56 per session.
To find out after how many sessions the cost will be the same, we can set up an equation, representing the total cost for each plan as a function of the number of sessions (n).
Plan A: Total cost = $80 (initial consultation) + $55n (cost per session)
Plan B: Total cost = $37 (initial consultation) + $56n (cost per session)
We want the total cost of Plan A to equal the total cost of Plan B:
$80 + $55n = $37 + $56n
Solving for n, we subtract $55n from both sides and then subtract $37 from both sides to get:
$80 - $37 = $56n - $55n
$43 = n
So, after 43 sessions, the two plans will cost the same amount.
Now to find the total cost at this point, we plug the number of sessions (n = 43) into the total cost equation of either plan:
Total cost for 43 sessions in Plan A = $80 + $55 * 43
We can calculate this to find the cost:
Total cost for 43 sessions in Plan A = $80 + $55 * 43 = $80 + $2365 = $2445
Therefore, after 43 sessions, both plans will have a total cost of $2445.