Rodrigo just became a personal trainer and is finalizing his pricing plans. For one plan, Rodrigo charges $39 for the initial consultation and $42 per session. For another plan, he... Rodrigo just became a personal trainer and is finalizing his pricing plans. For one plan, Rodrigo charges $39 for the initial consultation and $42 per session. For another plan, he charges $54 for the consultation and $39 per session. Rodrigo realizes that the two plans have the same cost for a certain number of sessions. What is that cost? How many sessions is that?
Understand the Problem
The question involves calculating the cost of two different training plans that share the same total cost after a certain number of sessions. It requires setting up a system of equations to find both the cost of the plans and the number of sessions needed to reach that cost.
Answer
Both plans cost $249 for 5 sessions.
Answer for screen readers
Both plans cost $249 for 5 sessions.
Steps to Solve
- Define the variables
Let ( x ) be the number of sessions.
Let ( C_1 ) be the total cost of the first training plan, and ( C_2 ) be the total cost of the second training plan.
- Set up the costs for each plan
For the first plan: $$ C_1 = 54 + 39x $$
For the second plan: $$ C_2 = 39 + 42x $$
- Set the costs equal
Since both plans cost the same after ( x ) sessions: $$ 54 + 39x = 39 + 42x $$
- Solve the equation for ( x )
First, rearrange the equation:
$$ 54 - 39 = 42x - 39x $$
This simplifies to:
$$ 15 = 3x $$
Now divide both sides by 3: $$ x = 5 $$
- Calculate the total cost for either plan
Substitute ( x = 5 ) into either cost equation. Using the first plan: $$ C_1 = 54 + 39(5) = 54 + 195 = 249 $$
So, both plans cost $249 for 5 sessions.
Both plans cost $249 for 5 sessions.
More Information
Rodrigo's pricing strategy highlights how different plans can ultimately lead to the same total cost depending on the number of sessions chosen. This scenario showcases the importance of analyzing pricing structures in various service industries.
Tips
- Forgetting the initial consultation fee: Many might overlook the fact that an initial fee is included in the total cost. Make sure to include it in your calculations.
- Misaligning the variable in the equation: Ensure that the session variable ( x ) is consistently applied across both equations.
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