Josiah just became a personal trainer and is finalizing his pricing plans. One plan is to charge $77 for the initial consultation and $71 per session. Another plan is to charge $61... Josiah just became a personal trainer and is finalizing his pricing plans. One plan is to charge $77 for the initial consultation and $71 per session. Another plan is to charge $61 for the consultation and $73 per session. Josiah 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 is asking us to write a system of equations to model the pricing plans of a personal trainer named Josiah and find the costs and number of sessions where the two plans are equal. We will set up the equations based on the provided information and solve for the unknowns.
Answer
The cost on either plan is $645 for 8 sessions.
Answer for screen readers
The cost on either plan is $645 for 8 sessions.
Steps to Solve
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Define the Variables
Let $x$ be the number of sessions. We need to write the cost equations for both pricing plans based on the provided information. -
Write the Cost Equations
- For the first plan (initial consultation of $77 and $71 per session):
The cost can be represented as:
$$ C_1 = 77 + 71x $$ - For the second plan (initial consultation of $61 and $73 per session):
The cost can be represented as:
$$ C_2 = 61 + 73x $$
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Set the Equations Equal
To find the number of sessions where both plans cost the same, set the two cost equations equal to each other:
$$ 77 + 71x = 61 + 73x $$ -
Solve for x
Rearranging the equation:
$$ 77 - 61 = 73x - 71x $$
$$ 16 = 2x $$
Dividing both sides by 2 gives:
$$ x = 8 $$ -
Calculate the Costs
Substituting $x = 8$ into either cost equation to find the total cost:
Using $C_1$:
$$ C_1 = 77 + 71(8) = 77 + 568 = 645 $$
The cost on either plan is $645 for 8 sessions.
More Information
The two plans become equal in cost after 8 sessions, where both plans will charge $645. This scenario helps show how different pricing structures can lead to equal costs at certain quantities.
Tips
- Forgetting to set the equations equal: Ensure that both cost expressions are set to each other for solving.
- Incorrectly simplifying equations: Be careful with arithmetic when moving terms around.
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