A personal trainer charges a fixed fee for the first 3 hours and an hourly fee for each additional hour. If the fee charged for 6 hours is $495 and the fee charged for 9 hours is $720, what is the fee charged, in dollars, for 4 hours of training?
Answer (2)
Define the fixed fee as f and the hourly fee as h .
Set up a system of equations based on the given information: f + 3 h = 495 and f + 6 h = 720 .
Solve for h by subtracting the equations: h = 75 .
Solve for f by substituting h into one of the equations: f = 270 .
Calculate the fee for 4 hours: f + h = 345 .
The fee charged for 4 hours of training is 345 .
Explanation
Define variables Let f be the fixed fee for the first 3 hours and h be the hourly fee for each additional hour.
Fee for 6 hours For 6 hours, the fee is f + ( 6 − 3 ) h = f + 3 h = 495 .
Fee for 9 hours For 9 hours, the fee is f + ( 9 − 3 ) h = f + 6 h = 720 .
System of equations We now have a system of two equations:
f + 3 h = 495
f + 6 h = 720
Solve for hourly fee Subtract the first equation from the second equation to eliminate f and solve for h :
( f + 6 h ) − ( f + 3 h ) = 720 − 495
3 h = 225
h = 3 225 = 75
Solve for fixed fee Substitute the value of h back into the first equation to solve for f :
f + 3 ( 75 ) = 495
f + 225 = 495
f = 495 − 225 = 270
Calculate fee for 4 hours For 4 hours, the fee is f + ( 4 − 3 ) h = f + h = 270 + 75 = 345 .
Final Answer Therefore, the fee charged for 4 hours of training is $345.
Examples
Imagine you're planning a pizza party. The trainer's fee structure is similar to how some pizza places charge: a base fee for the first few pizzas and then an additional cost per pizza after that. Knowing how to calculate these costs helps you budget effectively. Similarly, understanding fixed and variable costs is crucial in many real-life scenarios, such as managing a business, planning events, or even understanding your utility bills, where you often have a fixed service charge plus a variable charge based on usage. This problem demonstrates how to break down costs and predict expenses based on different usage levels.
The fee charged for 4 hours of training is $345, calculated using a fixed fee of $270 and an hourly fee of $75. This was determined through a system of equations based on the fees for 6 and 9 hours of training. By solving these equations, we found both the fixed and hourly charges.
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