Question

Usa the trainer has two solo workout plans that she offers her dients: Plan A and Plan B. Each client does elther one or the other (not both). On Wednesday there were 4 clients who did Plan A and 8 who did Plan B. On Thursday there were 2 clients who did Plan A and 3 who did Plan B. Lisa trained her Wednesday dients for a total of 17 hours and her Thursday clients for a total of 7 hours. How long does each of the workout plans last?

Answer 1

To solve this problem, let's define two variables:

• Let \( x \) be the duration (in hours) of Plan A.
• Let \( y \) be the duration (in hours) of Plan B.

For Wednesday

On Wednesday, there were:

• 4 clients doing Plan A
• 8 clients doing Plan B

The total hours spent on Wednesday can be represented by the equation: \[ 4x + 8y = 17 \]

For Thursday

On Thursday, there were:

• 2 clients doing Plan A
• 3 clients doing Plan B

The total hours spent on Thursday can be represented by the equation: \[ 2x + 3y = 7 \]

Now we have a system of equations:

1. \( 4x + 8y = 17 \) (Equation 1)
2. \( 2x + 3y = 7 \) (Equation 2)

Step 1: Simplifying the equations

First, we can simplify Equation 1 by dividing all terms by 4: \[ x + 2y = \frac{17}{4} \] or, \[ x + 2y = 4.25 \quad \text{(Equation 3)} \]

Step 2: Solve the system of equations

Now, we can use Equation 2 and Equation 3 together.

From Equation 3: \[ x = 4.25 - 2y \]

Substituting \( x \) in Equation 2: \[ 2(4.25 - 2y) + 3y = 7 \] \[ 8.5 - 4y + 3y = 7 \] \[ 8.5 - y = 7 \] \[ -y = 7 - 8.5 \] \[ -y = -1.5 \] \[ y = 1.5 \]

Step 3: Find \( x \)

Now we can use the value of \( y \) to find \( x \): \[ x = 4.25 - 2(1.5) \] \[ x = 4.25 - 3 \] \[ x = 1.25 \]

Conclusion

The durations of the workout plans are:

• Plan A lasts for 1.25 hours (or 1 hour and 15 minutes).
• Plan B lasts for 1.5 hours (or 1 hour and 30 minutes).