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).