Question

austin, the trainer, has two workout plans that he offers his clients: plan a and plan b. each client does either one or the other (not both). on monday there were 3 clients who did plan a and 2 who did plan b. on thursday there were 8 clients who did plan a and 4 who did plan b. austin trained his monday clients for a total of 7 hours and his thursday clients for a total of 17 hours. how long does each of the workout plans last?
length of each plan a workout: □ hour(s)
length of each plan b workout: □ hour(s)

Explanation:

Step1: Define variables

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

Step2: Set up equations from given data

From the first day: \( 3x + 2y = 7 \) (3 clients for Plan A, 2 for Plan B, total 7 hours).
From Thursday: \( 8x + 4y = 17 \) (8 clients for Plan A, 4 for Plan B, total 17 hours).

Step3: Simplify the second equation

Divide the second equation \( 8x + 4y = 17 \) by 2: \( 4x + 2y = \frac{17}{2} = 8.5 \).

Step4: Subtract the first equation from the simplified second equation

\( (4x + 2y) - (3x + 2y) = 8.5 - 7 \)
Simplify: \( x = 1.5 \).

Step5: Substitute \( x = 1.5 \) into the first equation

\( 3(1.5) + 2y = 7 \)
Calculate \( 3(1.5) = 4.5 \), so \( 4.5 + 2y = 7 \).
Subtract 4.5 from both sides: \( 2y = 7 - 4.5 = 2.5 \).
Divide by 2: \( y = \frac{2.5}{2} = 1.25 \).

Answer:

Duration of each Plan A workout: \( 1.5 \) hour(s)
Duration of each Plan B workout: \( 1.25 \) hour(s)