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A personal trainer designs exercise plans based on a combination of strength-training and aerobic exercise. A beginner plan has 15 minutes per session of strength-training and 20 minutes per session of aerobic exercise for a total of 90 minutes of exercise in a week. An advanced plan has 20 minutes per session of strength-training and 30 minutes of aerobic exercise for a total of 130 minutes of exercise in a week.

Which statement describes when the plans are based on the same number of aerobic exercise sessions?

Each plan utilizes a combination of 2 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 2 aerobic exercise sessions per week.
Each plan utilizes a combination of 3 strength-training sessions and 3 aerobic exercise sessions per week.

1 answer

To determine when the beginner and advanced plans are based on the same number of aerobic exercise sessions, we will first analyze the structure of each plan.

Beginner Plan:

  • Strength-training: 15 minutes/session
  • Aerobic exercise: 20 minutes/session
  • Total weekly exercise: 90 minutes

Let:

  • \( x \) = number of strength-training sessions
  • \( y \) = number of aerobic exercise sessions

Since the total time for the beginner plan is 90 minutes, we can write: \[ 15x + 20y = 90 \]

Advanced Plan:

  • Strength-training: 20 minutes/session
  • Aerobic exercise: 30 minutes/session
  • Total weekly exercise: 130 minutes

Similar to the beginner plan, we write: \[ 20x + 30y = 130 \]

Now we will simplify both equations.

For the beginner plan: \[ 15x + 20y = 90 \quad \rightarrow \quad 3x + 4y = 18 \quad \text{(dividing the equation by 5)} \]

For the advanced plan: \[ 20x + 30y = 130 \quad \rightarrow \quad 2x + 3y = 13 \quad \text{(dividing the equation by 10)} \]

Next, let’s test each option to see which combination yields the same number of aerobic exercise sessions.

  1. 2 strength-training sessions and 2 aerobic exercise sessions: \[ \text{Beginner:} \quad 3(2) + 4(2) = 6 + 8 = 14 \quad (\text{does NOT fit the total of 18}) \] \[ \text{Advanced:} \quad 2(2) + 3(2) = 4 + 6 = 10 \quad (\text{does NOT fit the total of 13}) \]

  2. 2 strength-training sessions and 3 aerobic exercise sessions: \[ \text{Beginner:} \quad 3(2) + 4(3) = 6 + 12 = 18 \quad (\text{fits the total of 18}) \] \[ \text{Advanced:} \quad 2(2) + 3(3) = 4 + 9 = 13 \quad (\text{fits the total of 13}) \]

  3. 3 strength-training sessions and 2 aerobic exercise sessions: \[ \text{Beginner:} \quad 3(3) + 4(2) = 9 + 8 = 17 \quad (\text{does NOT fit the total of 18}) \] \[ \text{Advanced:} \quad 2(3) + 3(2) = 6 + 6 = 12 \quad (\text{does NOT fit the total of 13}) \]

  4. 3 strength-training sessions and 3 aerobic exercise sessions: \[ \text{Beginner:} \quad 3(3) + 4(3) = 9 + 12 = 21 \quad (\text{does NOT fit the total of 18}) \] \[ \text{Advanced:} \quad 2(3) + 3(3) = 6 + 9 = 15 \quad (\text{does NOT fit the total of 13}) \]

After analyzing the options, we find that only the combination of 2 strength-training sessions and 3 aerobic exercise sessions yields correct totals for both plans.

Thus, the correct statement is: Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week.