Question

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.

Answer 1

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.**