The tables below show the number of jumping jacks completed after a given period of time in minutes.

Question

Kimberly

Time (min) # Jumping Jacks
3 28
8 58
12 82
16 106
Katrina

Time (min) # Jumping Jacks
2 16
5 40
12 96
20 160

The person doing the most jumping jacks per minute

The person showing a proportional relationship between time and # of jumping jacks

The person that did the most jumping jacks initially before the time started

Options, Katrina, Kimberly, Neither, Both

Answer 1

Let's analyze the data for both Kimberly and Katrina to answer each part of the prompt.

Jumping Jacks Per Minute Calculation

Kimberly:

• For 3 minutes: 28 jumping jacks ⇒ (28 / 3) ≈ 9.33 jacks/min
• For 8 minutes: 58 jumping jacks ⇒ (58 / 8) = 7.25 jacks/min
• For 12 minutes: 82 jumping jacks ⇒ (82 / 12) ≈ 6.83 jacks/min
• For 16 minutes: 106 jumping jacks ⇒ (106 / 16) = 6.625 jacks/min

Maximum for Kimberly: 9.33 jacks/min (at 3 minutes)

Katrina:

• For 2 minutes: 16 jumping jacks ⇒ (16 / 2) = 8 jacks/min
• For 5 minutes: 40 jumping jacks ⇒ (40 / 5) = 8 jacks/min
• For 12 minutes: 96 jumping jacks ⇒ (96 / 12) = 8 jacks/min
• For 20 minutes: 160 jumping jacks ⇒ (160 / 20) = 8 jacks/min

Maximum for Katrina: 8 jacks/min

Conclusion on Jumping Jacks Per Minute:
Kimberly is doing the most jumping jacks per minute (9.33 jacks/min).

Proportional Relationship

To find out if there's a proportional relationship, we check if the ratio of jumping jacks to time is constant.

Kimberly:

• \( \frac{28}{3} \), \( \frac{58}{8} \), \( \frac{82}{12} \), and \( \frac{106}{16} \) are not constant ratios.

Katrina:

• \( \frac{16}{2} = 8 \), \( \frac{40}{5} = 8 \), \( \frac{96}{12} = 8 \), and \( \frac{160}{20} = 8 \) are consistent ratios and thus show a proportional relationship.

Conclusion on Proportional Relationship:
Katrina shows a proportional relationship.

Most Jumping Jacks Initially

For the time before starting, we check the initial values.

• Kimberly: Doesn't have a data point before 3 minutes (no initial jumps).
• Katrina: At 2 minutes, she starts with 16 jumping jacks.

Conclusion on Initial Jumping Jacks:
Katrina did the most jumping jacks initially.

Summary:

1. The person doing the most jumping jacks per minute: Kimberly
2. The person showing a proportional relationship: Katrina
3. The person that did the most jumping jacks initially before the time started: Katrina

Thus, the answers for the questions posed are:

• Most Jumping Jacks Per Minute: Kimberly
• Proportional Relationship: Katrina
• Most Jumping Jacks Initially: Katrina

For the options provided:

• Neither, Both, Katrina, Kimberly:
• Final answers: Katrina (for proportional relationship and initial jumps), Kimberly (for most per minute).