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


(3 points)

The person doing the most jumping jacks per minute

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

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

options: Kimberly, Neither, Katrina, Both

Answer 1

Let's analyze the data for each person based on the criteria provided.

Jumping Jacks per Minute

Kimberly:

• At 3 minutes: 28 jumping jacks → \( \frac{28}{3} \approx 9.33 \) jacks/min
• At 8 minutes: 58 jumping jacks → \( \frac{58}{8} = 7.25 \) jacks/min
• At 12 minutes: 82 jumping jacks → \( \frac{82}{12} \approx 6.83 \) jacks/min
• At 16 minutes: 106 jumping jacks → \( \frac{106}{16} = 6.625 \) jacks/min

Katrina:

• At 2 minutes: 16 jumping jacks → \( \frac{16}{2} = 8 \) jacks/min
• At 5 minutes: 40 jumping jacks → \( \frac{40}{5} = 8 \) jacks/min
• At 12 minutes: 96 jumping jacks → \( \frac{96}{12} = 8 \) jacks/min
• At 20 minutes: 160 jumping jacks → \( \frac{160}{20} = 8 \) jacks/min

Conclusion for Jumping Jacks per Minute:

• The highest rate is from Kimberly at the 3-minute mark, which is approximately 9.33, while Katrina remains constant at 8 jacks/min.
• So, Kimberly is doing the most jumping jacks per minute.

Most Jumping Jacks Initially

• Kimberly: starts with 28 jumping jacks at 3 minutes.
• Katrina: starts with 16 jumping jacks at 2 minutes.

Conclusion for Most Jumping Jacks Initially: Kimberly initially did the most jumping jacks (28).

Proportional Relationship Between Time and # of Jumping Jacks

To determine if there is a proportional relationship, we can check if the ratios of jumping jacks to time are constant.

For Kimberly:

• The jumping jacks do not increase proportionally with time. For example, between the 8-minute and 12-minute mark, the difference in jumping jacks does not correspond linearly with time.

For Katrina:

• Values seem to increase proportionally:
  • From 2 to 5 minutes (16 to 40): increases by 24 jacks for 3 minutes.
  • From 5 to 12 minutes (40 to 96): increases by 56 jacks for 7 minutes.
  • From 12 to 20 minutes (96 to 160): increases by 64 jacks for 8 minutes.

Even though the intervals differ, the ratios of jumping jacks to time remain consistent as you proceed with the intervals.

Conclusion for Proportional Relationship: Katrina shows a proportional relationship between time and the number of jumping jacks.

Final Answers:

• The person doing the most jumping jacks per minute: Kimberly
• The person that did the most jumping jacks initially before the time started: Kimberly
• The person showing a proportional relationship between time and # of jumping jacks: Katrina

Thus, for the options provided:

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