The tables below show the number of jumping jacks completed after a given period of time in seconds. Kimberly: Time (seconds) Jumping Jacks 3 17 8 37 12 53 16 69 Katrina: Time (seconds) Jumping Jacks 2 10 5 25 12 60 20 100(3 points) a. Which person is doing more jumping jacks per second? b. Which person had done more jumping jacks initially before the timer started? c. Which person shows a proportional relationship?

Question

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Answer 1

To analyze the data provided for Kimberly and Katrina, we need to calculate the jumping jacks per second for each individual, identify their initial jumping jacks before timers started, and assess whether their relationships are proportional.

a. Jumping Jacks per Second

To find the jumping jacks per second for each person, divide the number of jumping jacks by the time taken in seconds.

Kimberly:

• At 3 seconds: \( \frac{17}{3} \approx 5.67 \)
• At 8 seconds: \( \frac{37}{8} = 4.625 \)
• At 12 seconds: \( \frac{53}{12} \approx 4.42 \)
• At 16 seconds: \( \frac{69}{16} \approx 4.31 \)

Average for Kimberly: \[ \text{Average} = \frac{5.67 + 4.625 + 4.42 + 4.31}{4} \approx 4.77 \text{ jumping jacks/second} \]

Katrina:

• At 2 seconds: \( \frac{10}{2} = 5 \)
• At 5 seconds: \( \frac{25}{5} = 5 \)
• At 12 seconds: \( \frac{60}{12} = 5 \)
• At 20 seconds: \( \frac{100}{20} = 5 \)

Average for Katrina: \[ \text{Average} = \frac{5 + 5 + 5 + 5}{4} = 5 \text{ jumping jacks/second} \]

Conclusion for Part a: Katrina is doing more jumping jacks per second (5 jacks/second) compared to Kimberly (average of 4.77 jacks/second).

b. Initial Jumping Jacks

To determine how many jumping jacks each person had completed before the timer started, we look at their performance at 0 seconds. Since there’s no explicit information provided, we can deduce from the given values:

• Kimberly: Assuming she started at 0 seconds: 0 jumping jacks before timer
• Katrina: Assuming she also started timer at 0 seconds: 0 jumping jacks before timer

Conclusion for Part b: Both persons had completed 0 jumping jacks before the timer started.

c. Proportional Relationship

A relationship is considered proportional if the ratio of the quantities remains constant.

• Kimberly's Data:

  • \( \frac{17}{3}, \frac{37}{8}, \frac{53}{12}, \frac{69}{16} \) - The ratios are different.

• Katrina's Data:

  • \( \frac{10}{2} = 5, \frac{25}{5} = 5, \frac{60}{12} = 5, \frac{100}{20} = 5 \) - The ratios are consistent and equal (5).

Conclusion for Part c: Katrina shows a proportional relationship since her jumping jacks per second remains constant at 5, unlike Kimberly's varying ratios.

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To summarize:

• a. Katrina is doing more jumping jacks per second.
• b. Both had completed 0 jumping jacks initially.
• c. Katrina shows a proportional relationship.

Answer 2

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
The person doing the most jumping jacks per minute

Answer 3

Let's analyze the data for Kimberly and Katrina based on your new questions regarding jumping jacks.

1. The Person that Did the Most Jumping Jacks Initially

The tables provided do not explicitly mention the number of jumping jacks done before the timer started. However, if we consider the assumption that both individuals started at 0 jumping jacks before the timer began, it suggests that:

• Kimberly: 0 jumping jacks initially
• Katrina: 0 jumping jacks initially

Conclusion: Neither did any jumping jacks initially; both had completed 0 jumping jacks before the timer started.

2. The Person Showing a Proportional Relationship Between Time and Number of Jumping Jacks

A relationship is proportional if the ratio of the two quantities is constant.

Kimberly's Data:

• Time (seconds): 3, 8, 12, 16
• Jumping Jacks: 17, 37, 53, 69

Calculating ratios:

• \( \frac{17}{3} \approx 5.67 \)
• \( \frac{37}{8} = 4.625 \)
• \( \frac{53}{12} \approx 4.42 \)
• \( \frac{69}{16} \approx 4.31 \)

These values are not consistent; they vary.

Katrina's Data:

• Time (seconds): 2, 5, 12, 20
• Jumping Jacks: 10, 25, 60, 100

Calculating ratios:

• \( \frac{10}{2} = 5 \)
• \( \frac{25}{5} = 5 \)
• \( \frac{60}{12} = 5 \)
• \( \frac{100}{20} = 5 \)

These values are consistent and equal.

Conclusion: Katrina is the person showing a proportional relationship between time and the number of jumping jacks.

3. The Person Doing the Most Jumping Jacks Per Minute

To find out who did the most jumping jacks per minute, we will convert the number of jumping jacks done in the provided seconds into jumps per minute.

Conversion: To find jumps per minute, we can use the formula: \[ \text{Jumping Jacks per minute} = \left( \frac{\text{Jumping Jacks}}{\text{Time (seconds)}} \right) \times 60 \]

Kimberly:

• At 3 seconds: \( \left( \frac{17}{3} \right) \times 60 \approx 340 \) jumping jacks/minute
• At 8 seconds: \( \left( \frac{37}{8} \right) \times 60 \approx 277.5 \) jumping jacks/minute
• At 12 seconds: \( \left( \frac{53}{12} \right) \times 60 \approx 265 \) jumping jacks/minute
• At 16 seconds: \( \left( \frac{69}{16} \right) \times 60 \approx 258.75 \) jumping jacks/minute

Average for Kimberly: \[ \text{Average} = \frac{340 + 277.5 + 265 + 258.75}{4} \approx 285.56 \text{ jumping jacks/minute} \]

Katrina:

• At 2 seconds: \( \left( \frac{10}{2} \right) \times 60 = 300 \) jumping jacks/minute
• At 5 seconds: \( \left( \frac{25}{5} \right) \times 60 = 300 \) jumping jacks/minute
• At 12 seconds: \( \left( \frac{60}{12} \right) \times 60 = 300 \) jumping jacks/minute
• At 20 seconds: \( \left( \frac{100}{20} \right) \times 60 = 300 \) jumping jacks/minute

Average for Katrina: \[ \text{Average} = \frac{300 + 300 + 300 + 300}{4} = 300 \text{ jumping jacks/minute} \]

Final Summary

• Most jumping jacks initially before timer started: Both had 0 jumping jacks initially.
• Person showing a proportional relationship: Katrina.
• The person doing the most jumping jacks per minute: Katrina (300 jumping jacks/minute) compared to Kimberly's average of approximately 285.56 jacks/minute.