coach is comparing data about the number of miles Jacob and Otto jogged and the time they spent jogging on week days.

Question

Jacob jogged 3 miles in 30 minutes on Wednesday and 5 miles in 50 minutes on Thursday. Otto jogged 4 miles in 32 minutes on Wednesday and 6 miles in 50 minutes on Thursday.

Whose data shows a proportional relationship between the number of miles jogged and the time spent jogging?

Answer 1

To determine whether the relationships are proportional, we can compare the ratios of miles jogged to time spent jogging for both Jacob and Otto.

For Jacob:

1. Wednesday:

   • Miles: 3
   • Time: 30 minutes
   • Ratio: \( \frac{3 \text{ miles}}{30 \text{ minutes}} = \frac{1}{10} \)

2. Thursday:

   • Miles: 5
   • Time: 50 minutes
   • Ratio: \( \frac{5 \text{ miles}}{50 \text{ minutes}} = \frac{1}{10} \)

Both of Jacob's ratios are equal to \( \frac{1}{10} \). Therefore, Jacob's data shows a proportional relationship.

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For Otto:

1. Wednesday:

   • Miles: 4
   • Time: 32 minutes
   • Ratio: \( \frac{4 \text{ miles}}{32 \text{ minutes}} = \frac{1}{8} \)

2. Thursday:

   • Miles: 6
   • Time: 50 minutes
   • Ratio: \( \frac{6 \text{ miles}}{50 \text{ minutes}} = \frac{3}{25} \)

The ratios for Otto do not equal each other (\( \frac{1}{8} \) and \( \frac{3}{25} \) are not the same), indicating that Otto's data does not show a proportional relationship.

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Conclusion: Jacob's data shows a proportional relationship between the number of miles jogged and the time spent jogging, while Otto's data does not.