11. Higher Order Thinking Mark is training for a min triathlon. He rode his bike \( \frac{3}{4} \) mile, ran \( \frac{2}{4} \) mile, and swam \( \frac{1}{4} \) mile each day. How does the distance he biked in 3 days compare to the distance he swam in 3 days? In 5 days? In 6 days? Why?

- Daily biking distance: \( \frac{3}{4} \) mile - Daily swimming distance: \( \frac{1}{4} \) mile **Over 3 days:** - Total biking distance: \( 3 \times \frac{3}{4} = \frac{9}{4} \) miles - Total swimming distance: \( 3 \times \frac{1}{4} = \frac{3}{4} \) miles Comparing the two: \[ \frac{\frac{9}{4}}{\frac{3}{4}} = 3 \] Thus, the distance biked is 3 times the distance swam. **Over 5 days:** - Total biking distance: \( 5 \times \frac{3}{4} = \frac{15}{4} \) miles - Total swimming distance: \( 5 \times \frac{1}{4} = \frac{5}{4} \) miles Comparing the two: \[ \frac{\frac{15}{4}}{\frac{5}{4}} = 3 \] The biking distance is again 3 times the swimming distance. **Over 6 days:** - Total biking distance: \( 6 \times \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \) miles - Total swimming distance: \( 6 \times \frac{1}{4} = \frac{6}{4} = \frac{3}{2} \) miles Comparing the two: \[ \frac{\frac{9}{2}}{\frac{3}{2}} = 3 \] Once more, the biking distance is 3 times the swimming distance. **Why?** Each day, the distance biked is 3 times the distance swam (\( \frac{3}{4} \) vs. \( \frac{1}{4} \)). Multiplying both daily distances by the same number of days preserves this ratio, so for any number of days the biking distance will always be 3 times the swimming distance.