Let's break down each statement step-by-step using the rates calculated from the given data as a reference.
### Information given:
- Kelly's rate before snack: It took her [tex]\(\frac{1}{14}\)[/tex] hour to walk [tex]\(\frac{1}{4}\)[/tex] mile.
- Kelly's rate after snack: She walked [tex]\(\frac{1}{6}\)[/tex] mile in [tex]\(\frac{1}{16}\)[/tex] hour.
### Calculations:
#### 1. Rate Before Snack:
Kelly's walking rate before her snack is:
[tex]\[ \text{Rate Before} = \frac{\frac{1}{4} \text{ miles}}{\frac{1}{14} \text{ hour}} = \frac{1/4}{1/14} = \frac{1}{4} \times \frac{14}{1} = 3.5 \text{ miles per hour} \][/tex]
#### 2. Rate After Snack:
Kelly's walking rate after her snack is:
[tex]\[ \text{Rate After} = \frac{\frac{1}{6} \text{ miles}}{\frac{1}{16} \text{ hour}} = \frac{1/6}{1/16} = \frac{1}{6} \times \frac{16}{1} = \frac{16}{6} = \frac{8}{3} = 2.\overline{666} \text{ miles per hour} \][/tex]
### Statements:
#### Statement A:
Before her snack, Kelly walked at a rate of [tex]\(\frac{4}{14}\)[/tex] miles per hour.
[tex]\[ \frac{4}{14} = \frac{2}{7} \text{ miles per hour} \][/tex]
Since [tex]\(3.5 \text{ miles per hour}\)[/tex] is not equal to [tex]\(\frac{2}{7} \text{ miles per hour}\)[/tex], this statement is False.
#### Statement B:
For the second part of her hike, Kelly walked at a rate of [tex]\(2 \frac{2}{3}\)[/tex] miles per hour.
Since [tex]\(2 \frac{2}{3} = 2.666...\)[/tex] and this matches with Kelly's calculated rate after snack ([tex]\(2.666...\)[/tex]), this statement is True.
#### Statement C:
It took Kelly 2 hours longer to walk [tex]\(\frac{1}{6}\)[/tex] mile than it did for her to walk [tex]\(\frac{1}{4}\)[/tex] mile.
We need to compare the time it took for each segment. Kelly took:
- [tex]\(\frac{1}{14}\)[/tex] hour to walk [tex]\(\frac{1}{4}\)[/tex] mile.
- [tex]\(\frac{1}{16}\)[/tex] hour to walk [tex]\(\frac{1}{6}\)[/tex] mile.
The difference in time is:
[tex]\[ \text{Time Difference} = \left( \frac{1}{16} - \frac{1}{14} \right) = -0.008928571428571425 \text{ hours approx.} \][/tex]
Since the difference is not 2 hours, this statement is False.
#### Statement D:
Kelly walked over 30% faster before her snack than she did after her snack.
[tex]\[ \text{Rate After} \times 1.3 = 2.666 \times 1.3 = 3.466 \text{ miles per hour} \][/tex]
Since [tex]\(3.5 \text{ miles per hour}\)[/tex] is greater than [tex]\(3.466 \text{ miles per hour}\)[/tex], this statement is True.
### Summary of Statements:
A - False\
B - True\
C - False\
D - True
