Let's solve the problem step by step:

Problem:

Penny and Arthur both bike at a pace of $3 \frac{1}{2}$ miles in $\frac{1}{4}$ of an hour. We need to find their speed in miles per hour.

Solution:

1. Convert $3 \frac{1}{2}$ to an improper fraction: $3 \frac{1}{2} = \frac{7}{2}$

2. Speed is calculated as: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

   Substituting the given values: $\text{Speed} = \frac{\frac{7}{2}}{\frac{1}{4}}$

3. Dividing fractions: $\frac{\frac{7}{2}}{\frac{1}{4}} = \frac{7}{2} \times \frac{4}{1} = \frac{28}{2} = 14$

Thus, their speed is 14 miles per hour.

Would you like a detailed explanation for any part of this calculation? Here are some additional related questions for practice:

1. How would their speed change if the distance was $4$ miles instead of $3 \frac{1}{2}$?
2. How can you calculate time if you know speed and distance?
3. What is the formula for speed in general terms?
4. How would the speed change if it took $\frac{1}{2}$ an hour to travel $3 \frac{1}{2}$ miles?
5. What is their speed in kilometers per hour if $1$ mile = (