Let's analyze the problem:

Information Given:

1. Grid dimensions:

   • Each east-west block = $\frac{1}{8} \, \text{mile}$
   • Each north-south block = $\frac{1}{5} \, \text{mile}$

2. Task:

   • Find the shortest walking distance (by traveling only along the blocks).
   • Find the straight-line distance between the Theater and Library (using the Pythagorean theorem).

Step 1: Count Blocks on the Grid

From the Theater to the Library:

• Horizontal (east-west) blocks: 5 blocks
• Vertical (north-south) blocks: 2 blocks

Step 2: Calculate Walking Distance

Walking distance involves traveling along both horizontal and vertical paths: $\text{Walking Distance} = (\text{Horizontal distance}) + (\text{Vertical distance})$

• Horizontal distance: $5 \times \frac{1}{8} = \frac{5}{8} \, \text{miles}$
• Vertical distance: $2 \times \frac{1}{5} = \frac{2}{5} \, \text{miles}$

Convert $\frac{2}{5}$ to a fraction with denominator 40: $\frac{2}{5} = \frac{16}{40} \quad \text{and} \quad \frac{5}{8} = \frac{25}{40}$

Add: $\text{Walking Distance} = \frac{25}{40} + \frac{16}{40} = \frac{41}{40} \, \text{miles} = 1.025 \, \text{miles}$

Step 3: Calculate Straight-Line Distance

Straight-line distance can be found using the Pythagorean theorem: $d = \sqrt{(\text{horizontal distance})^2 + (\text{vertical distance})^2}$

• Horizontal distance = $\frac{5}{8} = 0.625 \, \text{miles}$
• Vertical distance = $\frac{2}{5} = 0.4 \, \text{miles}$

$d = \sqrt{(0.625)^2 + (0.4)^2} = \sqrt{0.390625 + 0.16} = \sqrt{0.550625} \approx 0.742 \, \text{miles}$

Final Answers:

1. Shortest walking distance: $1.025 \, \text{miles}$
2. Straight-line distance: $0.742 \, \text{miles}$

Let me know if you’d like further clarification or details! Here are related questions for deeper understanding:

1. How do different block sizes affect walking distance?
2. Can other routes along the grid provide the same walking distance?
3. How does the straight-line distance compare to walking distance as blocks increase?
4. What is the effect of block angles on the total distance?
5. How does rounding errors affect the precision of straight-line calculations?

Tip: Always double-check units when converting distances to avoid calculation errors!