27. The scale on a map is 1 in.: 50 mi . a. The actual distance between two cities is 312.5 miles. What is the distance between the cities on the map? (be sure to include correct measurement with your anwer) b. Two cities are 4.75 inches apart on a map. What is the actual distance between these two cities?(be sure to include correct measurement with your anwer)

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To solve problems involving map scales, it's essential to understand the relationship between the map distance and the actual distance. The scale given here is **1 inch : 50 miles**, which means that **1 inch on the map represents 50 actual miles**.

### a. Finding the Distance on the Map

**Given:**
- Actual distance between two cities = **312.5 miles**
- Scale = **1 inch : 50 miles**

**To Find:**
- Distance between the cities on the map in inches.

**Solution:**

1. **Use the Scale to Set Up the Proportion:**

$$
   \frac{1\ \text{inch}}{50\ \text{miles}} = \frac{x\ \text{inches}}{312.5\ \text{miles}}
   $$

2. **Solve for $$ x $$:**

$$
   x = \frac{312.5\ \text{miles} \times 1\ \text{inch}}{50\ \text{miles}} = \frac{312.5}{50} = 6.25\ \text{inches}
   $$

**Answer:**
- The distance between the two cities on the map is **6.25 inches**.

---

### b. Finding the Actual Distance

**Given:**
- Distance between two cities on the map = **4.75 inches**
- Scale = **1 inch : 50 miles**

**To Find:**
- Actual distance between the two cities in miles.

**Solution:**

1. **Use the Scale to Set Up the Calculation:**

$$
   \text{Actual Distance} = \text{Map Distance} \times \text{Scale Factor}
   $$

2. **Multiply the Map Distance by the Scale Factor:**

$$
   \text{Actual Distance} = 4.75\ \text{inches} \times 50\ \text{miles/inch} = 237.5\ \text{miles}
   $$

**Answer:**
- The actual distance between the two cities is **237.5 miles**.

---

**Summary:**
- **a.** 6.25 inches on the map
- **b.** 237.5 miles actual distance
a. The distance between the cities on the map is 6.25 inches. b. The actual distance between the two cities is 237.5 miles.

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