Question: The scale on a map is $\frac{1}{2}$ in. equals 8 mi. What is the actual distance between two points that are $1\frac{1}{4}$ in. apart on the map?
1. **State the problem:** We are given a map scale where $\frac{1}{2}$ inch represents 8 miles. We need to find the actual distance between two points that are $1\frac{1}{4}$ inches apart on the map.
2. **Formula and rules:** The scale means that every $\frac{1}{2}$ inch on the map corresponds to 8 miles in reality. To find the actual distance, we use the proportion:
$$\text{Actual distance} = \left(\frac{\text{Actual miles}}{\text{Map inches}}\right) \times \text{Given map distance}$$
3. **Convert mixed number to improper fraction:**
$$1\frac{1}{4} = \frac{5}{4}$$ inches
4. **Set up the proportion:**
$$\frac{1}{2} \text{ in.} = 8 \text{ mi}$$
So,
$$1 \text{ in.} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \text{ mi}$$
5. **Calculate actual distance:**
$$\text{Actual distance} = 16 \times \frac{5}{4} = \frac{16 \times 5}{4} = \frac{80}{4}$$
6. **Simplify the fraction:**
$$\frac{80}{4} = 20$$
7. **Final answer:** The actual distance between the two points is $20$ miles.