Question
Reasoning On a map, 1 inch equals 11.2 miles. Two houses are 3.5 inches apart on the map. What is the actual distance between the houses? Use pencil and paper. Show how you can represent the scale with two different ratios. What ratio is more helpful for solving the problem? Explain.
Question content area bottom
Part 1
The actual distance between the houses is
enter your response here miles.
Question content area bottom
Part 1
The actual distance between the houses is
enter your response here miles.
Answers
GPT-4o mini
To find the actual distance between the two houses represented on the map, we can use the scale provided: 1 inch equals 11.2 miles.
Given that the two houses are 3.5 inches apart on the map, we can find the actual distance by multiplying the distance on the map by the scale factor.
### Calculation:
1. **Convert inches to miles**:
\[
\text{Actual distance} = \text{Distance on map} \times \text{Scale factor}
\]
\[
\text{Actual distance} = 3.5 \text{ inches} \times 11.2 \text{ miles/inch}
\]
2. **Calculate**:
\[
\text{Actual distance} = 3.5 \times 11.2 = 39.2 \text{ miles}
\]
### Part 1
Thus, the actual distance between the houses is **39.2 miles**.
### Part 2: Representing the Scale with Two Ratios
1. **Ratio 1 (inches to miles)**:
\[
\text{Ratio} = \frac{1 \text{ inch}}{11.2 \text{ miles}}
\]
2. **Ratio 2 (miles to inches)**:
\[
\text{Ratio} = \frac{11.2 \text{ miles}}{1 \text{ inch}}
\]
(This is the inverse of the first ratio).
### Helpful Ratio
The first ratio \( \frac{1 \text{ inch}}{11.2 \text{ miles}} \) is more helpful for solving this problem because it directly relates inches from the map to the actual distance in miles. This means we can easily apply this ratio when we know the distance measured on the map (in inches) to find the corresponding distance in miles.
The second ratio, while mathematically equivalent, is less intuitive for the purpose of this specific calculation since it presents miles relative to inches.
Given that the two houses are 3.5 inches apart on the map, we can find the actual distance by multiplying the distance on the map by the scale factor.
### Calculation:
1. **Convert inches to miles**:
\[
\text{Actual distance} = \text{Distance on map} \times \text{Scale factor}
\]
\[
\text{Actual distance} = 3.5 \text{ inches} \times 11.2 \text{ miles/inch}
\]
2. **Calculate**:
\[
\text{Actual distance} = 3.5 \times 11.2 = 39.2 \text{ miles}
\]
### Part 1
Thus, the actual distance between the houses is **39.2 miles**.
### Part 2: Representing the Scale with Two Ratios
1. **Ratio 1 (inches to miles)**:
\[
\text{Ratio} = \frac{1 \text{ inch}}{11.2 \text{ miles}}
\]
2. **Ratio 2 (miles to inches)**:
\[
\text{Ratio} = \frac{11.2 \text{ miles}}{1 \text{ inch}}
\]
(This is the inverse of the first ratio).
### Helpful Ratio
The first ratio \( \frac{1 \text{ inch}}{11.2 \text{ miles}} \) is more helpful for solving this problem because it directly relates inches from the map to the actual distance in miles. This means we can easily apply this ratio when we know the distance measured on the map (in inches) to find the corresponding distance in miles.
The second ratio, while mathematically equivalent, is less intuitive for the purpose of this specific calculation since it presents miles relative to inches.