Question
Reasoning On a map, 1 inch equals 7.4 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 given that they are 3.5 inches apart on the map and the scale of the map is 1 inch = 7.4 miles, we'll start by using the scale to compute the actual distance.
### Step 1: Calculate the Actual Distance
If 1 inch represents 7.4 miles, then:
\[
\text{Actual distance} = \text{Distance on map} \times \text{Scale factor}
\]
Substituting the values, we have:
\[
\text{Actual distance} = 3.5 \, \text{inches} \times 7.4 \, \text{miles/inch}
\]
Now we can compute the multiplication:
\[
\text{Actual distance} = 3.5 \times 7.4 = 25.9 \, \text{miles}
\]
### Part 1 Answer:
The actual distance between the houses is **25.9 miles**.
### Part 2: Represent the Scale with Two Different Ratios
To represent the scale 1 inch = 7.4 miles as ratios, we can express it in two different ways:
1. **Ratio of Inches to Miles**:
- The ratio can be expressed as:
\[
1 \, \text{inch} : 7.4 \, \text{miles}
\]
2. **Ratio of Miles to Inches**:
- This can also be expressed as:
\[
7.4 \, \text{miles} : 1 \, \text{inch}
\]
### Which Ratio is More Helpful?
For this specific problem, the **first ratio (1 inch : 7.4 miles)** is more helpful. This is because we are starting with a distance on the map (in inches) and we want to convert it to a real-world distance (in miles).
Using the first ratio allows us to perform a straightforward multiplication of the distance on the map by the scale factor to find the actual distance, making it more intuitive for solving the problem. In contrast, the second ratio would require us to manipulate the numbers more to find the distance.
### Step 1: Calculate the Actual Distance
If 1 inch represents 7.4 miles, then:
\[
\text{Actual distance} = \text{Distance on map} \times \text{Scale factor}
\]
Substituting the values, we have:
\[
\text{Actual distance} = 3.5 \, \text{inches} \times 7.4 \, \text{miles/inch}
\]
Now we can compute the multiplication:
\[
\text{Actual distance} = 3.5 \times 7.4 = 25.9 \, \text{miles}
\]
### Part 1 Answer:
The actual distance between the houses is **25.9 miles**.
### Part 2: Represent the Scale with Two Different Ratios
To represent the scale 1 inch = 7.4 miles as ratios, we can express it in two different ways:
1. **Ratio of Inches to Miles**:
- The ratio can be expressed as:
\[
1 \, \text{inch} : 7.4 \, \text{miles}
\]
2. **Ratio of Miles to Inches**:
- This can also be expressed as:
\[
7.4 \, \text{miles} : 1 \, \text{inch}
\]
### Which Ratio is More Helpful?
For this specific problem, the **first ratio (1 inch : 7.4 miles)** is more helpful. This is because we are starting with a distance on the map (in inches) and we want to convert it to a real-world distance (in miles).
Using the first ratio allows us to perform a straightforward multiplication of the distance on the map by the scale factor to find the actual distance, making it more intuitive for solving the problem. In contrast, the second ratio would require us to manipulate the numbers more to find the distance.