Question
Reasoning On a map, 1 inch equals 11.4 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 based on the given map scale, we can start by calculating the distance using the conversion factor provided.
### Step 1: Calculate Actual Distance
Since 1 inch on the map represents 11.4 miles, we can set up the calculation for the distance between the two houses:
1. Measure the distance on the map: 3.5 inches.
2. Multiply the map distance by the conversion factor (11.4 miles/inch):
\[
\text{Actual Distance} = \text{Map Distance} \times \text{Scale Factor}
\]
\[
\text{Actual Distance} = 3.5 \, \text{inches} \times 11.4 \, \text{miles/inch}
\]
Calculating this gives:
\[
\text{Actual Distance} = 39.9 \, \text{miles}
\]
### Step 2: Representing the Scale with Ratios
We can represent the scale in two different forms:
1. **Fraction Ratio**: This gives a direct comparison of the map versus actual distances.
\[
\text{Scale Ratio} = \frac{1 \, \text{inch}}{11.4 \, \text{miles}}
\]
2. **Whole Number Ratio**: This can be simplified by converting miles into inches using the conversion factors (1 mile = 63,360 inches):
\[
\text{Scale} = \frac{11.4 \, \text{miles}}{1 \, \text{inch}} = \frac{11.4 \times 63,360 \, \text{inches}}{1} = \frac{724,224 \, \text{inches}}{1 \, \text{inch}} = 724,224:1
\]
### Which Ratio is More Helpful?
The first ratio (\(\frac{1 \, \text{inch}}{11.4 \, \text{miles}}\)) is more helpful for solving the problem because it allows for straightforward multiplication to find the actual distance when the measured distance on the map is known. The second ratio involves unnecessarily large numbers, which can complicate calculations.
### Final Answer
Thus, the actual distance between the houses is:
\[
\text{Actual Distance} = 39.9 \, \text{miles}
\]
### Step 1: Calculate Actual Distance
Since 1 inch on the map represents 11.4 miles, we can set up the calculation for the distance between the two houses:
1. Measure the distance on the map: 3.5 inches.
2. Multiply the map distance by the conversion factor (11.4 miles/inch):
\[
\text{Actual Distance} = \text{Map Distance} \times \text{Scale Factor}
\]
\[
\text{Actual Distance} = 3.5 \, \text{inches} \times 11.4 \, \text{miles/inch}
\]
Calculating this gives:
\[
\text{Actual Distance} = 39.9 \, \text{miles}
\]
### Step 2: Representing the Scale with Ratios
We can represent the scale in two different forms:
1. **Fraction Ratio**: This gives a direct comparison of the map versus actual distances.
\[
\text{Scale Ratio} = \frac{1 \, \text{inch}}{11.4 \, \text{miles}}
\]
2. **Whole Number Ratio**: This can be simplified by converting miles into inches using the conversion factors (1 mile = 63,360 inches):
\[
\text{Scale} = \frac{11.4 \, \text{miles}}{1 \, \text{inch}} = \frac{11.4 \times 63,360 \, \text{inches}}{1} = \frac{724,224 \, \text{inches}}{1 \, \text{inch}} = 724,224:1
\]
### Which Ratio is More Helpful?
The first ratio (\(\frac{1 \, \text{inch}}{11.4 \, \text{miles}}\)) is more helpful for solving the problem because it allows for straightforward multiplication to find the actual distance when the measured distance on the map is known. The second ratio involves unnecessarily large numbers, which can complicate calculations.
### Final Answer
Thus, the actual distance between the houses is:
\[
\text{Actual Distance} = 39.9 \, \text{miles}
\]