Answer :
To determine the correct scale that Lyle could include on his map, we need to find out how many miles are represented by one inch on the map.
We know that the distance between Lyle's house and his school on the map is [tex]\(\frac{1}{5}\)[/tex] of an inch and the actual distance is [tex]\(\frac{3}{4}\)[/tex] of a mile. Thus, we can calculate the scale in miles per inch as follows:
1. Calculate the scale:
[tex]\[ \text{scale (miles per inch)} = \frac{\text{actual distance in miles}}{\text{map distance in inches}} = \frac{\frac{3}{4} \text{ miles}}{\frac{1}{5} \text{ inches}} \][/tex]
2. To divide by a fraction, we multiply by its reciprocal:
[tex]\[ \text{scale (miles per inch)} = \frac{3}{4} \times \frac{5}{1} = \frac{3 \times 5}{4 \times 1} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
We now have the scale of 3.75 miles per inch. Let's analyze the given options to see which scales match this calculation.
### Checking each option:
1. [tex]\(\frac{1}{5}\)[/tex] inch [tex]\(=\frac{3}{4}\)[/tex] mile:
- This implies the same distances given in the problem and doesn't provide a new, meaningful scale.
- False
2. [tex]\(\frac{3}{4}\)[/tex] inch [tex]\(=\frac{1}{5}\)[/tex] mile:
- This would imply a different scale. Converting it into a scale in miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{\frac{1}{5} \text{ miles}}{\frac{3}{4} \text{ inches}} = \frac{1}{5} \times \frac{4}{3} = \frac{4}{15} \approx 0.2667 \text{ miles per inch} \][/tex]
- This does not match Lyle's calculated scale of 3.75 miles per inch.
- False
3. [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{3 \text{ miles}}{\frac{4}{5} \text{ inches}} = 3 \times \frac{5}{4} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
- This matches the calculated scale of 3.75 miles per inch.
- True
4. 3 inches [tex]\(= 12\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{12 \text{ miles}}{3 \text{ inches}} = 4 \text{ miles per inch} \][/tex]
- This does not match the calculated scale of 3.75 miles per inch.
- False
5. 4 inches [tex]\(= 15\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{15 \text{ miles}}{4 \text{ inches}} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
- This matches the calculated scale of 3.75 miles per inch.
- True
### Summary of Correct Options
The correct options that match the calculated scale of 3.75 miles per inch are:
- [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles
- 4 inches [tex]\(= 15\)[/tex] miles
So, Lyle could include the following scales on his map:
- [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles
- 4 inches [tex]\(= 15\)[/tex] miles
We know that the distance between Lyle's house and his school on the map is [tex]\(\frac{1}{5}\)[/tex] of an inch and the actual distance is [tex]\(\frac{3}{4}\)[/tex] of a mile. Thus, we can calculate the scale in miles per inch as follows:
1. Calculate the scale:
[tex]\[ \text{scale (miles per inch)} = \frac{\text{actual distance in miles}}{\text{map distance in inches}} = \frac{\frac{3}{4} \text{ miles}}{\frac{1}{5} \text{ inches}} \][/tex]
2. To divide by a fraction, we multiply by its reciprocal:
[tex]\[ \text{scale (miles per inch)} = \frac{3}{4} \times \frac{5}{1} = \frac{3 \times 5}{4 \times 1} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
We now have the scale of 3.75 miles per inch. Let's analyze the given options to see which scales match this calculation.
### Checking each option:
1. [tex]\(\frac{1}{5}\)[/tex] inch [tex]\(=\frac{3}{4}\)[/tex] mile:
- This implies the same distances given in the problem and doesn't provide a new, meaningful scale.
- False
2. [tex]\(\frac{3}{4}\)[/tex] inch [tex]\(=\frac{1}{5}\)[/tex] mile:
- This would imply a different scale. Converting it into a scale in miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{\frac{1}{5} \text{ miles}}{\frac{3}{4} \text{ inches}} = \frac{1}{5} \times \frac{4}{3} = \frac{4}{15} \approx 0.2667 \text{ miles per inch} \][/tex]
- This does not match Lyle's calculated scale of 3.75 miles per inch.
- False
3. [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{3 \text{ miles}}{\frac{4}{5} \text{ inches}} = 3 \times \frac{5}{4} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
- This matches the calculated scale of 3.75 miles per inch.
- True
4. 3 inches [tex]\(= 12\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{12 \text{ miles}}{3 \text{ inches}} = 4 \text{ miles per inch} \][/tex]
- This does not match the calculated scale of 3.75 miles per inch.
- False
5. 4 inches [tex]\(= 15\)[/tex] miles:
- Converting this into miles per inch:
[tex]\[ \text{scale (miles per inch)} = \frac{15 \text{ miles}}{4 \text{ inches}} = \frac{15}{4} = 3.75 \text{ miles per inch} \][/tex]
- This matches the calculated scale of 3.75 miles per inch.
- True
### Summary of Correct Options
The correct options that match the calculated scale of 3.75 miles per inch are:
- [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles
- 4 inches [tex]\(= 15\)[/tex] miles
So, Lyle could include the following scales on his map:
- [tex]\(\frac{4}{5}\)[/tex] inch [tex]\(= 3\)[/tex] miles
- 4 inches [tex]\(= 15\)[/tex] miles
