One inch represents 20text{ miles} on Map K and one inch represents 30text{ miles} on Map L. An area of...

1. Translate the problem requirements: We need to find how many more square miles are represented by 3 square inches on Map L compared to Map K. This requires understanding that linear scale affects area by the square of the scale factor.
2. Calculate area representation for each map: Convert the linear scales (\(\mathrm{20\,miles/inch}\) for Map K, \(\mathrm{30\,miles/inch}\) for Map L) into area scales by squaring the linear relationships.
3. Find the area difference: Calculate how many square miles 3 square inches represents on each map, then find the difference.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to find. We have two maps with different scales:

• Map K: \(\mathrm{1\,inch\,represents\,20\,miles}\)
• Map L: \(\mathrm{1\,inch\,represents\,30\,miles}\)

The question asks: "An area of 3 square inches represents how many more square miles on Map L than on Map K?"

This means we need to:

1. Find how many square miles 3 square inches represents on Map K
2. Find how many square miles 3 square inches represents on Map L
3. Calculate the difference (Map L minus Map K)

The key insight here is that when we're dealing with areas (square inches to square miles), the scale factor gets squared. If \(\mathrm{1\,inch = 20\,miles}\), then \(\mathrm{1\,square\,inch = 20 \times 20 = 400\,square\,miles}\).

2. Calculate area representation for each map

Now let's figure out the area relationships for each map.

For Map K:

Since \(\mathrm{1\,inch = 20\,miles}\), when we think about area:

\(\mathrm{1\,square\,inch = 20\,miles \times 20\,miles = 400\,square\,miles}\)

Therefore: \(\mathrm{3\,square\,inches = 3 \times 400 = 1,200\,square\,miles}\)

For Map L:

Since \(\mathrm{1\,inch = 30\,miles}\), when we think about area:

\(\mathrm{1\,square\,inch = 30\,miles \times 30\,miles = 900\,square\,miles}\)

Therefore: \(\mathrm{3\,square\,inches = 3 \times 900 = 2,700\,square\,miles}\)

Think of it this way: imagine a square that's 1 inch by 1 inch on the map. On Map K, this represents a real-world square that's 20 miles by 20 miles. On Map L, the same 1-inch square represents a larger real-world area of 30 miles by 30 miles.

3. Find the area difference

Now we can find how many more square miles are represented on Map L compared to Map K:

Difference = Area on Map L - Area on Map K

Difference = \(\mathrm{2,700\,square\,miles - 1,200\,square\,miles = 1,500\,square\,miles}\)

So 3 square inches represents 1,500 more square miles on Map L than on Map K.

4. Final Answer

The answer is 1,500 square miles, which corresponds to choice E.

Verification:

• Map K: \(\mathrm{3\,square\,inches = 1,200\,square\,miles}\)
• Map L: \(\mathrm{3\,square\,inches = 2,700\,square\,miles}\)
• Difference: \(\mathrm{2,700 - 1,200 = 1,500\,square\,miles}\)

This matches answer choice E: 1,500

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the scale conversion from linear to area: Students often forget that when converting from linear scale to area scale, the scale factor must be squared. They might think that if \(\mathrm{1\,inch = 20\,miles}\), then \(\mathrm{1\,square\,inch = 20\,square\,miles}\), instead of correctly calculating \(\mathrm{1\,square\,inch = 20 \times 20 = 400\,square\,miles}\).

2. Missing the comparative nature of the question: The question asks "how many MORE square miles on Map L than on Map K" but students might overlook the word "more" and simply calculate the area representation for one map instead of finding the difference between both maps.

3. Confusing which map has the larger scale: Students might incorrectly assume that Map K (\(\mathrm{1\,inch = 20\,miles}\)) represents larger areas than Map L (\(\mathrm{1\,inch = 30\,miles}\)), when actually Map L represents larger real-world areas because 30 miles per inch is a larger scale than 20 miles per inch.

Errors while executing the approach

1. Arithmetic errors when squaring the scale factors: Students might make calculation mistakes when computing \(\mathrm{20^2 = 400}\) or \(\mathrm{30^2 = 900}\), or when multiplying these by 3 to get the total areas (1,200 and 2,700 respectively).

2. Incorrect subtraction order: When finding the difference, students might subtract in the wrong order (Map K - Map L instead of Map L - Map K), leading to a negative result of -1,500 instead of the correct positive 1,500.

Errors while selecting the answer

1. Selecting an intermediate calculation instead of the final answer: Students might select 400 (choice B) which represents the square miles per square inch for Map K, or 900 (choice D) which represents the square miles per square inch for Map L, instead of the final difference of 1,500.