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Chapter 1: Problem 1

Two rectangular rugs are on display in a showroom. If one rug is twice as long as the other, does this necessarily mean that its area is also twice as large as that of the second? Explain.

Short Answer

Expert verified
Answer: Yes, when the widths of both rugs are the same, the area of the first rug with a length twice as long as the second rug will indeed be twice as large as the area of the second rug.

Step by step solution

01

Defining the given dimensions

Let the length of the smaller rug be L and its width be W. Then, the dimensions of the larger rug would be 2L (since it's twice as long) and W (assuming the widths of both rugs are the same).
02

Calculating the areas of the two rugs

To calculate the area of a rectangle, we must multiply its length by its width. Let's calculate the areas for both rugs: 1. Area of the smaller rug: Area1 = L * W 2. Area of the larger rug: Area2 = 2L * W
03

Comparing the areas

Now we will analyze the areas calculated above. Area2 = 2L * W Area1 = L * W Divide Area2 by Area1 to see if it's exactly twice the size of Area1: Area2/Area1 = (2L * W) / (L * W) Here, L * W in the numerator and denominator cancel out, leaving us with: Area2/Area1 = 2
04

Arriving at the conclusion

By comparing the areas of the two rugs, we can now see that the area of the larger rug is indeed twice as large as the area of the smaller rug. Therefore, the statement given in the problem is true for this specific context, where we assumed the widths of both rugs are the same. When the widths are different, the areas can be different as well. However, keep in mind that this is true only because we assumed the widths of both rugs were the same in this scenario. If the dimensions were different, the relationship between the areas would also be different.

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