SOLUTION: What can be concluded about the width of a rectangle if the ratio length to perimeter is 1 to 3? Make sketches and explain your reasoning.

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Question 471461: What can be concluded about the width of a rectangle if the ratio length to perimeter is 1 to 3? Make sketches and explain your reasoning.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
You'll have to do the sketches, but here's some information to help you.
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For a rectangle, the perimeter equals the sum of the four sides. Because you're working with a rectangle you know that the opposite sides are equal. You also know that one pair of the opposite sides, the Lengths, are longer than the other pair of opposite sides, the Widths. Use the letter L to represent the Length and the letter W to represent the Width. The equation that defines the Perimeter of a rectangle is that the Perimeter (call it P) is equal to the sum of the four sides. In equation form this is:
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P+=+L+%2B+W+%2B+L+%2B+W
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and when you add like terms on the right side this equation becomes:
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P+=+2L+%2B+2W
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The problem tells you that the ratio of the Length (L) to the Perimeter (P) is 1 to 3. This means that L is to P as 1 is to 3 and in equation form this is:
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L%2FP+=+1%2F3
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If you multiply both sides by P this becomes:
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L+=+P%2F3
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Then multiply both sides by 3 and you end up with:
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3L+=+P
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and by simply transposing sides this equation becomes:
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P+=+3L
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You can return to the equation that defines the Perimeter of a rectangle:
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P+=+2L+%2B+2W
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and substitute 3L for P to get:
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3L+=+2L+%2B+2W
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Get rid of the 2L on the right side by subtracting 2L from both sides of this equation. When you do, you are left with:
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L+=+2W
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This means that the Length is twice as long as the Width.
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This is a general equation for any rectangle in which the ratio of the Length to the Perimeter is 1 to 3. Therefore, you can conclude that for any rectangle if you make the Width equal to half the Length (or make the Length equal to twice the Width), the ratio of the Length of that rectangle to its Perimeter will be 1 to 3. You can also conclude that if are given the Perimeter of a rectangle and its Length is such that the ratio of the Length to the Perimeter is 1 to 3, then the Width of the rectangle will be half of its Length.
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Hope this helps you to understand the problem.