Finding Rectangle Dimensions Using a 'More Than Twice' Relationship
Apply the geometry problem-solving strategy when one dimension of a rectangle is described using both multiplication and addition relative to the other — a step up from problems where the relationship involves only addition or subtraction.
Problem: The length of a rectangle is four centimeters more than twice the width. The perimeter is centimeters. Find the length and width.
- Read: A rectangle has cm, and its length is four more than twice its width.
- Identify: The length and width of the rectangle.
- Name: Let = the width. The phrase "four more than twice the width" combines two operations: twice signals multiplication by , and four more than signals adding . So the length is . Draw and label the rectangle with width and length .
- Translate: Write the perimeter formula and substitute:
- Solve: Distribute: . Combine like terms: . Subtract from both sides: . Divide both sides by : . The width is cm. Find the length: . The length is cm.
- Check: , and
- Answer: The length is cm and the width is cm.
Unlike problems where one dimension differs from the other by a simple constant (e.g., "two less than the length"), this problem's relationship involves both multiplication and addition — the expression captures both operations. When is distributed during the perimeter substitution, it produces , which then combines with to yield a variable coefficient of rather than . This larger coefficient is what distinguishes 'more than twice' rectangle problems from simpler additive ones.
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