An (x + 4) by (x + 6) rectangle is compared with an (x + 5) by (x + 5) square. Which area is larger?
Answer and explanation
Correct answer: The square is larger by 1
The governing concept is algebraic comparison of equivalent area expressions. The rectangle has dimensions x + 4 and x + 6, so its area is (x+4)(x+6) = x² + 10x + 24. The square has side x + 5, so its area is (x+5)² = x² + 10x + 25. Subtracting the rectangle’s area from the square’s area gives (x²+10x+25) − (x²+10x+24) = 1. Thus the square is larger by exactly one square unit whenever the stated dimensions are valid lengths. This also illustrates that numbers equally spaced around x+5 produce a slightly smaller product than the square of their average. Therefore option B is correct. The x² and 10x terms cancel, so options C and D cannot be correct.
Frequently asked questions
What is the correct answer to this question?
The square is larger by 1
Why is this the correct answer?
The governing concept is algebraic comparison of equivalent area expressions. The rectangle has dimensions x + 4 and x + 6, so its area is (x+4)(x+6) = x² + 10x + 24. The square has side x + 5, so its area is (x+5)² = x² + 10x + 25. Subtracting the rectangle’s area from the square’s area gives (x²+10x+25) − (x²+10x+24) = 1. Thus the square is larger by exactly one square unit whenever the stated dimensions are valid lengths. This also illustrates that numbers equally spaced around x+5 produce a slightly smaller product than the square of their average. Therefore option B is correct. The x² and 10x terms cancel, so options C and D cannot be correct.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Visual models of identities.
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