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Ralph is mowing a yard 16 m long and 12 m wide. He mows continuously around the yard working towards the centre. He wonders how wide a strip must be cut before he is half done.

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I'm going to assume that the problem intends the following:

Of a 16-by-12 rectangle, a border of width $x$ is shaded. If the area of the shaded border is equal to the unshaded rectangular region in the middle, what is $x$?

Under that interpretation, the unshaded rectangle in the middle has dimensions $16-2x$ and $12-2x$, so area $(16-2x)(12-2x)$, and should have half the area of the original rectangle, so $$(16-2x)(12-2x)=\frac{1}{2}\cdot 16\cdot 12.$$ Solving gives $x=2$ or $x=12$ (which is out of range given the context), so a border of width 2 m.

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The answer to the question is correct but can someone explain the reasoning for the 2x and the rest of the equation – user9697 Apr 17 '11 at 21:06
If you draw a picture of a rectangle inside a 16x12 with the inside sides offset by $x$, you should be able to convince yourself that the inner one is $(16-2x)\times(12-2x)$. Then he says this small rectangle. is half the area of the 16x12 – Ross Millikan Apr 17 '11 at 21:09

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