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Problem-Solving and Data Analysis / Ratios, rates, proportional relationships, and units Difficulty: Medium

For a certain rectangular region, the ratio of its length to its width is $35$ to $10$ . If the width of the rectangular region increases by $7$ units, how must the length change to maintain this ratio?

It must decrease by $24.5$ units.

It must increase by $24.5$ units.

It must decrease by $7$ units.

It must increase by $7$ units.

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Explanation

Choice B is correct. It’s given that the ratio of the rectangular region’s length to its width is $35$ to $10$ . This can be written as a proportion: $\frac{length}{width} = \frac{35}{10}$ , or $\frac{l}{w} = \frac{35}{10}$ . This proportion can be rewritten as $10 script l equals 35 w$ , or $script l equals 3.5 w$ . If the width of the rectangular region increases by $7$ , then the length will increase by some number $x$ in order to maintain this ratio. The value of $x$ can be found by replacing $l$ with $l + x$ and $w$ with $w plus 7$ in the equation, which gives $l + x = 3.5 (w + 7)$ . This equation can be rewritten using the distributive property as $l + x = 3.5 w + 24.5$ . Since $script l equals 3.5 w$ , the right-hand side of this equation can be rewritten by substituting $l$ for $3.5 w$ , which gives $l + x = l + 24.5$ , or $x equals 24.5$ . Therefore, if the width of the rectangular region increases by $7$ units, the length must increase by $24.5$ units in order to maintain the given ratio.

Choice A is incorrect. If the width of the rectangular region increases, the length must also increase, not decrease.

Choice C is incorrect. If the width of the rectangular region increases, the length must also increase, not decrease.

Choice D is incorrect. Since the ratio of the length to the width of the rectangular region is $35$ to $10$ , if the width of the rectangular region increases by $7$ units, the length would have to increase by a proportional amount, which would have to be greater than $7$ units.