sat-dog

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: $\frac{\text{length}}{\text{width}}=\frac{35}{10}$, or $\frac{l}{w}=\frac{35}{10}$. This proportion can be rewritten as $10l=35w$, or $l=3.5w$. 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 $w+7$ in the equation, which gives $l+x=3.5\left(w+7\right)$. This equation can be rewritten using the distributive property as $l+x=3.5w+24.5$. Since $l=3.5w$, the right-hand side of this equation can be rewritten by substituting $l$ for $3.5w$, which gives $l+x=l+24.5$, or $x=24.5$. 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.