"The number of sports": Are we assuming that the total area of G is equal to that of T ?

Link

https://gre.magoosh.com/questions/3414

Question

In arriving at the correct answer I assumed that the total metropolitan area of Goldsville is equal to that of Terrera.

Then I set up an "equation" based on the second sentence of the passage. Because I consider the area to be equal for each of the two cities, I substituted the relation deduced from the first sentence and arrived at correct answer choice D.

Relation 1 :-

(Total no of cars in downtown of T) / Downtown area of T = (Total no of cars in downtown of G) / Downtown area of G + 40 %

Relation 2 :-

(Total no of cars in suburbs + exurbs + downtown of G) / Metropolitan area of G = (Total no of cars in suburbs + exurbs + downtown of T) / Metropolitan area of T + 20 %

Substituting ,

(Total no of cars in suburbs + exurbs of G) / Metropolitan area of G = (Total no of cars in suburbs + exurbs of T) / Metropolitan area of T + (40 + 20 ) %

This is supported by answer choice D.

I further assumed that area of downtown = area of suburb = area of exurb.

I hope I am clear.

Kindly comment on the methodology.

Answer

While your answer did get you to the correct answer, it has a couple of flaws.

• We cannot assume that the areas of the two towns are equal, no. Although, we don't need to--we're only looking at the ratios of car/area. The areas can vary greatly, as long as the numbers of cars increase or decrease appropriately, the ratios will be the same. What you set equal didn't require equal area, only equal ratios. If we include the percentage difference, then it's fine (you'll see what I mean below).
• There's some confusion with the math you've used. First, we need to keep in mind what the %s represent--that is, they are (40% __* G cars / G area__) and (20% __* T cars / T area). We also can't substitute out the values that you did so easily.

If we made two equations using this information, it would look like this:

(Cars in downtown T) / (Area of downtown T) = 1.4 * (Cars in downtown G) / (Area of downtown G)

1.2 * (Cars in downtown T + Cars out of downtown T) / (Area of downtown T + Area out of downtown T) = (Cars in downtown G + Cars out of downtown G) / (Area of downtown G + Area out of downtown G)

I'm going to abbreviate those to make them easier to read.

(Cdt) / (Adt) = 1.4 * (Cdg) / (Adg)

1.2 * (Cdt + Cot) / (Adt + Aot) = (Cdg + Cog) / (Adg + Aog)

That second equation is important. It's different from

1.2 * (Cdt/Adt + Cot/Aot) = Cdg/Adg + Cog/Aog

That would make substitution a bit easier, and I think you tried to use it, but it's not a valid equation.

That being said, we can still infer from these

(Cdt) / (Adt) > (Cdg) / (Adg)

(Cdt + Cot) / (Adt + Aot) < (Cdg + Cog) / (Adg + Aog)

That Cot/Aot is less than Cog/Aog, which is what our answer choice D states.

But really, the best way to tackle this isn't exactly mathematically, but rather logically. If Terrera has comparatively many cars/mile in the downtown, but comparatively few cars/mile over all, than it must have relatively *few* cars/mile outside of the downtown. If, hypothetically, it had comparatively many cars/mile downtown AND comparatively many cars/mile outside downtown, it would have comparatively many cars/mile overall (and we know that's not true).

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