Online Word Problems #10

[Carla]

Online Question Bank : Word Problems : #10

I had trouble with statement (2) for this example.

[Guest]

Of all the houses on Kermit Lane, 20 have front porches, 20 have front yards, and 40 have back yards. How many houses are on Kermit Lane?
(1) No house on Kermit Lane is without a back yard.

(2) Each house on Kermit Lane that has a front porch does not have a front yard.

[StaceyKoprince]

I can't draw a Venn diagram here - but you can look at the diagram in the official explanation if you need to visualize.

20 = front porches
20 = front yards
40 = back yards

So, at the least, there are 40 houses. A house could have 1, 2, or all 3 of the above characteristics, or it could have none of them.

(You said you had trouble with statement 2, so I'll only address that one.)
Statement 2 says that, if a house has a front porch, then it does not have a front yard. So the first two categories, above, don't overlap. I can conclude that I have at least 40 houses (20 with front porches and 20 with front yards - and these could overlap completely with the 40 that have backyards to give 40 houses total) but I could also have, say 80 houses: 20 with only front porches, 20 more with only front yards, and 40 more with only back yards. Or some other combo - but I've just found two different amounts, so I can stop here because I can't answer the question with this statement.

[RubenM342]

As a follow-up question, I initially tried to solve this problem using the 2*2 overlapping set matrix--assuming the following possible choices (Front Yard, Back Yard, Front Porch, Back Porch) but figured this was not the right method after plugging in the values (ie. 20 front yards and 40 back yards which would give you a total of 60 houses on Kermit Lane (from the question stem) assuming these were exhaustive and mutually exclusive).

Therefore, how do you determine when it is best to use the venn diagram v. overlapping set matrix? I recall the book saying if you have 3 or more choices for one particular set, but in this case I automatically categorized "Yard" and "Porches" as two different sets with two possible choices. Any clarity would be appreciated. Will a question usually explicitly state/hint when options are mutually exclusive? Thanks

[Sage Pearce-Higgins]

A double-set matrix wouldn't work out here at all (where did back porches come from!?). The basic idea of a set matrix is that you have Yes / No possibilities, e.g. front porch and no front porch as two columns. Here, since you have 3 possible features (apparently in any combination), then you have a 3 set problem, and using a Venn diagram would be suitable here. Check out chapter 10 of the Word Problems strategy guide for more on this.

However, don't let strategies get in the way of basic logic. Statement 1 tells you something pretty important: all the houses on the street have back yards. And statement 2 doesn't tell you anything about houses with no features. Thinking of it in those terms is probably easier.

[annec607]

Hi! I was hoping someone could provide a bit more on statement 1:

If all the houses on kermit lane have a backyard then does that mean the 20 houses with front porches and 20 houses with front yards also have backyards?

Why then can we not add up all the categories:

20 with front porches + 20 with front yard + 40 with backyards = 80 houses on kermit lane

I appreciate the response!

[Sage Pearce-Higgins]

You've got the right answer (that statement (1) is sufficient), but using unreliable reasoning. If you add up the categories then you're assuming that the 20 houses with front porches don't have back yards. This contradicts the information that we're told in statement (1).

This problem follows the wording that you'll find in GMAT problems involving sets. In the question stem, we're told that 20 houses have front porches. Now, be careful not to interpret this to mean 20 houses have only front porches (i.e. they don't have front yards or back yards). Sure, 20 houses have front porches, but these 20 houses may or may not have front yards and back yards. With this kind of problem, I encourage you to think of the "overlap" of the sets. In fact, when I read the question stem, my first thought is 'perhaps some houses have front porches, front yards, and back yards'.