On her way home from work, Janet drives through several

We are given that Janet drives through several tollbooths on her way home from work. We need to determine whether there is a pair of tollbooths that are less than 10 miles apart.

Statement One Alone:

The first tollbooth and the last tollbooth are 25 miles apart.

Since we do not know the number of tollbooths between her home and her work, we cannot determine whether there is a pair of these tollbooths that are less than 10 miles apart.

For example, if there are only two tollbooths, then they are 25 miles apart and thus greater than 10 miles apart. However, if there are three tollbooths, there could be a pair of tollbooths that are less than 10 miles apart. Statement one alone is not sufficient. We can eliminate answer choice A.

Statement Two Alone:

Janet drives through 4 tollbooths on her way home from work.

Since we do not know the distance between her home and her work, we cannot determine whether there is a pair of tollbooths that are less than 10 miles apart. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

From statements one and two, we know that the distance between the 1st and 4th tollbooths is 25 miles, which means on average, approximately 8 miles are between each consecutive tollbooth. For example, it’s possible that the distances between the 1st and 2nd tollbooths, 2nd and 3rd tollbooths, and 3rd and 4th tollbooths are, respectively, 8 miles, 8 miles, and 9 miles. In this case, we have a pair (actually three pairs) of tollbooths that are less than 10 miles apart. Furthermore, regardless of how we adjust the number of miles for the three distances, there must be at least one pair of tollbooths less than 10 miles apart. For example, let’s say the distance between the 1st and 2nd tollbooths was increased to 10 miles and the distance between the 2nd and 3rd tollbooths was increased to 12 miles; then, the distance between the 3rd and 4th tollbooths would have to be 3 miles, which is less than 10 miles. Thus, the two statements together are sufficient.

Answer: C
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On her way home from work, Janet drives through several tollbooths. Is there a pair of these tollbooths that are less than 10 miles apart?

(1) The first tollbooth and the last tollbooth are 25 miles apart --> even if we take several to mean more than 2, still (1) is not sufficient as we can have 3 tollbooths as shown below:
|---(12.5)---|---(12.5)---| (answer No) OR |-(2.5)-|-----(22.5)-----| (answer Yes).

(2) Janet drives through 4 tollbooths on her way home from work. Clearly insufficient.

(1)+(2) We know that the first tollbooth and the last tollbooth are 25 miles apart AND that there are total of 4 tollbooths:
first-----(25)-----last
Now, we cannot place 2 more tollbooths between the first one and the last one (the fourth) so that at least one pair of tollbooths won't be less than 10 miles apart (no matter whether they are evenly spaced or not). Average distance, in case tollbooths are evenly spaced will be 8.(3), so at least one pair must be less than or equal to this distance as if all pairs are more than 8.(3) miles apart then the distance between the first and the last will be more than 25 miles. Sufficient.

Answer: C.
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I got this question from the GMATprep and scored 45 on the math section. I felt this question was tricky because I didn't know whether to assume that the tollbooths were evenly spaced. It would be a straight forward question if the tollbooths are evenly spaced. Should we assume that?

On her way home from work, Janet drives through several tollbooths. Is there a pair of these tollbooths that are less than 10 miles apart?

(1) The first tollbooth and the last tollbooth are 25 miles apart

(2) Janet drives through 4 tollbooths on her way home from work.


1) alone doesnt answer the question as i donot know the no of booths shw passes on her way
2) alone doesnt anwer the ques as i donot know the distance she travels

both 1) and 2) taken together will also not answer the question as i dont know if the distance between the booths

so answer is cannot be answered

You dont have to assume here that the distance is equal.
(1) not suff
(2) not suf
(1) (2) we know that The first tollbooth and the last tollbooth are 25 miles apart and there are 4 tollbooths on the way. Try to imagine the toolbooths: 2 at 11 miles, remain 14 miles but there is another tollbooths to put. So it has to be a pair with less than 10 miles between them.

Should not assume anything if the text doesnt tell you!

Tricky question, I was also lured to pick C and this could happen, if these kinds of questions come at the end of GMAT questions. You are right sondenso

Is it just me, or does several indicate more than 2? Given that reasoning my answer is A.
I know the OA is C.

Any thoughts?

(1) 2 tollbooths=no. 50 toll booths between the 1st and last=yes (that's a lot of stopping!). INSUFFICIENT, eliminate A and D.
(2) 4 tollbooths each 11 miles apart=no. 4 toolbooths each 9 miles apart=yes. INSUFFICIENT, eliminate B.
(1)+(2) There's no way to partition 4 tollbooths where the first and last are 25 miles apart without putting 2 within 10 miles of each other. Eliminate E.

Answer: C.


My interpretation of several is 2 or more.

Since it is not mentioned in the question that the tollbooths are evenly spaced, you cannot assume it. Answer will be (E).
If it is indeed a GMAT Prep question, it would be a very very old question with no relevance now.
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Oh yes of course. I got so lost in the 'evenly spaced' debate, I didn't focus on the actual question!
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Answer is C, evenly spaced or not.
There are total 25 miles and 4 tool both, so one of them has to be <10mile apart.

Given: On her way home from work, Janet drives through several tollbooths.

Target question: Is there a pair of these tollbooths that are less than 10 miles apart?

Statement 1: The first tollbooth and the last tollbooth are 25 miles apart.
There are several scenarios that satisfy statement 1. Here are two:

Case a: There are exactly 2 toll booths, and they are 25 miles apart. In this case, the answer to the target question is NO, there are NOT two toll booths that are less than 10 miles apart
Case b: There are exactly 3 toll booths (A, B and C). Their distances are: A......(5 miles)...B.......(20 miles).....C. In this case, the answer to the target question is YES, there ARE two toll booths that are less than 10 miles apart
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Janet drives through 4 tollbooths on her way home from work
There are several scenarios that satisfy statement 2. Here are two:
Let the toll booths be A, B, C and D.

Case a: A......(5 miles)....B.......(5 miles).....C.......(5 miles)....D In this case, the answer to the target question is NO, there are NOT two toll booths that are less than 10 miles apart
Case b: A......(15 miles)....B.......(15 miles).....C.......(15 miles)....D In this case, the answer to the target question is YES, there ARE two toll booths that are less than 10 miles apart
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Let's see if it is possible to satisfy both statements such that NO two toll booths are less than 10 miles apart.
So let's see what happens if we make every toll booth exactly 10 miles apart.
We get: A......(10 miles)....B.......(10 miles).....C.......(10 miles)....D
Since the total distance from the first and last toll booth is 30 miles (and not 25 miles as statement 1 suggests), we can be certain that at least one of the distance is above (between two adjacent tollbooths) MUST be less than 10 miles.
So, it must be the case that the answer to the target question is YES, there ARE two toll booths that are less than 10 miles apart

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Interesting Ques.

To check: Is there a pair of these tollbooths that are less than 10 miles apart?

Statement 1: The first tollbooth and the last tollbooth are 25 miles apart.
Considering several as min 3 tollbooths, we can have a pair 12.5 miles apart OR 5-20 miles apart. No sufficient answer we are getting..... Insufficient

Statement 2: Janet drives through 4 tollbooths on her way home from work
That's it, no distance total or among booths given...only this much info. Can't deduce anything.......InSufficient

Combine S1 and S2:
Total tollbooths now: 4
Distance between first and last: 25 miles

- Assume all are equidistant, each would be ~8.3 miles apart --> Yes, there is a/are pair(s) of these tollbooths that are less than 10 miles apart.
- Assume one is more than 10, say 15 miles.. other would be total 10 or less than 10 miles each apart. ---> Yes, there is a/are pair(s) of these tollbooths that are less than 10 miles apart.
- Assume another case, distance of two booths is 12, 12... in this case last one would be 1 mile -->Yes, there is a/are pair(s) of these tollbooths that are less than 10 miles apart.

So overall, there is a pair of these tollbooths that are less than 10 miles apart..... Sufficient

Option C
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Any of the 4 toll booths can be considered a pair (1st & 3rd, 2nd & 4th, 1st & 4th). It wasn't explicitly stated if we are considering a pair of consecutive toll booths. Are we really sure that the answer is C based on the statement alone? Correct me if my logic is incorrect. Thanks

Consecutive or not, the question asks whether there is ATLEAST one pair that is less than 10 miles apart

We have a 25 miles distance and 4 toll booths. If we average out the distance between the booths it would be 8 Miles. However we adjust the spacing between the toll booths at least one of them has to be less than 10 miles. The question would have been trickier if the distance would have been 30 miles or more rather than 25 miles.

Hence, the answer is C

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She passes thorough 4 tollbooths, that doesn't mean there are total 4 tollbooths. Statement 1 says -distance between 1st and last tollbooth is 25 miles not 1st and last tollbooth she passes through is 25 miles. 2nd says she crossed 4 but how to conclude that there are total 4 tollbooths?

Hello Karishma. But no-where it is mentioned that all tollbooths are in the same line. How can we assume that?

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