## tollbooths less than 10 miles

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AmunaGmat
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### tollbooths less than 10 miles

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

1. The first and the last tollbooths are 25 miles apart.

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

I am confused about C and E in this question. Gmatprep's answer is C.

What is the highlight in this question that the distance between the towns is additive or cumulative? To me, they can all be 10 miles apart. When I got this question in GMATprep it reminded me of 1) Question 40 DS, OG12 page 276. In this question before I looked at numbers correctly, i thought the distance from R to U is additive. 2) The RTD question 104 DS, OG12 page 282, if I am not told that the average speed of the train was constant, never assume.

Why does the same reasoning not work here?

Cheers!!!
tim
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### Re: tollbooths less than 10 miles

if the tollbooths are all 10 miles or more apart, you have to have 3 gaps in between 4 tollbooths, which means 30 miles or more. once we bring in statement 1, this is impossible. so some of the booths must be closer than 10 miles apart..
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rachelhong2012
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### Re: tollbooths less than 10 miles

There's an easy way to do this type of problem if you turn it into an equation and approach it algebraically.

assuming you're debating between C and E, which you are, combine these two statements. You can draw a line, put down four dots, and name each gap between two dots as X, Y, Z.

So you have this equation:

X+Y+Z = 25

Test different combinations of #'s for these variables and you will see that there's no way you can have a variable that is NOT less than 10.
tim
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### Re: tollbooths less than 10 miles

thanks!
Tim Sanders
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PavanS722
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### Re: tollbooths less than 10 miles

rachelhong2012 wrote:There's an easy way to do this type of problem if you turn it into an equation and approach it algebraically.

assuming you're debating between C and E, which you are, combine these two statements. You can draw a line, put down four dots, and name each gap between two dots as X, Y, Z.

So you have this equation:

X+Y+Z = 25

Test different combinations of #'s for these variables and you will see that there's no way you can have a variable that is NOT less than 10.

X=8, Y=8 and Z=9 ?
RonPurewal
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### Re: tollbooths less than 10 miles

That poster didn't phrase her thoughts in the right way, but it's clear to me that she was trying to say "At least one of the numbers will have to be less than 10". That's the goal of the problem, after all.
JohannaH678
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### Re: tollbooths less than 10 miles

Hi there,

I understand Tim's explanation from Jan 2012, however from what I understand he is assuming that all the toll booths are equidistant. No where in the question or the statements is that stated. For 1+2) Trying to find a way to make it NOT work: If this was plotted on a number line and the the first is at 0 miles and the last is at 25 miles and there are 4 total booths then perhaps there are tollbooths at 0 miles,1 mi, 24mi , and 25mi. In this case the 2nd and 3rd are in fact >10mi apart. If on the other hand they were at 0, 10, 11, 25 then the first pair and the last pair are both 10 mi apart. To me 1+2) Is NS. Am I missing something here? Why is the equidistant assumption valid?
RonPurewal
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### Re: tollbooths less than 10 miles

the question is:
Is there a pair that are less than 10 miles apart?

the answer to this question is "yes" for BOTH of the examples you gave... and it's ALWAYS going to be "yes", if you are putting 4 items into a linear span of 25 miles.

the only way to get "no" would be to put MORE than 10 miles between EVERY pair of consecutive itemsâ€”but that's not possible unless you have more than thirty miles of distance to work with.