Problem Solving

Each dot in the mileage table above represents an entry indicating the distance between a pair of the five cities. If the table were extended to represent the distances of 30 cities and each distance were to be represented by only one entry, how many entries would the table then have?

(A) 60
(B) 435
(C) 450
(D) 465
(E) 900


The answer is B 900-30=870

870/2=435

The part I don't understand is - why dividing by 2? And if have to put only one dot for one pair of cities, wouldn't it extremely decrease the number of dots (entries). So, my question is why dividing by 2?

There are two ways to represent the distance information. You either represent it by putting a dot in A(row)B(column) or you put a dot in B(row)A(column). Since the question wants only one half of the table full, you divide 870 by 2; meaning you choose to fill only one half of the table and represent each distance by only one entry, as the question asks.

Dividing by 2 decreases the number of dots significantly but does not take away anything from the information of the table.

Pharo wrote:There are two ways to represent the distance information. You either represent it by putting a dot in A(row)B(column) or you put a dot in B(row)A(column). Since the question wants only one half of the table full, you divide 870 by 2; meaning you choose to fill only one half of the table and represent each distance by only one entry, as the question asks.

Dividing by 2 decreases the number of dots significantly but does not take away anything from the information of the table.

Does this mean that in column for city E (example) it would be two dots instead of four. Or it would me only one dot? I am not sure about this term "only one entry" - does it mean only one distance between one and other cities (one distance from city E to all other cities)?

Pharo wrote:There are two ways to represent the distance information. You either represent it by putting a dot in A(row)B(column) or you put a dot in B(row)A(column). Since the question wants only one half of the table full, you divide 870 by 2; meaning you choose to fill only one half of the table and represent each distance by only one entry, as the question asks.

Dividing by 2 decreases the number of dots significantly but does not take away anything from the information of the table.

@latte

Distance between E and A , for example, will be same as distance between A to E
The question asks us to find ONLY ONE entry for the same

hope that helps

mmslf75 wrote:

Pharo wrote:There are two ways to represent the distance information. You either represent it by putting a dot in A(row)B(column) or you put a dot in B(row)A(column). Since the question wants only one half of the table full, you divide 870 by 2; meaning you choose to fill only one half of the table and represent each distance by only one entry, as the question asks.

Dividing by 2 decreases the number of dots significantly but does not take away anything from the information of the table.

@latte

Distance between E and A , for example, will be same as distance between A to E
The question asks us to find ONLY ONE entry for the same

hope that helps

I see, and only one distance for example A-E is found in the first table isn't it? There are no few distances from the same cities - so I just ask what's the difference when we look at the first table?

Its a simple sum of AP...

Sn = n/2(2a+(n-1)d)
= 29/2(2+28)
=435

IMO B

ksundar wrote:Its a simple sum of AP...

Sn = n/2(2a+(n-1)d)
= 29/2(2+28)
=435

IMO B

Can you please explain to me the difference between this first table and the table that is to make (except additional it is 30x30 and not 5x5) ? There is already one entry for every city...?

Lattefah84 wrote:

ksundar wrote:Its a simple sum of AP...

Sn = n/2(2a+(n-1)d)
= 29/2(2+28)
=435

IMO B

Can you please explain to me the difference between this first table and the table that is to make (except additional it is 30x30 and not 5x5) ? There is already one entry for every city...?

There is no difference except for extending the table to be 30X30.

I was working on this question in the GMAT Official Guide, and there's a way to get this question right in about 30 seconds with almost no math, so I thought I'd share. I've been training to be aware of the answer options as I'm breaking the question down, and here, as is so often the case, it's clutch:

(A) 60
(B) 435
(C) 450
(D) 465
(E) 900

So, if our 30 city table has 900 cells, is there any way all 900 cells would have an entry (dot)? No way, right?
Then, we can even use the example table as a reference. Notice about half of the cells have dots, but is it exactly half? No. So C is out. In fact, we can see from both the description and the table that there are no entries from a town to itself. I mean do we care about the distance from Town A to Town A? 0? No. So notice, that we're talking just a little bit less than a half. Which option is the only option that's a little bit less than a half of 900. B. This is not guessing. It's guaranteed that logically, option B has to be correct.