Six countries in a certain region sent a total of 75 representatives

Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives. Of the six countries, if Country A sent the second greatest number of representatives, did Country A send at least 10 representatives?

Given: \(x_1<x_2<x_3<x_4<A<x_6\) and \(x_1+x_2+x_3+x_4+A+x_6=75\). Q: is \(A\geq{10}\)


(1) One of the six countries sent 41 representatives to the congress --> obviously \(x_6=41\) --> \(x_1+x_2+x_3+x_4+A=34\).

Can \(A\geq{10}\)? Yes. For example: \(x_1=2\), \(x_2=3\), \(x_3=8\), \(x_4=10\), \(A=11\) --> \(sum=34\) (answer to the question YES);
Can \(A<{10}\)? Yes. For example: \(x_1=4\), \(x_2=6\), \(x_3=7\), \(x_4=8\), \(A=9\) --> \(sum=34\) (answer to the question NO).


(2) Country A sent fewer than 12 representatives to the congress --> \(A<{12}\).

The same breakdown works here as well:
Can \(12>A\geq{10}\)? Yes. For example: \(x_1=2\), \(x_2=3\), \(x_3=8\), \(x_4=10\), \(A=11\), \(x_6=41\) --> \(sum=75\) (answer to the question YES);
Can \(A<{10}\)? Yes. For example: \(x_1=4\), \(x_2=6\), \(x_3=7\), \(x_4=8\), \(A=9\), \(x_6=41\) --> \(sum=75\) (answer to the question NO).


(1)+(2) The given examples fit in both statements and A in one is more than 10 and in another less than 10. Not sufficient.

Answer: E.
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It is when you have a plan! You should know how to manipulate numbers and examples.
Let me explain.

6 countries, 75 people. No 2 countries sent the same number of people.

Stmnt 1: One country sent 41 people.
The other 5 together sent 75 - 41 = 34 people. 'A' sent the most number of people from the remaining 5 countries. Does A need to send atleast 10?
34 divided by 5 is approximately 7. On average, every country sent 7 people so we can split it like 4, 6, 7, 8, 9 (try and split around 7 so that the average stays apprx 7). A sent 9 people here.
A could have sent more than 10 people of course (say if the other 4 countries sent 1, 2, 3 and 4)
Not sufficient.

Stmnt 2: A sent fewer than 12
A could have sent 9 people (example above) or 11 (Split is 2, 6, 7, 8, 11. Try to make minimum changes to get what you want so that you can minimize the chances of error. Since I want to increase the last number, I just reduce the first one appropriately).
Again, not sufficient.

Using both statements together, A could have sent 9 or 11 people so not sufficient.

Answer (E)
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I took one good look at this problem and had the word "guess" in my head. After doing this problem again, I realized that it was not as difficult as it first looked. The GMAT is definitely beatable! Not giving up.

Excellent explanation. Thanks very much.

Why and how is it obvious that obviously \(x_6=41\)

Only the country which sent greatest number of representatives (\(x_6\)) could have sent 41, as 41 is more than half of the total (75).
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I understand by putting the values we can get but during exam its tough to out so many values and get it. I am thinking can there any other simpler and quick solution.

Putting values like 4,6,7,8 and 11 is not easy.

Great explanation, I need to spend time checking to prove and disprove these type of questions. For some reason I have a bad habit of rushing DS questions and not thinking through.

Bunuel,
Sorry :s for making this question again hehe:
Is this a word problem question or another type of question? I cannot recognize it using the MGMAT guides.
Could you provide me more links related or tell me how to find them?
Thank you!

Yes, it's a word problem but also a bit like min/max type of question.

You can find all kinds of questions in our question banks here: viewforumtags.php

DS min/max questions: search.php?search_id=tag&tag_id=42
PS min/max questions: search.php?search_id=tag&tag_id=63

Hope it helps.
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Thx for the answer!
But I still have a question,,,,,,,
Since the question we need to anwer is : did A at least send 10 representitives?
so if I can conclude tht : yes,A at least sent 10. it is sufficient.
if I conclude tht: no, A didn't not at least send 10. it is also sufficient.
Because GMAC says if we can hve one specific answer, it is sufficient.

Well, when I consider(1), clearly I can say that: no, A didn't not at least send 10.

then why not choose A??????

In a Yes/No Data Sufficiency questions, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

For the first statement we can have an YES as well as a No answer to the question, which means that the statement is NOT sufficient.

For example, check here: six-countries-in-a-certain-region-sent-a-total-of-93368.html#p718376 or here: six-countries-in-a-certain-region-sent-a-total-of-93368.html#p1075895

Hope it helps.
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Well this question is really a good one!

I got E, but I wasn't too sure about this answer. The best way to do such questions is asking yourself whether it is possible for you to answer what has been asked.

Bunuel's approach is excellent, but I think it would require at least 3-3.5 mins of working.

Hi guys, I'm certainly no math"magician" and just started studying for the GMAT and as such perhaps why I did not take the same approach to solve the question but it seems to be the answer of "E" is wrong, can anyone help figure out if I'm just insane ?

I went with a basic trial & error approach and stated the following:

75 Reps = R
Country A or A # of R cannot be = to B, C, D, E ,F
A must be # 2 in terms of high # of R and need to find out if A is = or greater than 10 ?

Condition 1: One of the six countries sent 41 representatives to the congress

So filling in the blanks by trial & error:

Country # R Ranking

A = 10 2
B = 41 1

which we know must be respected, so 75 - 51 = 24 R left, if we randomly assign countries C, D, E, F, #'s of R that are not the same, lets say like :

C = 9 3
D = 8 4
E = 4 5
F = 3 6

Then the total = 75 which makes condition # 1 work and is true ??

If we sub in condition # 2: Country A sent fewer than 12 representatives to the congress

this still holds true and you could make A = 11 and C = 9, D = 8, E= 4, F = 2 and this would still work and add up to 75 ? as well as many other combination allowing for condition 1 & 2 to hold true so it seems as though each of them alone is enough to satisfy the questions ? Am I missing something ? If someone can help clarify it would be highly appreciated

Matt

Hi Matt

Let me try to see if I can help. Firstly, in DS questions, even if we are unable to calculate a unique value, there should at least be a unique proper answer to the question asked. Eg, if a question asks 'Is XYZ true or no' then the answer needs to be either a clear-cut YES or a clear-cut NO - then only we can say that we have got our answer.

In the given question, 'did Country A send at least 10 representatives', we can be sure that we have our answer in two situations:

Either if we can logically conclude that there were at least 10 representatives - in this case we have our answer as YES to the question asked
Or if we can logically conclude that there were less than 10 representatives for sure - in this case we have our answer as NO to the question asked

But if we cannot conclude any one (and only one) of the above two things then our conclusion would be that 'the data is insufficient to answer the question asked'.
Lets now analyse the statements.

(1) One country sent 41, if country A sent 10, then remaining 75-51 = 24; which can be divided among 4 different countries (say 3, 4, 8, 9 as you depicted).
But if country A sent 9, then remaining 75-50 = 25; this too can be divided among 4 different countries (say 4, 6, 7, 8).

So, the given condition of statement 1 is possible with country A having 10 R, and it is also possible with country A having 9 R.
Thus, we cannot say with surety whether country A has sent at least 10 R or not. That is why this condition is not sufficient to give either a clear-cut YES or a clear-cut NO to the question asked.

(2) As you have yourself said, it is possible with country A sending 11 R. But it is also possible with country A sending 4 R or 9 R (you can make various combinations as you have yourself written).
Again, since it is possible for this condition of statement 2 to hold true with both A having >= 10 R, and also possible with A having < 10 R, we cannot answer with a clear-cut YES or a clear-cut NO to the question asked. Not sufficient.

When we combine the two statements, still we could make a case with country A having 10 R or 11 R, but we could also make a case with country A having 9 R. This data is still insufficient to give us either a proper YES or a proper NO to the asked question (which was 'did Country A send at least 10 representatives'). That is why answer is E.

Here is how I approached this one :

we know total 6 countries sent 75 reps

Statement1: One of the countries sent 41

So we can deduce that 5 countries sent 34 reps (75-41)

I quickly drew 5 dashes and then started :

9 8 7 6 4 = 34

So we can say country A did send less than 10 representatives


Or we could also say,

15 10 5 2 1 = 34

So not sufficient.

statement 2 Is also of no help

hence E

Hi @Karisham

When solving this problem - the bit in ed and purple

Are you actually doing this all in your head (i.e. mentally /logically in your mind) or are you actually writing out the list

If i write lists like this that are so close -- i wil definitely make a mistake

Thank you !

Denote the arrangement of the six countries in ascending order (lowest to highest). B sent the lowest number of reps and F sent the highest number of reps. The letters are arbitrary, except for A.


B C D E A F
(B = Lowest) (A = Second highest) (F = Highest)


(1) One of the six countries sent 41 representatives to the congress

We can identify who could have sent 41 reps. If we remove the 41 from consideration, we will have 75 - 41 = 34 reps to be distributed among the remaining five countries such that A is the second highest.

B, C, D and E could not have sent 41 reps (each) because then A will not be the second highest. Now if A sent 41 reps, F cannot be the highest (only 34 reps left for 5 countries). So, we can let F = 41.

Check whether A can be less than 10.

If A = 9, F = 41, we have to distribute remaining 25 among 4 countries such that each of them sent unique number of reps. So we could have: B = 4, C = 6, D = 7, E = 8, A = 9, F = 41.

Check whether A can be greater than or equal to 10.

If A = 10, F = 41, we have to distribute remaining 24 reps among 4 countries such that each of them sent unique number of reps. So we could have: B = 3, C = 6, D = 7, E = 8, A = 10, F = 41.

Since we have two different possibilities for A, this statement is INSUFFICIENT.


(2) Country A sent fewer than 12 representatives to the congress

We can resuse the plausible assignments from the first statement and determine that there are at least two different arrangements:

B = 4, C = 6, D = 7, E = 8, A = 9, F = 41.

B = 3, C = 6, D = 7, E = 8, A = 10, F = 41.

This statement is INSUFFICIENT.


Statements (1) and (2) together:

One of the six countries sent 41 representatives to the congress AND Country A sent fewer than 12 representatives to the congress.

We can resuse the plausible assignments from the first statement and determine that there are at least two different arrangements:

B = 4, C = 6, D = 7, E = 8, A = 9, F = 41.

B = 3, C = 6, D = 7, E = 8, A = 10, F = 41.

Statements (1) and (2) taken together are INSUFFICIENT.

ANSWER: (E)

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