This question is an odd amalgam of a IR 2PA question with a DS question. I don't believe I have ever seen an official question in this particular format.

(A) The total number of fruits received by each student is divisible by 3 but not divisible by 8.
By itself, this says nothing.
With statement #1, the portions add up to 1+3+7=11, so the total will always be a multiple of 11 --- so it could be 33, 66, 99, 132, etc. All four of those first numbers are divisible by 3, not by 8, so there are too many possibilities. This statement doesn't determine anything. alone or with statement #1

(B) The teacher distributed a total of 12 apples, 36 bananas and 84 cherries.
(D) The teacher distributed more than 120 fruit items in total.
For both of these, we don't know the total number of students, so how many the teacher distributed in total tells us zilch about what an individual student received. either alone or in combination with statement #1.

(C) The teacher distributed 3 times as many bananas as apples and distributed more than 2 apples per student but less than 10 bananas per student.
By itself ---- more than two apples, so apples could be {3, 4, 5, ....}, and three times as many bananas, so bananas could be {9, 12, 15, ...} --- but if number of bananas must be less than 10 --- only the first case works, so, apples must be 3 and bananas must be 9. This one, by itself, determines the number of apples. This is the correct answer to the second column.

(E) Each student received no more than 40 cherries but no less than 15 bananas.
By itself, this doesn't tell us bupkis about apples.
With statement #1 --- no more than 40 cherries, and it must be a multiple of 7, so cherries could be {7, 14, 21, 28, 35}. According to the ratio, this would have the corresponding numbers of banana, respectively, (3, 6, 9, 12, 15} --- but if bananas can't be lower than 15, only the last possibility can be accepted. That would be 35 cherries, 15 bananas, and 3 apples.
With statement #1, this allows us to determine the number of apples. This is the correct answer to the first column.

Column #1: (E)
Column #2: (C)

BTW, in addition to presenting a format not supported by official material, this question also produces different numerical results for case (C) and case (E) ---- that falls short of the standard the GMAT maintains on Data Sufficiency question, in which all statements are consistent with a single set of answers.

Those are my thoughts.
Mike
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The question is flawed.
Option C says that the teacher distributed less than 10 bananas per student.
Option E says that each student received no less than 15 bananas.

This cannot happen. The number of bananas distributed is b and it cannot vary with the option.

Anyway, given the correct data, you would want to first identify the statement which is sufficient alone. It would be easy to figure it out since all the required data will be in a single statement.

A. The total number of fruits received by each student is divisible by 3 but not divisible by 8. - Not Sufficient
B. The teacher distributed a total of 12 apples, 36 bananas and 84 cherries. - Not Sufficient. We don't know the number of students.
C. The teacher distributed 3 times as many bananas as apples and distributed more than 2 apples per student but less than 10 bananas per student. - Apple:Banana = 1:3. No of apples = 3/4/5/6... and Number of bananas = 9/12/15. Since number of bananas is less than 10, number of bananas must be 9. This statement alone is sufficient.


Now, no other statement can be sufficient alone. So analyze each of the 4 remaining statements keeping statement 1 in mind:

Statement 1: The teacher handed out apples, bananas and cherries in the ratio of 1:3:7, respectively.
Each student received fruits in the ratio 1:3:7 i.e. number of fruits he/she received = 11/22/33/44/55...

A. The total number of fruits received by each student is divisible by 3 but not divisible by 8. - Each student could have received 33/66/99 etc fruits. Not sufficient.
B. The teacher distributed a total of 12 apples, 36 bananas and 84 cherries. - Statement 1 is already included in this information. Not sufficient.
D. The teacher distributed more than 120 fruit items in total. - Not sufficient.
E. Each student received no more than 40 cherries but no less than 15 bananas. - No of cherries received could be 35/28/21... and number of bananas will be 15/12/9 etc. Since number of bananas cannot be less than 15, the n umber of bananas must be 15. Sufficient along with statement 1.

Sufficient along with statement 1: (E)
Sufficient alone: (C)
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