Let the number of students in five sections, F, G, H, J and K are f, g, h , j & k respectively .
Thus average test score on all the students in the five sections = {(avg score of F, G & H)*(f+g+h) +(avg score of J & K)*(j+k)}/(f+g+h+j+k)
For simplicity , say, (f+g+h)=x & (j+k)=y , hence above expression ={(avg score of F, G & H)x +(avg score of J & K)y}/(x+y)

(1) The average of the final exam scores of students in sections F, G, and H was 72.
Hence the expression : {72x +(avg score of J & K)y}/(x+y).... no info for x, y or (avg score of J & K)..... Thus insufficient.

(2) The average of the final exam scores of students in sections J and K was 78.
Hence the expression : {(avg score of F, G & H)x +78y}/(x+y).... no info for x, y or (avg score of F, G & H)..... Thus insufficient.

(1) +(2) ===> The expression becomes 72x +78y)/(x+y).....................still no info for x, y......................... Thus insufficient.

Hence I would go for option E.
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Isn't x+y =5 in this case?

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The Q-Stem states : What is the average test score on all the students in the five sections ?

Thus we can not assume x+y =5... as that will in tern mean that all the sections have equal number of students .
And no where in the question the same is mentioned.
Hope this helps .
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The question stem does not tell us how many students are in each of the 5 sections. Statement 1 tells us the average of the
F,G and H is 72. This means we can represent the score of each student in the 3 sections by 72. Since all students and all sections have the same averages , the average of the 3 sections will be 72. However adding the scores of the other studentsin sections J and K may increase or decrease the average so we need to know the averages of the other 2 sections.INSUFFICIENT

Statement is just like statement 1 ,the averages of 2 sections and hence all students in J and K can be represented by 78 We do not know if the scores in the other 3 sections will bring the average down or up.We need to have their average scores as well. Taking both statements together, we get the averages for F,G,H, J and K as 72,72,72,78 and 78 BUT we do not know how many students are in each section even though we have the marks for each student If there is one student each in each section we get the marks to be 72,72,72,78,78 and hence an average of 74.4. If there are 2 students in section F and 1 student in the rest we get 72,72,72,72,78,78 which has an average of 72. BOTH statements are insufficient