SOLUTION: some students appeared for mathematics test . average marks scored by successful students were 78 and those who failed in that test scored average marks 60 . if the average marks o

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Question 709418: some students appeared for mathematics test . average marks scored by successful students were 78 and those who failed in that test scored average marks 60 . if the average marks of all students who appeared for this exam were 71, then find the number of students who appeared for this exam ?
Answer by KMST(5328) (Show Source):
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= number of successful students
= number of students who failed
We will remember, during all algebraic manipulations, that and are positive integers.
= number of students who took the test (also a positive integer).

The average score for a group of students is calculated by adding the scores of all students, and dividing the sum of scores obtained by the number of students in the group.
So the sum of the scores for successful students was
and the sum of the scores for students who failed was .
The sum of the scores for all students was .

We have one linear equation:
<--> <--> <-->
With another linear equation, we would have a system of linear equations that could have a unique solution.
With just we can just say that the number of students must have been a multiple of 12.
Because is and also , that number is a multiple of 5 and 7.
So must be a multiple of 7
and must be a multiple of 5.
We can write is a multiple of 5 as for some integer .
Then --> -->
and -->
Without any more information, all we can say is that the number of students who took the test was a multiple of 12.
If we were told that less than 20 students appeared for the test,
we would know that it was and