document.write( "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 ? \n" ); document.write( "

Algebra.Com's Answer #436572 by KMST(5328) $\"\"$ $\"About$

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$\"x\"$ = number of successful students
\n" ); document.write( " $\"y\"$ = number of students who failed
\n" ); document.write( "We will remember, during all algebraic manipulations, that $\"x\"$ and $\"y\"$ are positive integers.
\n" ); document.write( " $\"x%2By\"$ = number of students who took the test (also a positive integer).
\n" ); document.write( "
\n" ); document.write( "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.
\n" ); document.write( "So the sum of the scores for successful students was $\"78x\"$
\n" ); document.write( "and the sum of the scores for students who failed was $\"60x\"$ .
\n" ); document.write( "The sum of the scores for all students was $\"71%28x%2By%29=78x%2B60x\"$ .
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\n" ); document.write( "We have one linear equation:
\n" ); document.write( " $\"71%28x%2By%29=78x%2B60x\"$ <--> $\"71x%2B71y=78x%2B60y\"$ <--> $\"71y-60y=78x-71x\"$ <--> $\"5y=7x\"$
\n" ); document.write( "With another linear equation, we would have a system of linear equations that could have a unique solution.
\n" ); document.write( "With just $\"5y=7x\"$ we can just say that the number of students must have been a multiple of 12.
\n" ); document.write( "Because $\"5y=7x\"$ is $\"5y\"$ and also $\"7x\"$ , that number is a multiple of 5 and 7.
\n" ); document.write( "So $\"y\"$ must be a multiple of 7
\n" ); document.write( "and $\"x\"$ must be a multiple of 5.
\n" ); document.write( "We can write $\"x\"$ is a multiple of 5 as $\"x=5n\"$ for some integer $\"n\"$ .
\n" ); document.write( "Then $\"5y=7%285n%29\"$ --> $\"y=7%2A5%2An%2F5\"$ --> $\"y=7n\"$
\n" ); document.write( "and $\"x%2By=5n%2B7n\"$ --> $\"x%2By=12n\"$
\n" ); document.write( "Without any more information, all we can say is that the number of students who took the test was a multiple of 12.
\n" ); document.write( "If we were told that less than 20 students appeared for the test,
\n" ); document.write( "we would know that it was $\"n=1\"$ and $\"x%2By=12\"$ \n" ); document.write( "

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