document.write( "Question 1169857: A group of 14 people spent $28 for admission tickets to Cinema Uno. The theater charges $2.50 for adults , $1.50 for students, and $1 for children. If the same people had attended Cinema Doz, which charges $4 for adults, $2 for students, and $1 for children, they would have spent $42 for admission tickets . How many adults, how many students, and how many children were in the group? \n" ); document.write( "

Algebra.Com's Answer #794694 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "The given information is not sufficient to find a single answer.

\n" ); document.write( "x = # adults
\n" ); document.write( "y = # students
\n" ); document.write( "z = # children

\n" ); document.write( "(1) $\"x%2By%2Bz+=+14\"$ [number of people in the group is 14]
\n" ); document.write( "(2) $\"2.5x%2B1.5y%2Bz+=+28\"$ [cost at Cinema Uno is $28]
\n" ); document.write( "(3) $\"4x%2B2y%2Bz+=+42\"$ [cost at Cinema Doz is $42]

\n" ); document.write( "Subtract (1) from (2):
\n" ); document.write( "(4) $\"1.5x%2B0.5y+=+14\"$
\n" ); document.write( " $\"3x%2By+=+28\"$

\n" ); document.write( "Subtract (2) from (3):
\n" ); document.write( "{5) $\"1.5x%2B0.5y+=+14\"$
\n" ); document.write( " $\"3x%2By+=+28\"$

\n" ); document.write( "Equations (4) and (5) are the same; that tells us there will NOT be a single solution to the set of equations.

\n" ); document.write( "Since the problem requires solutions in non-negative integers, there might be a single solution; but there might be more than one.

\n" ); document.write( "We know that
\n" ); document.write( " $\"3x%2By+=+28\"$

\n" ); document.write( "and, since the total number in the group is 14, we know that
\n" ); document.write( " $\"x%2By+%3C=+14\"$

\n" ); document.write( "Comparing those two, we know that
\n" ); document.write( " $\"2x+%3E=+14\"$
\n" ); document.write( " $\"x+%3E=+7\"$

\n" ); document.write( "So the number of adults is at least 7.

\n" ); document.write( "Now find all solutions, knowing that
\n" ); document.write( "(1) x >= 7
\n" ); document.write( "(2) y = 28-3x
\n" ); document.write( "(3) x+y+z = 14:

\r\n" );
document.write( "   x  y  z\r\n" );
document.write( "  ---------\r\n" );
document.write( "   7  7  0\r\n" );
document.write( "   8  4  2\r\n" );
document.write( "   9  1  4

\n" ); document.write( "Larger values of x will result in negative values for y, so these are all the solutions.

\n" ); document.write( "ANSWERS:
\n" ); document.write( "(1) 7 adults and 7 students; OR
\n" ); document.write( "(2) 8 adults, 4 students, and 2 children; OR
\n" ); document.write( "(3) 9 adults, 1 student, and 4 children

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

\n" );