document.write( "Question 1108299: a dance recital had 100 people in the audience. Gina attended with both her parents and found that tickets prices were 50 cents for children, $3 for parents, and $10 for others. If $100 was collected from the 100 people, find out how many parents attended \n" ); document.write( "
Algebra.Com's Answer #723397 by greenestamps(13203)\"\" \"About 
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\n" ); document.write( "Let x be the number of children
\n" ); document.write( "Let y be the number of parents
\n" ); document.write( "Let z be the number of other people

\n" ); document.write( "Note that the statement of the problem requires y to be at least 2, since both Gina's parents attended.
\n" ); document.write( "Then...

\n" ); document.write( "(1) \"x%2By%2Bz+=+100\" there were 100 people in attendance
\n" ); document.write( "(2) \".5x%2B3y%2B10z+=+100\" the total cost of the tickets was $100

\n" ); document.write( "We have three unknowns but only two equations; but we can find the answer to the problem (or in some similar problems a small family of solutions) using the fact that the variables have whole number values.

\n" ); document.write( "To solve this kind of system of equations, we eliminate one variable and use the requirement that the solutions be whole numbers to find the answer.

\n" ); document.write( "Double equation (2) and subtract equation (1) from the result:

\n" ); document.write( "\"x%2B6y%2B20z+=+200\"
\n" ); document.write( "\"x%2By%2Bz+=+100\"
\n" ); document.write( "\"5y%2B19z+=+100\"

\n" ); document.write( "Solve this equation for one variable in terms of the other:

\n" ); document.write( "\"19z+=+100-5y\"
\n" ); document.write( "\"z+=+%28100-5y%29%2F19+=+%285%2820-y%29%29%2F19\"

\n" ); document.write( "The only whole number value of y that makes z also a whole number is y=1. That makes z=5; and then x=94, because the total number of people was 100.

\n" ); document.write( "But that means there were 94 children, 1 parent, and 5 other people in attendance -- and the problem requires the number of parents to be at least 2.

\n" ); document.write( "So THERE IS NO SOLUTION to the problem as stated.

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

\n" ); document.write( "Tutor ikleyn is right....

\n" ); document.write( "The problem as stated requires that there be at least one child and two parents; but it does not require that there be any other people. So in my equation
\n" ); document.write( "\"z+=+%28100-5y%29%2F19+=+%285%2820-y%29%29%2F19\"
\n" ); document.write( "y=20 is valid, making z=0 and x=80.

\n" ); document.write( "I carelessly read the problem as requiring that there be at least one of each kind of person.
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