document.write( "Question 1208467: A movie hall sold tickets to one of its shows in two denominations, $11 and $7. A fourth of all those who bought a ticket also spent $4 each on refreshments at the movie hall. If the total collections from tickets and refreshments for the show was $124, how many $7 tickets were sold? Note: The number of $11 tickets sold is different from the number of $7 tickets sold.
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Algebra.Com's Answer #846889 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A movie hall sold tickets to one of its shows in two denominations, $11 and $7.
\n" ); document.write( "A fourth of all those who bought a ticket also spent $4 each on refreshments at the movie hall.
\n" ); document.write( "If the total collections from tickets and refreshments for the show was $124,
\n" ); document.write( "how many $7 tickets were sold?
\n" ); document.write( "Note: The number of $11 tickets sold is different from the number of $7 tickets sold.
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document.write( "Let x be the number of movie tickets sold at $11.\r\n" );
document.write( "Let y be the number of movie tickets sold at  $7.\r\n" );
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document.write( "We know that x+y, the total number of visitors, is a multiple of 4.\r\n" );
document.write( "Also, we know that the total revenue was $124 and that x =/= y.\r\n" );
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document.write( "So, we write this equation for the total revenue\r\n" );
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document.write( "    11x + 7y +  = 124,   (1)\r\n" );
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document.write( "and we look for a solution to it in non-negative integer numbers such that and x=/= y and x+y is a multiple of 4.\r\n" );
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document.write( "We simplify equation (1)\r\n" );
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document.write( "    12x + 8y = 124.\r\n" );
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document.write( "We re-write (2) in an equivalent form\r\n" );
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document.write( "    x =  = .\r\n" );
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document.write( "31 - 2y  is divisible by 3 for y = 2, 5, 8, 11, 14, giving for x these values, respectively\r\n" );
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document.write( "                                   9, 7, 5,  3,  1.\r\n" );
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document.write( "Thus we see that these pairs (x,y) = (9,2), (7,5), (5,8), (3,11), (1,14) are the potential solutions.\r\n" );
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document.write( "But then we check for the sum x+y to be a multiple of 4,  and we see that the only a pair,\r\n" );
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document.write( "satisfying this condition, is (x,y) = (7,5).\r\n" );
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document.write( "ANSWER.  7 tickets at $11  and  5 tickets at $7 dollars.\r\n" );
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document.write( "CHECK.   7*11 + 5*7 +  = 77 + 35 + 12 = 124 dollars as the total revenue, including refreshments.  ! correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that the condition x=/= y is excessive: it is not used in the solution.\r
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