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 #846892 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "x = number of $11 tickets sold
\n" ); document.write( "y = number of $7 tickets sold
\n" ); document.write( "x+y = total number of tickets
\n" ); document.write( "0.25(x+y) = number of people who bought a refreshment\r
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\n" ); document.write( "\n" ); document.write( "A = 11x = amount spent on the $11 tickets
\n" ); document.write( "B = 7y = amount spent on the $7 tickets
\n" ); document.write( "C = 4*0.25(x+y) = x+y = amount spent on refreshments\r
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\n" ); document.write( "\n" ); document.write( "A+B+C = total amount spent
\n" ); document.write( "A+B+C = 124
\n" ); document.write( "11x+7y+x+y = 124
\n" ); document.write( "12x+8y = 124
\n" ); document.write( "4(3x+2y) = 124
\n" ); document.write( "3x+2y = 124/4
\n" ); document.write( "3x+2y = 31
\n" ); document.write( "Using trial-and-error, a graph, or the Extended Euclidean Algorithm, you will find these nonnegative integer solutions

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document.write( "(x, y) = (1, 14)\r\n" );
document.write( "(x, y) = (3, 11)\r\n" );
document.write( "(x, y) = (5, 8)\r\n" );
document.write( "(x, y) = (7, 5)\r\n" );
document.write( "(x, y) = (9, 2)

Each time x goes up by 2, y decreases by 3.\r
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\n" ); document.write( "\n" ); document.write( "Plug each of those coordinates into 0.25(x+y) to see which results in an integer.
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document.write( "(x,y) = (1, 14) ----> 0.25(x+y) = 0.25(1+14) = 0.25(15) = 3.75\r\n" );
document.write( "(x,y) = (3, 11) ----> 0.25(x+y) = 0.25(3+11) = 0.25(14) = 3.5\r\n" );
document.write( "(x,y) = (5, 8) -----> 0.25(x+y) = 0.25(5+8) = 0.25(13) = 3.25\r\n" );
document.write( "(x,y) = (7, 5) -----> 0.25(x+y) = 0.25(7+5) = 0.25(12) = 3\r\n" );
document.write( "(x,y) = (9, 2) -----> 0.25(x+y) = 0.25(9+2) = 0.25(11) = 2.75\r\n" );
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Of those results only 3 is an integer. Everything else is a decimal value.
\n" ); document.write( "The result 3 corresponds to (x,y) = (7,5)
\n" ); document.write( "Another way to find this is to note that x+y = 7+5 = 12 is a multiple of 4. Every other sum is not a multiple of 4 (eg: x+y=1+14 = 15).\r
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\n" ); document.write( "\n" ); document.write( "Question: how many $7 tickets were sold?
\n" ); document.write( "Answer: 5
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