document.write( "Question 1122239: The spring concert at a certain high school sold 159 tickets. Students were charged $6 each and adults $8 each. The income from the sale of tickets was $1142. How many student and how many adults bought tickets? \n" ); document.write( "
Algebra.Com's Answer #738304 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "        There are different methods to solve the problem.
\n" ); document.write( "        I will show you 3 (three) basic methods in this post.\r
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\n" ); document.write( "\n" ); document.write( "Solution  1   (the system of  2  equations approach)\r
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document.write( "Let x be the number of adults attended and\r\n" );
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document.write( "let y be the number of students attended.\r\n" );
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document.write( "Then you have these two equations\r\n" );
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document.write( " x +  y =  159,           (1)     (counting people)\r\n" );
document.write( "8x + 6y = 1142  dollars   (2)     (counting dollars)\r\n" );
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document.write( "These are your basic equations, and as soon as you got this system, the setup is completed.\r\n" );
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document.write( "There are different methods of solving such system (Substitution, Elimination, using determinants). \r\n" );
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document.write( "I will use the Substitution method here. \r\n" );
document.write( "From equation (1) express y = 159-x and substitute it into equation (2). You will get a single equation for the unknown x\r\n" );
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document.write( "8x + 6(159-x) = 1142.     (3)\r\n" );
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document.write( "8x + 6*159 - 6x = 1142  ====>  (8-2)x = 1142 - 6*159  ====>  x =  \"%281142-6%2A159%29%2F%288-6%29\" = 94.    (4)\r\n" );
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document.write( "Thus, 94 adult tickets were sold.\r\n" );
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document.write( "Hence the number of student tickets was 159 - 94 = 65.\r\n" );
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document.write( "Check.  8*94 + 6*65 = 1142 dollars.   ! Correct. !\r\n" );
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document.write( "Answer.  94 adults and 65 students.\r\n" );
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document.write( "Let x be the number of adults attended.\r\n" );
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document.write( "Then the number of students attended is (159-x),  according to the condition.\r\n" );
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document.write( "The adults payed 8x dollars for their tickets.\r\n" );
document.write( "The students payed 6*(159-x) for their tickets.\r\n" );
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document.write( "Summing up these money, you get the revenue equation\r\n" );
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document.write( "8x + 6(159-x) = 1142.     (5)\r\n" );
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document.write( "It is your basic equation in the frame of this approach,\r\n" );
document.write( "and as soon as you got this equation, the setup is completed.\r\n" );
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document.write( "Notice that this equation (5) coincides exactly with the equation (3) of the solution 1 above.\r\n" );
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document.write( "Solve equation (5) by the same method as it was done in the Solution 1.\r\n" );
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document.write( "Surely, you will get the same answer.\r\n" );
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document.write( "Let assume for a minute that all 159 people were students.\r\n" );
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document.write( "Then the total revenue would be 6*159 = 954 dollars.\r\n" );
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document.write( "It is by 1142 - 954 = 188 dollars less than the given revenue value.\r\n" );
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document.write( "Why did we get the difference ?  - But of course, because we counted adult tickets as the student tickets.\r\n" );
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document.write( "To compensate the difference, we must replace some number of 159 \"student\" tickets by the adult tickets.\r\n" );
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document.write( "At each replacement, we diminish the difference of $188 by $2 (which is the difference $8-$6 between the adult and the student tickets price).\r\n" );
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document.write( "Then it is clear that the number of adult tickets is  \"188%2F2\" = 94.\r\n" );
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document.write( "Hence, the number of students is  159 - 94 = 65,  and you got the same answer as in solutions 1 and 2 above.\r\n" );
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document.write( "    Notice, that this logic works EXACTLY in accordance with the formula\r\n" );
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document.write( "        adults = \"%281142-6%2A159%29%2F%288-6%29\"    (6)\r\n" );
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document.write( "    which is EXACTLY THE SAME as the formula (4).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Congratulations !   You are now familiar with 3 methods for ticket problems solution.\r
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\n" ); document.write( "\n" ); document.write( "I suggest that algebraic methods will be your basic methods for such problems,\r
\n" ); document.write( "\n" ); document.write( "and the logical analysis method will allow you to solve the problems  QUICKLY  without using equations.\r
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\n" ); document.write( "To see other similar solved problems on tickets,  look into the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Using systems of equations to solve problems on tickets \r
\n" ); document.write( "\n" ); document.write( "    - Three methods for solving standard (typical) problems on tickets\r
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\n" ); document.write( "\n" ); document.write( "To see how the logical method works for other similar problems, look into the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Problem on two-wheel and three-wheel bicycles\r
\n" ); document.write( "\n" ); document.write( "    - Problem on animals at a farm\r
\n" ); document.write( "\n" ); document.write( "    - Problem on pills in containers\r
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\n" ); document.write( "\n" ); document.write( "The lesson to learn from my presentation :\r
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document.write( "     The typical ticket problem is THIS :\r\n" );
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document.write( "         - You are given the total number of cheap and expensive tickets;\r\n" );
document.write( "         - you are given the prices for tickets of these two categories;\r\n" );
document.write( "         - you are given the total revenue;\r\n" );
document.write( "         - you are asked to find the number of tickets in each of the two categories.\r\n" );
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document.write( "     When you solve such a problem, you may use ANY of the three methods of this presentation.\r\n" );
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document.write( "     But you also can write the solution in one line as a fraction whose numerator is the difference \r\n" );
document.write( "     between the given revenue and the hypothetical revenue assuming that all tickets are for cheap price, \r\n" );
document.write( "     and whose denominator is the difference (positive difference) between expensive and cheap prices.\r\n" );
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document.write( "    The logic explained in the Solution 3 will help you to go in this way and will prevent you of making errors or steps aside.\r\n" );
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document.write( "    And when the formula is implemented and all the given numbers are plug-in, all you need to do is to copy and paste the formula\r\n" );
document.write( "    into MS Excel in your computer (or into other similar software) and press the \"Enter\" button to get the final number (solution; answer).\r\n" );
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document.write( "    And it is, if you want, the fourth method of solving such problems - the shortest and the quickest. \r\n" );
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