document.write( "Question 1183541: The Knave of Hearts stole some tarts. He ate half of them, and half a tart more.
\n" ); document.write( "The Knave of Diamonds ate half of what was left, and half a tart more.
\n" ); document.write( "Then the Knave of Clubs ate half of what remained, and half a tart more.
\n" ); document.write( "This left just one tart for the Knave of Spades.
\n" ); document.write( "How many tarts did the Knave of Hearts steal? \n" ); document.write( "

Algebra.Com's Answer #813912 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "The Knave of Hearts stole some tarts. He ate half of them, and half a tart more.
\n" ); document.write( "The Knave of Diamonds ate half of what was left, and half a tart more.
\n" ); document.write( "Then the Knave of Clubs ate half of what remained, and half a tart more.
\n" ); document.write( "This left just one tart for the Knave of Spades.
\n" ); document.write( "How many tarts did the Knave of Hearts steal?
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document.write( "I will solve the problem  from the ending state to the starting step. \r\n" );
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document.write( "I will organize my logic, my solution and my calculations using this table below.\r\n" );
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document.write( "    Name           starting       consumed         ending\r\n" );
document.write( "   (step)           amount         amount          amount\r\n" );
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document.write( "  H (Heart)           x           0.5x+0.5        h = x - (0.5x+0.5) = 0.5x-0.5     (1)\r\n" );
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document.write( "  D (diamond)         h           0.5h+0.5        d = h - (0.5h+0.5) = 0.5h-0.5     (2)\r\n" );
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document.write( "  C (club)            d           0.5d+0.5        c = d - (0.5d+0.5) = 0.5d-0.5     (3)\r\n" );
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document.write( "  S (spade)           c = 1                                                         (4)\r\n" );
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document.write( "        So, I have 4 equations  (1),  (2),  (3)  and  (4)  for  4  unknowns  x, h,  d  and  c.\r\n" );
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document.write( "        Now I will move backward from (4) to (1) to determine the unknowns one after another.\r\n" );
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document.write( "The last equation  c = 1  (4)  in the table gives me the opportunity to find \"d\" from (3)\r\n" );
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document.write( "    1 = 0.5d-0.5  --->  1 + 0.5 = 0.5d  --->  1.5 = 0.5d  --->  d = 1.5/0.5 = 3;  d = 3.       (5)\r\n" );
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document.write( "Now, knowing d= 3, I substitute it into equation (2) and find h\r\n" );
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document.write( "    3 = 0.5h-0.5  --->  3 + 0.5 = 0.5h  --->  3.5 = 0.5h  --->  h = 3.5/0.5 = 7;  h = 7.       (6)\r\n" );
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document.write( "Finally, knowing h= 7, I substitute it into equation (1) and find x\r\n" );
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document.write( "    7 = 0.5x-0.5  --->  7 + 0.5 = 0.5x  --->  7.5 = 0.5x  --->  x = 7.5/0.5 = 15;  x = 15.     (7)\r\n" );
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document.write( "Thus the problem is just solved, and the ANSWER is\r\n" );
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document.write( "      +-----------------------------------------+\r\n" );
document.write( "      |   The Knave of Hearts stole 15 tarts.   |\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "This method is one of standard methods of solving such problems.\r
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\n" ); document.write( "\n" ); document.write( "What you will meet rarely in other solutions, is perfect organization of logic and data.\r
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\n" ); document.write( "\n" ); document.write( "This problem is good for a Math Circle session.\r
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\n" ); document.write( "\n" ); document.write( "Do not forget to post your \"THANKS\" to me for my teaching.\r
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