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\n" ); document.write( "Arthur starts with a certain number of rare trading cards. First, Arthur gives 4 1/2 cards to a local collector to complete a set. After this, the ratio of the cards Arthur has left to the cards he started with is exactly 3:4. Next, Ben takes half of the remaining cards plus another half of a card. Then, Cole takes half of what Ben left behind plus another half of a card. Finally, Dan takes half of what Cole left behind plus another half of a card. After Dan is finished, there are exactly 2 cards left. How many trading cards did Arthur start with?
\n" ); document.write( "The second and third sentences make no sense for a couple of reasons.
\n" ); document.write( "(1) Rare trading cards become much less valuable if you cut them in half. Arthur giving 4 1/2 cards to another collector is nonsense in the real world.
\n" ); document.write( "(2) If you work the problem in the \"forwards\" direction, the second and third sentences together mean Arthur started with 18 cards, and after giving 4 1/2 to another collector he had 13 1/2 cards left. Working from there, the fractions of cards get even smaller, leading to nonsensical numbers (and Arthur does NOT end up with 2 cards).
\n" ); document.write( "So we have to ignore those first few sentences to work the problem.
\n" ); document.write( "And then the problem is much easier to solve by working backwards, starting with the 2 cards Arthur finished with.
\n" ); document.write( "In each transaction, the remaining number of cards was cut in half, and then the number of cards remaining was reduced by another one half of a card.
\n" ); document.write( "To work each transaction backwards -- i.e. \"undo\" each transaction --, we need to add one half of a card and then double the number of cards. (The opposite (inverse) of \"cut in half and subtract one half\" is \"add one half and double\").
\n" ); document.write( "These transactions occur three times. To find the number of cards Arthur started with, we need to start with the 2 cards he finished with and do the \"undo\" transaction three times.
\n" ); document.write( "2 cards plus half a card is 2 1/2 cards; doubled is 5 cards
\n" ); document.write( "5 cards plus half a card is 5 1/2 cards; doubled is 11 cards
\n" ); document.write( "11 cards plus half a card is 11 1/2 cards; doubled is 23 cards
\n" ); document.write( "Arthur started with 23 cards (and at no point did any of the rare cards have to be cut in half!)
\n" ); document.write( "ANSWER: 23
\n" ); document.write( "Note that answer again ignores, as it must, the first part of the statement of the problem that says the first thing Arthur did was give 4 1/2 cards to someone else.
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