SOLUTION: 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 lef

Question 1210592: 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?
Answer by greenestamps(13325) (Show Source): You can put this solution on YOUR website!

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?

The second and third sentences make no sense for a couple of reasons.

(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.
(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).

So we have to ignore those first few sentences to work the problem.

And then the problem is much easier to solve by working backwards, starting with the 2 cards Arthur finished with.

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.

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").

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.

2 cards plus half a card is 2 1/2 cards; doubled is 5 cards
5 cards plus half a card is 5 1/2 cards; doubled is 11 cards
11 cards plus half a card is 11 1/2 cards; doubled is 23 cards

Arthur started with 23 cards (and at no point did any of the rare cards have to be cut in half!)

ANSWER: 23

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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