document.write( "Question 530630: A deck with 26 red and 26 black cards is separated into two unequal nonempty piles. The first pile contains 7 times as many black cards as red. The number of red cards in the second pile is a multiple of the number of black cards in that piles. How many red cards are in the first pile ? \n" ); document.write( "

Algebra.Com's Answer #350367 by bucky(2189) $\"\"$ $\"About$

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Let B represent the number of Black cards in the first pile, and let R represent the number of Red cards in the first pile.
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\n" ); document.write( "Since B in the first pile is given to be 7 times R, B must be divisible by 7. That means that R can only be one of 4 numbers: 0, 1, 2, or 3. If R were greater than 3, then 7 times R would require that there be more than 26 Black cards in the deck. Also, R cannot be zero. Why? Because if R were zero, then B (which is 7 times R) would also have to be zero. But if both R and B were zero, then the first pile would be empty, and the problem states that both of the piles are non-empty.
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\n" ); document.write( "That leaves us with the possibilities that R (the number of Red cards in the first pile) is 1, 2, or 3. This means that B (the number of Black cards in the first pile) is 7, 14, or 21 since it B equals 7 times R.
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\n" ); document.write( "Now let's look at the second pile. The number of Black cards in this pile is 26 less the number of Black cards in the first pile. Since we now know that the number of Black cards in the first pile is 7, 14, or 21, then the number of Black cards in the second pile is 26 - 7, 26 - 14, or 26 - 21. In other words, the number of Black cards in the second pile can only be 19, 12, or 5. We also know that the number of Red cards in the second pile must be 26 minus the number of Red cards in the first pile. And we know that the number of Red cards in the first pile can only be 1, 2, or 3. Therefore, the number of Red cards in the second pile can only be 26 - 1, 26 - 2, or 26 - 3. So the number of Red cards in the second pile can only be 25, 24, or 23.
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\n" ); document.write( "But the requirement is that for the second pile, the number of Red cards must be a multiple of the number of Black cards. Black card possibilities are 19, 12, or 5, and Red card possibilities are 25, 24, or 23. We can eliminate 23 as being the number of Red cards in the second pile because 23 is not a multiple of 19, 12, or 5, the possible number of Black cards.
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\n" ); document.write( "So we are left with the possibility that the number of Red cards in the second pile is either 24 or 25. This means the corresponding number of Black cards in the second pile is respectively either 12 or 5.
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\n" ); document.write( "If there are 24 Red cards in the second pile, there are 12 Black cards in that pile. This meets the requirement for the second pile because the number of Red cards in it is a multiple of the number of Black cards in it. This also means that the first pile has 2 Red cards (26 - 24) and 14 Black cards (26 - 12). That works, because the number of Black cards in the first pile would be 7 times the number of Red cards in that pile.
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\n" ); document.write( "But if there are 25 Red cards in the second pile, then there must be 5 Black cards in that pile so that the number of red cards is a multiple of the number of Black cards. This would require that the first pile contain 1 Red card (26 - 25) and 21 Black cards (26 - 5). That won't work because in the first pile, the number of Black cards must be 7 times the number of Red cards in that pile.
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\n" ); document.write( "So by the process of elimination, we have reached the point where we know the answer to this problem. The piles consist of the following:
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\n" ); document.write( "Pile 1 consists of 2 Red cards and 14 Black cards.
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\n" ); document.write( "Pile 2 consists of 24 Red cards and 12 Black cards.
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\n" ); document.write( "So the specific answer required by this problem is that the first pile has 2 Red cards.
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\n" ); document.write( "Hope this solution to your puzzle helps you to gain some insight into the role that analysis plays in solving math problems.
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