Mock AIME I 2015 Problems/Problem 1

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David, Justin, Richard, and Palmer are demonstrating a "math magic" concept in front of an audience. There are four boxes, labeled A, B, C, and D, and each one contains a different number. First, David pulls out the numbers in boxes A and B and reports that their product is $14$. Justin then claims that the product of the numbers in boxes B and C is $16$, and Richard states the product of the numbers in boxes C and D to be $18$. Finally, Palmer announces the product of the numbers in boxes D and A. If $k$ is the number that Palmer says, what is $20k$?


Associative Property! Note that the product of all four numbers can be written in two different ways, $(AB)(CD)$ and $(AD)(BC)$. Setting these equal to each other gives \[k=AD=\dfrac{(AB)(CD)}{BC}=\dfrac{14\times 18}{16}=\dfrac{63}4\,\,\implies\,\, 20k=\boxed{315}.\]

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