2015 Mock AIME I Problem 1: Picking Numbers from a Hat

Algebra Level 2

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 $20k$ ?

The answer is 315.

1 solution

Julian Liu
Sep 26, 2014

We have AB = 14, BC = 16, and CD = 18.

Multiplying the first and third equations gives us $ABCD = 252$ , while multiplying all three equations gives us $A(B^{2})(C^{2})D = 252 \times 16$ . Therefore, we can say that $252^{2} = 252 \times 16AB$ . Solving the equation gets us $AB = 63/4$ , so $20AB$ is $\boxed{315}$ .

2015 Mock AIME I Problem 1: Picking Numbers from a Hat

The answer is 315.

1 solution

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