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Question

Amy, Ben, Carl, and Debbie each have some coins. Ben has three times the number of coins that Amy has and a third of the number of coins that Carl has, and Debbie has two-thirds the number of coins that Carl has. The number of coins that Amy has, multiplied by the number of coins that Ben has, multiplied by the number of coins that Carl has, multiplied by the number of coins that Debbie has, is 162. How many coins do the four children have all together?

Accepted Answer

Let a be the number of coins Amy has, b the number Ben has, c the number Carl has, and d the number Debbie has. We can use the information in the problem to create the following system of linear equations: align* 3a &= b 3b &= c 23c &= d a b c d &= 162 align* We know that b = 3a. Also, c = 3b, so c must equal 9a. Then, d = 23c, so d = 6a. Substituting these quantities into our product gives a 3a 9a 6a = 162, which simplifies to 162a^4 = 162, or a^4 = 1, so a = 1. Given this value, we can find that b = 3, c = 9, and d = 6. Thus, the four children have 1 + 3 + 9 + 6 = 19 coins all together.