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jmc

geometry intermediate

Problem

Bonnie makes the frame of a cube out of 12 pieces of wire that are each six inches long. Meanwhile Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames that are not connected to each other. The total volume of Roark's cubes is the same as the volume of Bonnie's cube. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire? Express your answer as a common fraction.
problem
Solution
The total length of Bonnie's wire is inches, while her total volume is cubic inches. Each of Roark's unit cubes has volume cubic inch, so he needs cubes.

Since each cube has edges, each of Roark's cubes has inches of wire. So his cubes have a total of inches of wire.

So the desired fraction is .
Final answer
\dfrac{1}{36}