# math

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A floor-finishing machine uses bars of wax. Once the bar is worn down to 1/2 its original size, it must be replaced with a new bar. Only full-size bars can be put in the machine. The pieces are melted and recycled to make full-size bars. If you buy 24 bars and recycle all the pieces, how many full-size bars can you use? How much wax will be left over? Your goal is to have as little wasted wax as possible.

• math -

1st cycle:
equivalent of 12 bars used, 24 half-bars left over
recycled to make 12 new bars
2nd cycle:
equivalent of 6 bars used, 12 half-bars left over
recycled to make 6 new bars
3rd cycle:
equivalent of 3 bars used, 6 half-bars left over
recycled to make 3 new bars
4th cycle:
equivalent of 1.5 bars used, 3 half-bars left over
recycled to make 1.5 new bars

since only full-size bars can be used, only one more bar can be used

total used: 12+6+3+1.5+1 = 23.5
So 1/2 bar is left over.

• math -

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