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Brian takes an 81 inch piece of rope and cuts it three times. Each time he cuts the rope, he cuts off and discards the same fraction of the remaining length. When he is finished, the piece of rope is 3 inches long. Which fraction represents the amount of rope removed in each of the three cuts? The answer is 2/3, but I have not a clue why.
Let x be the common fraction removed with each cut.
Then after the first cut the amount of rope left is
81 - x*81
After the second cut then we have:
(81-X*81) - x*(81-x*81)
After the last cut we have:
(81-x*81) - x*(81-x*81) - x*((81-x*81) - x*(81-x*81))
Expanding this we have:
81*(1-x-x+x^2-x+x^2+x^2-x^3) = 3
81*(1-3x+3x^2-x^3) = 3
1-3x+3x^2-x^3 = 3/81 = 1/27
(-x+1)^3 = 1/27
-x+1 = 1/3
x = 1-1/3 = 2/3