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Jess and Brian are playing the following game: Jess thinks of a fraction, and Brian flips a coin. If the coin turns up heads, Jess multiplies the number she's thinking of by 5/6. If the coin turns up tails, she multiplies the number she's thinking of by 6/5. Brian flips the coin ten times, and after each flip Jess multiplies the number in her head by either 5/6 or 6/5, depending on the coin flip. The ten coin flips turn out to be: THTTHTHTTT, where H means 'heads' and T means 'tails.' What number is Jess thinking of at the end of the game if she starts out with the fraction 5/9?

 Jan 21, 2019
 #1
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\(f\left(\dfrac 5 6\right)^3 \left(\dfrac 6 5\right)^7 = \dfrac 5 9\\ f\left(\dfrac 6 5\right)^4 = \dfrac 5 9\\ f = \dfrac 5 9 \left(\dfrac 5 6\right)^4 = \dfrac{3125}{11664}\)

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 Jan 21, 2019
 #2
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(5/6)^3 x (6/5)^7 x 5/9 =72 / 625

 Jan 21, 2019

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