2023 SSMO Relay Round 4 Problems/Problem 1

Problem

Martha starts out with the number $7$ on her calculator, which has three buttons that multiply the current number by $\frac{1}{2}$ , $\frac{2}{3}$ , and $\frac{5}{6}$ respectively. She randomly presses one of the buttons four times. After these $4$ presses, she cubes the number. The expected value of the final number is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

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2023 SSMO Relay Round 4 Problems/Problem 1

Problem

Solution