Marbles

Probability Level 3

A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is 1 y \dfrac{1}{y} , compute the sum of all possible integer values of y.

This is an ARML problem


The answer is 69.

Math Man by math man , 20, Indonesia

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1 solution

With x x orange marbles, we require that

x x + 20 + 12 = 1 y ⟹ x + 32 = x y ⟹ x ( y − 1 ) = 32. \dfrac{x}{x + 20 + 12} = \dfrac{1}{y} \Longrightarrow x + 32 = xy \Longrightarrow x(y - 1) = 32.

Thus ( y − 1 ) (y - 1) can be any of the positive divisors of 32 32 , i.e., 1 , 2 , 4 , 8 , 16 , 32. 1,2,4,8,16,32.

The corresponding values for y y are 2 , 3 , 5 , 9 , 17 , 33 , 2,3,5,9,17,33, which add to 69 . \boxed{69}.

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