Bunuel wrote:

Amit and Ian paint a wall in alternating shifts. First Amit paints alone, then Ian paints alone, then Amit paints alone, etc. During each of his shifts, Amit paints 1/2 of the remaining unpainted area of the wall, while Ian paints 1/3 of the remaining unpainted area of the wall during each of his shifts. If Amit goes first, what fraction of the wall's area will remain unpainted after Amit has completed his 4th shift?

A. 1/27

B. 1/54

C. 1/81

D. 1/162

E. 1/216

Amit’s 1st shift: He paints ½ of the room, so ½ of the room is left unpainted.

Ian’s 1st shift: He paints ⅓(½) = ⅙ of the room, so ½ - ⅙ = ⅓ of the room is left unpainted.

Amit’s 2nd shift: He paints ½(⅓) = ⅙ of the room, so ⅓ - ⅙ = ⅙ of the room is left unpainted.

Ian’s 2nd shift: He paints ⅓(⅙) = 1/18 of the room, so ⅙ - 1/18 = 1/9 of the room is left unpainted.

Amit’s 3rd shift: He paints ½(1/9) = 1/18 of the room, so 1/9 - 1/18 = 1/18 of the room is left unpainted.

Ian’s 3rd shift: He paints ⅓(1/18) = 1/54 of the room, so 1/18 - 1/54 = 1/27 of the room is left unpainted.

Amit’s 4th shift: He paints ½(1/27) = 1/54 of the room, so 1/27 - 1/54 = 1/54 of the room is left unpainted.

Alternate Solution:

Let’s say the room is 6^3 = 216 ft^2. In his first shift, Amit will paint 216/2 = 108 ft^2 and there will be 216 - 108 = 108 ft^2 left to be painted.

In his first shift, Ian will paint 108/3 = 36 ft^2 and there will be 108 - 36 = 72 ft^2 left to be painted.

In his second shift, Amit will paint 72/2 = 36 ft^2 and there will be 72 - 36 = 36 ft^2 left to be painted.

In his second shift, Ian will paint 36/3 = 12 ft^2 and there will be 36 - 12 = 24 ft^2 left to be painted.

In his third shift, Amit will paint 24/2 = 12 ft^2 and there will be 24 - 12 = 12 ft^2 left to be painted.

In his third shift, Ian will paint 12/3 = 4 ft^2 and there will be 12 - 4 = 8 ft^2 left to be painted.

Finally, in his fourth shift, Amit will paint 8/2 = 4 ft^2 and there will be 8 - 4 = 4 ft^2 left to be painted.

Thus, after Amit’s fourth shift, 4/216 = 1/54 of the room will be left to be painted.

Answer: B

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