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 5th shift? Choose the correct choice from the following answers
A) 1/27	B) 1/54	C) 1/81	D) 1/162	E) 1/216
Fraction of area unpainted after FIRST shift of Amit = 1-(1/2) = 1/2   Fraction of area unpainted after FIRST shift of Ian = (1/2)-(1/3)(1/2) = (2/3)(1/2)   Fraction of area unpainted after SECON1/162 shift of Amit = (1/2)(2/3)(1/2)   Fraction of area unpainted after SECON1/162 shift of Ian = (2/3)(1/2)(2/3)(1/2)   Fraction of area unpainted after THIR1/162 shift of Amit = (1/2)(2/3)(1/2)(2/3)(1/2)   Fraction of area unpainted after THIR1/162 shift of Ian = (2/3)(1/2)(2/3)(1/2)(2/3)(1/2)   Fraction of area unpainted after FORTH shift of Amit = (1/2)(2/3)(1/2)(2/3)(1/2)(2/3)(1/2)   Fraction of area unpainted after FORTH shift of Ian = (2/3)(1/2)(2/3)(1/2)(2/3)(1/2)(2/3)(1/2)   Fraction of area unpainted after FIFTH shift of Amit = (1/2)(2/3)(1/2)(2/3)(1/2)(2/3)(1/2)(2/3)(1/2)=1/162   Answer: option 1/162
<result>D</result>
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 5th shift?