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# Amit and Ian paint a wall in alternating shifts. First Amit paints alo

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Math Expert
Joined: 02 Sep 2009
Posts: 58464
Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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17 Sep 2015, 00:51
2
10
00:00

Difficulty:

75% (hard)

Question Stats:

59% (02:46) correct 41% (03:03) wrong based on 173 sessions

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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

Kudos for a correct solution.

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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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17 Sep 2015, 13:21
5
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

Kudos for a correct solution.

Work remaining after amit completes his shift will be 1/2.
Ian will complete 1/3 of this work. So,work remaining after Ian completes his shift will be 2/3 of the work remaining after Amit completes his shift.
Fraction of the work which is unfinished after amit completes his 4th shift will be

$$\frac{1}{2}$$*$$\frac{2}{3}$$*$$\frac{1}{2}$$*$$\frac{2}{3}$$*$$\frac{1}{2}$$*$$\frac{2}{3}$$*$$\frac{1}{2}$$ = $$\frac{1}{54}$$
(Each color indicates different shift)

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Posts: 233
Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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17 Sep 2015, 01:12
1
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

Kudos for a correct solution.

Let total work is 1

amit's 1st shift: 1/2 (paint left: 1-1/2= 1/2)
Ian 1st shift: (1/2)*(1/3)= 1/6 (paint left: 1-(1/2 +1/6) = 1/3
Amint's 2nd shift: (1/3)*(1/2)= 1/6 (paint left: 1/3 - 1/6 = 1/6
Ian 2nd shift: 1/6*1/3= 1/18 (paint left: 1/6-1/18 = 1/9
amit's 3rd shift: 1/9*1/2=1/18 (paint left: 1/9-1/18 = 1/18
Ian 3rd shift:....... (paint left: 1/18*2/3 = 1/27
amit's 4th shift:.....(paint left: 1/27*1/2 = 1/54
Remain paintings: 1/54 ( i used short cut at the last sections)
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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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17 Sep 2015, 06:57
took me 3 minutes to solve this.any shortcuts??
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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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18 Sep 2015, 04:51
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

Kudos for a correct solution.

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 SECOND shift of Amit = (1/2)(2/3)(1/2)

Fraction of area unpainted after SECOND shift of Ian = (2/3)(1/2)(2/3)(1/2)

Fraction of area unpainted after THIRD shift of Amit = (1/2)(2/3)(1/2)(2/3)(1/2)

Fraction of area unpainted after THIRD 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) = 1/54

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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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20 Sep 2015, 21:55
2
2
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

Kudos for a correct solution.

ECONOMIST GMAT TUTOR OFFICIAL SOLUTION:

The issue of this question is (efficiently) finding out a term in a sequence. The terms in this sequence are the fractions of unpainted wall area. To find out how to calculate the terms, start writing them down and look for a pattern.

The first term - the fraction of unpainted area after Amit's first shift is 1/2.

In the next shift, Ian paints a 1/3 of the unpainted area, i.e., a 1/3 of 1/2. But how much is left unpainted? Ian leaves 2/3 of the 1/2 unpainted. Thus, the second term is (2/3)(1/2).

In the following shift, (Amit's second shift,) half of the unpainted area is painted, or, in other words, half of unpainted area remains unpainted, hence, the third term is (1/2)(2/3)(1/2). Continue the pattern from here until after Amit's 4th shift.

The final expression after Amit's 4th shift (labeled Amit (4)):

Amit(1) Ian Amit(2) Ian Amit(3) Ian Amit(4)
$$\frac{1}{2}*\frac{2}{3}*\frac{1}{2}*\frac{2}{3}*\frac{1}{2}*\frac{2}{3} *\frac{1}{2}=\frac{1}{54}$$.

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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo  [#permalink]

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04 Oct 2018, 19:44
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.

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Re: Amit and Ian paint a wall in alternating shifts. First Amit paints alo   [#permalink] 04 Oct 2018, 19:44
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