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Of the 150 houses in a certain development, 60 percent have

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Of the 150 houses in a certain development, 60 percent have [#permalink]

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Question Stats:

59% (02:01) correct 41% (02:07) wrong based on 546 sessions

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Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?

(A) 10
(B) 45
(C) 50
(D) 55
(E) 65
[Reveal] Spoiler: OA
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 29 May 2017, 05:24
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ganand wrote:
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?

(A) 10
(B) 45
(C) 50
(D) 55
(E) 65



This can be solved using Venn Diagram (refer the attachment)
AC = 60% of 150 = 90
Sunporch = 50% of 150 = 75
SP = 30% of 150 = 45

Using the venn diagram we can form the following equations
a+b+c+d+e+f+g+h = 150
a+b+c+d+f+g = 140 ----- (1) (as e = h = 5)

Taking 1 circle at a time
AC = a+b+d = 85 ---- (2) .. (as e = 5)
SunPorch = c+b+f = 70 ---- (3) .. (as e = 5)
SP = g+d+f = 40 ---- (4) .. (as e = 5)

equation (2) + (3) + (4)
a+c+g+ 2(d+b+f) = 195
subtracting this with equation (1)
d+b+f = 195 - 140 = 55

Answer option D
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 29 May 2017, 07:23
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ganand wrote:
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?

(A) 10
(B) 45
(C) 50
(D) 55
(E) 65



all three = neither =5 = 10/3 % of 150

working formula :-
Total = AC + sunporch + swimming pool - Exactly two - 2*all three + neither

100 = 60 + 50 + 30- Exactly two - 2*10/3 + 10/3
100 =140 - Exactly two - 10/3
Exactly two = 40 - 10/3 = (120-10)/3 = 110/3 %
110/3 % of 150 = 110*150/300 =55

Ans D
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Of the 150 houses in a certain development, 60 percent have [#permalink]

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ganand wrote:
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?


(A) 10
(B) 45
(C) 50
(D) 55
(E) 65


Hi Bunuel,

Why formula (1) is incorrect? https://gmatclub.com/forum/advanced-overlapping-sets-problems-144260.html

(i) In term of percentage

Total = A + B + C − (sum of 2−group overlaps) + (all three) + Neither
100 = 60 + 50 + 30 - (x) + (5/150)(100) + (5/150)(100)
100 = 140 + (20/3) - x
x = 40 + (20/3)
x = 140/3%

(x/150)(100) = (140/3)
x = (140/300)(150)
x = 70

(ii) In term of number

Total = A + B + C − (sum of 2−group overlaps) + (all three) + Neither
150 = 90 + 75 + 45 - x + 5 + 5
x = 70
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 28 Jun 2017, 05:58
hazelnut wrote:
ganand wrote:
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?


(A) 10
(B) 45
(C) 50
(D) 55
(E) 65


Hi Bunuel,

Why formula (1) is incorrect? https://gmatclub.com/forum/advanced-overlapping-sets-problems-144260.html

Total = A + B + C − (sum of 2−group overlaps) + (all three) + Neither
100 = 60 + 50 + 30 - (x) + (5/150)(100) + (5/150)(100)
100 = 140 + (20/3) - x
x = 40 + (20/3)
x = 140/3


We need to find exact two. The formula you have listed is for general sum of 2 group overlap. The total you have listed is 100 while in the question it is 150.

Formula for exact two is: Total = A + B + C + Neither - (exact 2) - 2 * (all three)

150=90+75+45+5-10-sum of exact 2.
205-150= sum of exact 2
sum of exact 2=55

D
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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Total = A + B +C - (exactly two) - 2(all three) + none

Refer to GMAT Club math book for more details.
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 07 Oct 2017, 09:29
I can solve this problem with the appropriate formula. Is there a way to solve this problem by backsolving
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 15 Nov 2017, 16:22
ganand wrote:
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?

(A) 10
(B) 45
(C) 50
(D) 55
(E) 65


We can create the following equation:

Total houses = number with air conditioning + number with sunporch + number with pool - number with only two of the three things - 2(number with all three things) + number with none of the three things

150 = 0.6(150) + 0.5(150) + 0.3(150) - D - 2(5) + 5

150 = 90 + 75 + 45 - D - 10 + 5

150 = 205 - D

D = 55

Answer: D
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Re: Of the 150 houses in a certain development, 60 percent have [#permalink]

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New post 15 Nov 2017, 20:28
150= 90+75+45-Both-10+5

Both = 55

This is a 30% level question imo.
Re: Of the 150 houses in a certain development, 60 percent have   [#permalink] 15 Nov 2017, 20:28
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