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

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Manager
Joined: 02 Aug 2018
Posts: 65
Location: India
Concentration: Strategy, Operations
GMAT 1: 590 Q45 V26
GPA: 3.5
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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03 Oct 2018, 09:20
ScottTargetTestPrep 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

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

Dear Scott

Thanks for a detailed response, I always like your responses.

I need your help though, if you could please advice, why are we using 2(5) in the equation. I understand 5 is for people who are doing all three activities but multiplication by 2?

Frankly, I have made similar error in other question as well, not having understood the secret of multiplication by 2?

Thank You
Abhinav
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10669
Location: United States (CA)
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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04 Oct 2018, 18:27
2
proabhinav wrote:
ScottTargetTestPrep 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

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

Dear Scott

Thanks for a detailed response, I always like your responses.

I need your help though, if you could please advice, why are we using 2(5) in the equation. I understand 5 is for people who are doing all three activities but multiplication by 2?

Frankly, I have made similar error in other question as well, not having understood the secret of multiplication by 2?

Thank You
Abhinav

Notice that in a 3-circle Venn diagram, when we add each individual circle, we include the “triple overlap” three times since each circle has the “triple overlap.” However, we only can count it once, so we have to subtract 2 times whatever number in the “triple overlap.”
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# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
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Director
Joined: 25 Jul 2018
Posts: 719
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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28 Jun 2019, 13:30
deethompson wrote:
Please kindly post a video solution. I cannot seem to understand where 2(5) came from. I seem to get the other workings but not sure about the 2(5).

JeffYin wrote:
If anyone is looking for 3-set Venn Diagram questions to practice on from the Official Guide, I have compiled a list at the following link:

On this question, while I would still draw the Venn Diagram myself, the key is to know the formula for three-set Venn Diagrams that has been mentioned in this thread:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + None

Then, all we need to do is plug in the numbers we are given (converting percents to numbers) and solve for the sum of 2-group overlaps, because that is what the question is asking us for.

Please let me know if you have any questions, or if you want me to post a video solution!

Hi,
I am sending the attachment of a solution. Here, you can see the detailed explanation of how 2(5) came.
Hope it helps
PFA
Attachments

File comment: sorry about handwriting.
hope you can get easily!

venn_diagram.jpg [ 185.47 KiB | Viewed 655 times ]

Manager
Joined: 01 Jan 2019
Posts: 107
Concentration: Finance, Entrepreneurship
GPA: 3.24
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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08 Aug 2019, 06:40
ScottTargetTestPrep 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

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

Hi Scott,
I follow your solutions very closely. And this is an optimum way to work out the problem. I only have a question that why have you taken all three houses i.e 5 multiplied by 2?
Is there something that I missed in the question?

Posted from my mobile device
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
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Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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12 Aug 2019, 17:59
Shef08 wrote:

Hi Scott,
I follow your solutions very closely. And this is an optimum way to work out the problem. I only have a question that why have you taken all three houses i.e 5 multiplied by 2?
Is there something that I missed in the question?

Posted from my mobile device

I am guessing it is clear that there are 90 houses with AC, 75 houses with sunporch and 45 houses with swimming pool. Some houses have two of the above amenities, some have all three while some have none. Let's see what happens when we add all three: The houses that have exactly two of the above amenities will be counted twice; for instance, when you add 90 houses with AC and 75 houses with sunporch and find 90 + 75 = 165, any house with both AC and sunporch has been double counted. The same is true for all other choices of two amenities. Since every house with exactly two of the amenities has been double counted, we subtract the sum of the houses with exactly two of the amenities so that the sum includes each house only once. The reason we multiply the number of houses with all three amenities by 2 before subtracting is very similar: Every house that has all three amenities has been represented three times in the sum; for instance, if house H has all three of the amenities, when you add 90 + 75 + 45, house H is one of the 90 houses with AC, one of the 75 houses with sunporch and one of the 45 houses with swimming pool. As you can see, house H is counted exactly three times, but we would like the summation to contain only one count of house H. That's why we subtract 2 times the number of houses with all three amenities from the sum 90 + 75 + 45.
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
202 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern
Joined: 28 Jan 2020
Posts: 27
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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23 Mar 2020, 15:48
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

When looking at a problem with more than 2 overlapping sets, its best to visualize as a Venn Diagram. We want to set up our equation as follows:

Total Number of Houses = Number with AC Units + Number with Sunporches + Number with Swimiming Pools - Number with 2 of These (x) - 2 * Number with 3 of these + Number with None of These

150 = 0.6*150 + 0.5 * 150 + 0.3 * 150 - X - 2*5 + 5
150 = 90 + 75 + 45 - X - 10 + 5
55 = X

Hope this helps!
Intern
Joined: 10 Oct 2018
Posts: 10
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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20 Apr 2020, 01:28
Hi! Can please someone explain why we multiply by 5 (all three options) by 2?
Math Expert
Joined: 02 Sep 2009
Posts: 64314
Re: Of the 150 houses in a certain development, 60 percent have  [#permalink]

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20 Apr 2020, 01:38
aaliyahkhalifa wrote:
Hi! Can please someone explain why we multiply by 5 (all three options) by 2?

That formula is explained in details here: ADVANCED OVERLAPPING SETS PROBLEMS.

Hope it helps.
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Re: Of the 150 houses in a certain development, 60 percent have   [#permalink] 20 Apr 2020, 01:38

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

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