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

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
_________________

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

Similar problem: of-the-300-subjects-who-participated-in-an-experiment-using

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

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
_________________

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

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

Refer to GMAT Club math book for more details.

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:

Three-set Venn Diagrams

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!
_________________

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).

Sure, deethompson, I would be happy to post a video solution! I should be able to get to that later this week. In the meantime, I think that the key thing is to understand why we are subtracting some numbers in the first place. We subtract the overlaps because we are double- or triple-counting some houses if we add up all of the houses that have each of the 3 features. We are double-counting those that have exactly 2 of the features, so we need to subtract those once to account for that. We are triple-counting those that have all 3 of the features, so we need to subtract 2X that number of houses to account for that. Because we are told that 5 houses have all 3 features, we need to subtract 2 times that number, or 2*5, to account for this triple-counting.

Let me know if you have any other questions that you want me to address in my video solution!
_________________

Hi deethompson, I have now posted my video solution to YouTube (see below).

You can also see more information about the question and links to similar questions on this page of my website, YinTutoring.com:

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

Please let me know if you have any more questions!


_________________

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.”
_________________

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.
_________________

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.
_________________

Total of 150

=> 60 percent have Air-conditioning: \(\frac{60}{100}\) * 150 = 90

=> 50 percent have a Sunporch: \(\frac{50}{100}\) * 150 = 75

=> 30 percent have a Swimming pool: \(\frac{30}{100}\)* 150 = 45

Total - Neither = Air-conditioning + Sunporch + Swimming pool - Exactly two - 2*(Air-conditioning + Sunporch + Swimming pool - all together)

=> 150 - 5 = 90 + 75 + 45 - Exactly two - 2(5)

=> 145 = 210 - 10 - Exactly two

=> Exactly two = 200 - 145 = 55

Answer D
_________________

Solution:

Given total number of houses =150

Number of houses that have air-conditioning =60% of 150 = 90

Number of houses that have a sun-porch = 50% of 150 =75

Number of houses that have a swimming pool =30% of 150 = 45

Number of houses that have all three of these amenities =5

Number of houses that have none of them = 5

We are asked houses have exactly two of these amenities.

Its a straight question of using the formulae on overlapping sets

Total = A + B+ C -(Exactly two) - 2(all) + None

Total houses = number with air conditioning + number with sun-porch + 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 = 90 + 75 + 45 - (Exactly two) - 10 + 5

=> Exactly two = 55 (option d)

Devmitra Sen
GMAT SME