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655-705 (Hard)|   Overlapping Sets|                        
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Of the 150 houses in a certain development, 60 percent have 06 Feb 2023, 10:47
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Want more help? http://linktr.ee/thegmatstrategy :cool:
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Of the 150 houses in a certain development, 60 percent have 12 Mar 2024, 11:41
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somtsat99
Total = A + B +C - (exactly two) - 2(all three) + none

Refer to GMAT Club math book for more details.
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­Hello,

Overlapping sets are a daunting task and this formula is very helpful. I want to make sure that I translated correctly for n sets with n≥3.

So, the formula is Total = a+b+..+n + none - (n-1)*n - nk where nk is the number of sets in which I want to find how many objects there are in EXACTLY nk sets ?

For instance, if I have a total of 260 objects with 4 sets (a,b,c,d) and each one has 25% of the total and 8 there aren't in none sets and I want to find how many there are in EXACTLY 3 sets then I'll do: 260 = 4*0.25*260 + 8 - (4-1)*3 - 3 = 256?

Thank you in advance!­
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Of the 150 houses in a certain development, 60 percent have 12 Mar 2024, 11:44
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Gmatguy007

somtsat99
Total = A + B +C - (exactly two) - 2(all three) + none

Refer to GMAT Club math book for more details.
­Hello,

Overlapping sets are a daunting task and this formula is very helpful. I want to make sure that I translated correctly for n sets with n≥3.

So, the formula is Total = a+b+..+n + none - (n-1)*n - nk where nk is the number of sets in which I want to find how many objects there are in EXACTLY nk sets ?

For instance, if I have a total of 260 objects with 4 sets (a,b,c,d) and each one has 25% of the total and 8 there aren't in none sets and I want to find how many there are in EXACTLY 3 sets then I'll do: 260 = 4*0.25*260 + 8 - (4-1)*3 - 3 = 256?

Thank you in advance!­
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­That formula is explained in details here: ADVANCED OVERLAPPING SETS PROBLEMS.

Hope it helps.­
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Of the 150 houses in a certain development, 60 percent have 02 May 2024, 02:32
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ganand

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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Hello to everyone and thanks for the explanation. I still got a doubt on the formula i highlighted, especially on the -D.
In the -D what we need to put, if we are request to find the -D? the sum of all the 3 group that overlaps with 2 element or only one of the 3 group that overlaps with 2 elements?

thanks 
­
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Of the 150 houses in a certain development, 60 percent have 02 May 2024, 02:38
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thexxx

ScottTargetTestPrep

ganand

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
Hello to everyone and thanks for the explanation. I still got a doubt on the formula i highlighted, especially on the -D.
In the -D what we need to put, if we are request to find the -D? the sum of all the 3 group that overlaps with 2 element or only one of the 3 group that overlaps with 2 elements?

thanks 
­
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­That formula is explained in details here: ADVANCED OVERLAPPING SETS PROBLEMS.
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Of the 150 houses in a certain development, 60 percent have 29 May 2024, 21:40
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­Classic three group overlapping sets. Just plug in and solve:
­
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Of the 150 houses in a certain development, 60 percent have 10 Jul 2024, 17:15
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Total homes = 150

60% of total homes have AC
A = 3/5(150) = 90

50% of total homes have a sunporch
B = 1/2(150) = 75

30% of total homes have a swimming pool
C = 3/10(150) = 45

Number of homes with all three amenities = 5

Number of homes with none of the amenities = 5

Since there is a mention of exactly two, which is what the question is asking for, we can use the following equation:

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

A = Number of homes with Air Conditioning
B = Number of homes with a sunporch
C = Number of homes with a swimming pool

We can assign variable x for number with exactly two of these amenities.

150 = 90 + 75 + 45 - x - 2(5) + 5

x = 205 - 150
x = 55

Option D

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Of the 150 houses in a certain development, 60 percent have 12 Nov 2024, 20:19
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Just adding my 2 cents, something which may be overlooked

(1) We are given all the houses with AC, Sunporch & Swimming Pool - this contains OVERLAPS
(2) We are given the TOTAL

As a result, we need to SUBTRACT OUT the overlaps

(1) Exactly 2 - your unknown variable
(2) 2 x 5 (houses with three amenities) - because these are counted 3 TIMES already
(3) Add back the neither for the total

Once you understand this the formula is easy to understand (vs trying to rote learn / memorise).
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Of the 150 houses in a certain development, 60 percent have 16 Nov 2025, 07:26
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Let's use inclusion–exclusion cleanly.
Total houses = 150.
Counts: AC = 60% of 150 = 0.60×150 = 90.
Sunporch = 50% of 150 = 75.
Pool = 30% of 150 = 45.
All three = 5. None = 5.
Let xxx = number with exactly two amenities, and yyy = number with exactly one.
We have none+exactly1+exactly2+exactly3=150 \text{none} + \text{exactly1} + \text{exactly2} + \text{exactly3} = 150none+exactly1+exactly2+exactly3=150, so
5+y+x+5=150⇒y+x=140.5 + y + x + 5 = 150 \Rightarrow y + x = 140.5+y+x+5=150⇒y+x=140. (1)
Also sum of single-amenity counts = 90+75+45=21090+75+45=21090+75+45=210. That equals
y+2x+3⋅5=y+2x+15.y + 2x + 3\cdot5 = y + 2x + 15.y+2x+3⋅5=y+2x+15. So
y+2x+15=210⇒y+2x=195.y + 2x + 15 = 210 \Rightarrow y + 2x = 195.y+2x+15=210⇒y+2x=195. (2)
Subtract (1) from (2): (y+2x)−(y+x)=x=195−140=55.(y+2x)-(y+x)=x=195-140=55.(y+2x)−(y+x)=x=195−140=55.
So 55 houses have exactly two amenities. Answer: (D) 55.

aaliyahkhalifa
Hi! Can please someone explain why we multiply by 5 (all three options) by 2?
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