In a group of 100 homeowners, x homeowners had an alarm security syste

Question

In a group of 100 homeowners, x homeowners had an alarm security system and y homeowners had deadbolt locks. If z homeowners had neither an alarm security system nor deadbolt locks, how many homeowners had both an alarm security system and deadbolt locks?

A. 100 – x – y – z

B. 100 – x – y + z

C. x – y – z + 100

D. x + y + z + 100

E. x + y + z – 100

A. 100 – x – y – z

B. 100 – x – y + z

C. x – y – z + 100

D. x + y + z + 100

E. x + y + z – 100

GMATinsight · Answer

There can be only 4 catagories

Home with alarm security only + Homes with deadbolt Lock + Home with Both + Home with none = 100%

x = Home with alarm security only+ Home with Both
y = Homes with deadbolt Lock + Home with Both
z = Home with none

x+y+z - Home with Both = 100

So Home with both = x + y + z – 100

Answer: option E

US09 · Answer

I have a question.
Will the equation not become: x+y+z - 2*Home with Both = 100? Why are we counting the "homes with both" only once here?

pushpitkc · Answer

Hey urvashis09 Think of it as a Venn diagram where the sum of x and y includes the value of both(marked in red) twice. So, we need to reduce it once as the total number of homes will include the home with both facilities once only. Home_VennDiag.png Hope this clears your confusion.

US09 · Answer

Oh, okay so that means subtracting it twice would then just refer to the non-shaded part, which would be wrong. Now I get it! Thank you so much!

ColumbiaEMBA · Answer

Alarm: x (not alarm only, therefore could include people with deadbolt)
Deadbolt: y (not deadbolt only, therefore could include people with alarm)
None: z
Both: call it  b

Equation: (x - b) + b + (y - b) + z = 100. This can be rewritten as: x + b + y + z = 100. We need b so rewrite as: b = x + y + z - 100. Answer is E.

blueviper · Answer

The answer should be A. 100 - x - y - z

JeffTargetTestPrep · Answer

We can create the equation:

Total = alarm + deadbolt - both + neither

100 = x + y - both + z

both = x + y + z - 100

Answer: E

AkshdeepS · Answer

This can be done by using formula :

Total = Group 1 + Group 2 - Both + Neither

Both = Group 1 + Group 2 + Neither - Total

Both = x + y + z - 100

(E)

raj2812 · Answer

TOTAL - NEITHER = A + B - EITHER (BOTH) 100 - Z = X + Y - EITHER EITHER = X + Y - ( 100 - Z ) EITHER = X + Y + Z - 100 ANS = E

BrentGMATPrepNow · Answer

Let's use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions)..

Here, we have a population of homeowners, and the two characteristics are: 
- has alarm security system or does NOT have alarm security system
- has deadbolt locks or does NOT have deadbolt locks

In a group of 100 homeowners, x homeowners had an alarm security system and y homeowners had deadbolt locks.   
We can set up our matrix as follows:
 https://i.imgur.com/6WRDYE4.png

z homeowners had neither an alarm security system nor deadbolt locks  
We get: 
 https://i.imgur.com/mVm5Vaf.png

Since the two boxes in the BOTTOM ROW must add to 100-y, we know that the missing box must be  100-y-z , since  100-y-z  + z = 100-y
   https://i.imgur.com/INJ7qG7.png

Finally, the two boxes in the LEFT-SIDE column must add to x
In other words, ? + (100-y-z) = x
Subtract (100-y-z) from both sides to get: ? =  x - (100-y-z) 
   https://i.imgur.com/ju0j2Wp.png

Take:  x - (100-y-z) 
Simplify to get: x - 100 + y + z
Rearrange to get: x + y + z - 100

Answer: E

This question type is  VERY COMMON  on the GMAT, so be sure to master the technique.

RELATED VIDEO 
 https://www.youtube.com/watch?v=jK-tiBrrf04

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In a group of 100 homeowners, x homeowners had an alarm security syste

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