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

We can create the equation:

Total = alarm + deadbolt - both + neither

100 = x + y - both + z

both = x + y + z - 100

Answer: E

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:



z homeowners had neither an alarm security system nor deadbolt locks
We get:



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



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)


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

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

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?

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!

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.

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)

TOTAL - NEITHER = A + B - EITHER (BOTH)

100 - Z = X + Y - EITHER

EITHER = X + Y - ( 100 - Z )

EITHER = X + Y + Z - 100

ANS = E

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