GMAT Math: Word Problem - Set Theory

GMAT Math: Word Problem - Set Theory

 In a village of hundred households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. Every village has at least one of these three devices. If X and Y are respectively the greatest and the lowest possible number of households that have all the three of these devices, X-Y is :

A) 65

B) 55

C) 45

D) 35

E) 25

Solution

Answer C

This is a really tricky one, you will encounter this on the actual GMAT if you are doing very good on the GMAT math section. This involves sets.

First we need to figure out what the maximum is. We know it is possible there are 55 with all three, since the 55 MP3 owners is smallest set of the three. (Max = # of MP3)

To calculate the fewest possible, we first need to figure out how many must have all three devices. (max possible - max without all three = minimum with all three).

Of the 80 cell phone owners, 25 cannot have an MP3 player 80 - 55 = 25.

Similarly, 20 of the DVD owners cannot have an MP3 player. 75 - 55 = 20.

If the owners who do not have all three are unique (no overlap), we get the maximum number of owners who do not own an MP3 player, and thus cannot own all three devices.

Max possible - maximum without all three = minimum number of people with all three devices.

55 - 45 = 10

Now we can solve:

55 - 10 = 45.