In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from:

(A) 30,000 to 90,000 inclusive
(B) 30,000 to 120,000 inclusive
(C) 60,000 to 90,000 inclusive
(D) 60,000 to 120,000 inclusive
(E) 90,000 to 120,000 inclusive

In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders

A survey was conducted to determine the popularity of 3 foods among students. The data collected from 75 students are summarized as below

48 like Pizza
45 like Hoagies
58 like tacos
28 like pizza and hoagies
37 like hoagies and tacos
40 like pizza and tacos
25 like all three food

What is the number of students who like none or only one of the foods?

(A) 4
(B) 16
(C) 17
(D) 20
(E) 23

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

(A) 65
(B) 55
(C) 45
(D) 35
(E) 25

You only have 100 households so there has to be overlap considering that the numbers, 75, 80, and 55 sum to 210. Let's look at the DVD Player and Cell Phone groups first because they are the biggest and will create the most obvious constraints. They sum to 155. Considering that there are only 100 households it must be that there are at least 55 overlaps, people who have both DVD and Cell. You could have the MP3 group completely contained in that 55. In that case, 55 people have three of the devices. That's the max that have all three of the devices.

Now let's try to minimize the people who have DVD, Cell Phone, and MP3 (I call these triples). With that in mind we want to make the MP3 people as independent as possible. There are 25 people who DON'T have a DVD (100 - 75) and 20 people who DON'T have a Cell Phone (100 - 80). That's 45 people total. Now, some of the 25 no DVD could be a part of the 20 Cell Phone but they don't have to be. The 20 and 25 could be independent and in this case we want those groups as unique as possible to maximize the number of people who are in ONLY MP3 player. Those 45 people could be ONLY MP3 player. So, because there are then ten leftover MP3 Player people (55 - 45) those 10 must be in all three groups. 55 (max in all three) - 10 (min in all three) = 45. Choose C.

You could also use the three group overlapping sets formulas:

Group A + Group B + Group C + None - Doubles - 2(Triples) = Total

210 - d - 2t = 100

-d - 2t = -110

d + 2t = 110

So now we know that that sum of d + 2t is 110. So the max t could be is 55 (2*55 = 110). A quick common sense check verifies that it is possible that all of the MP3 people are contained in the overlap between the DVD and Phone people. Now let's try to maximize the doubles. Can they be 110? Nope. You only have 100 households total. Can they be 100? Nope. Then you'd have 5 triples to balance out the equation. Again your max d + t is 100. Can they be 90? Then t is 10. That could work 90 + 10 = 100.

You can also visualize with Venn diagrams. For the most part I don't like Venns for three group (or two group) sets but for this one it can actually work relatively well. I'll circle back with a Venn Diagram solution.

GMAT Question of the Day - Data Sufficiency - Overlapping Sets

Of the leopards at a certain zoo, 20% are both spotted and mature. If all the leopards at the zoo are either spotted or not spotted, or mature or immature, is the ratio of the number of immature leopards who are spotted to the number of immature leopards who are not spotted greater than 1?

(1) If the number of immature spotted leopards doubled than there would be 102 total leopards at the zoo.

(2) If the number of immature but not spotted leopards were decreased by a half there would 86 total leopards at the zoo.

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GMAT Question of the Day Solution

This overlapping sets question might leave you feeling a bit empty. Thinking - what was the point of that? Well, the point of that was to define the information that you were given and then to realize that you didn't have enough to answer the questions. That's what DS questions with an "E" answer are like. Sometimes there is no further meaning than just plain old insufficient.

GMAT Question of the Day Overlapping Sets Solution

 

GMAT Question of the Day - Problem Solving - Overlapping Sets

A certain farmer has a field with 105 tomato plants that have genes from at least one of three varieties of tomato: Green Tomatoes, Red Tomatoes, and Purple Tomatoes. 19 of the plants have red and green genes, 17 of the plants have red and purple genes and 14 of the plants have green and purple genes. If 10 of the plants have all three types of genes then how many have only one type of gene?

A. 45

B. 55

C. 60

D. 65

E. 75

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GMAT Question of the Day Solution

 

 

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