GMAT Question Patterns: Overlapping Sets

Question

GMAT Question Patterns: Overlapping Sets

Table of Contents:

• Two-Set Basic Overlap

Basic Overlap (Solve for Total/Both/Neither)
Conditional Overlap (Subgroup Ratios)
Exactly One" or "At Least One"

• Three-Set Overlap
• Minimum/Maximum Overlap
• Theory
• Questions

Pattern #1: Two-Set Basic Overlap
- Involves two groups (A and B) with possible overlap. Solved using logic, Venn diagrams, or a double-set matrix.

1.1 Basic Overlap (Solve for Total/Both/Neither) - Direct application of Total = A + B - Both + Neither

Example 1.1 (Difficulty: 505)
In a class of 30 students, 18 students have used electronic mail to submit their homework and 9 students have used faxes to submit their homework. If 8 students have used neither electronic mail nor faxes to submit their homework, how many students have used both electronic mail and faxes to submit their homework?

A. 3
B. 4
C. 5
D. 9
E. 10

Question Discussion.

Example 1.2 (Difficulty: 505)
Of the 3,600 families in Township X, 1/2 subscribe to the Beacon, 1/4 subscribe to the Herald, and 1/5 subscribe to both papers. How many of the families in Township X do not subscribe to either the Beacon or the Herald?

A. 1,620
B. 1,200
C. 720
D. 540
E. 180

Question Discussion.

Example 1.3 (Difficulty: 505)
Of the 400 employees in a certain office, 260 have been given a watch and 80 have been given a cell phone. If the number of employees who have been given both a watch and a cell phone is 1/2 the number who have been given neither a watch nor a cell phone, how many employees have been give a watch but not a cell phone?

A. 90
B. 170
C. 180
D. 200
E. 220

Question Discussion.

Example 1.4 (Difficulty: 505)
In a survey on sales of soft drinks, 65 percent of the surveyed customers said that they buy brand A regularly, 60 percent said that they buy brand B regularly, and 32 percent said that they do not usually buy either of the two brands. What percent of the surveyed customers said that they buy both brands A and B regularly?

A. 37
B. 54
C. 57
D. 59
E. 61

Question Discussion.

Example 1.5 (Difficulty: 505)
In a survey of 200 people, 80 people read Magazine M and 96 people read magazine R. If the number of people surveyed who do not read either magazine is 3 times the number who read both magazines, how many of the people surveyed read both the magazines?

A. 8
B. 12
C. 16
D. 24
E. 50

Question Discussion.

Example 1.6 (Difficulty: 505)
In a study conducted on a sample of 360 economists, it was found that 75 percent of the economists had investment in stocks and 40 percent had investment in bonds. If 15 percent of the sample did not have investments in either stocks or bonds, how many economists in the sample had investment in stocks but not in bonds?

A. 108
B. 120
C. 126
D. 162
E. 180

Question Discussion.

Example 1.7 (Difficulty: 505)
Of the apartments in a certain building, 75 have a balcony, 80 have a fireplace, and 30 have neither a balcony nor a fireplace. How many of the apartments in the building that have a balcony do not have a fireplace?

(1) There is a total of 150 apartments in the building.
(2) 35 of the apartments in the building that have a balcony also have a fireplace.

Question Discussion.

Example 1.8 (Difficulty: 505)
Of the 240 coats at a certain store, how many are green and are not made of wool?

(1) 2/5 of the coats are neither green nor made of wool.
(2) 1/2 of the coats are made of wool and are not green.

Question Discussion.

Example 1.9 (Difficulty: 505)
In a product test of a common cold remedy, x percent of the patients tested experienced side effects from the use of the drug and y percent experienced relief of cold symptoms. What percent of the patients tested experienced both side effects and relief of cold symptoms?

(1) Of the 1,000 patients tested, 15 percent experienced neither side effects nor relief of cold symptoms.
(2) Of the patients tested, 30 percent experienced relief of cold symptoms without side effects.

Question Discussion.

Example 1.10 (Difficulty: 505)
Each of the dogs enrolled in an obedience course either passed or failed. What percent of the dogs enrolled in the course passed?

(1) Of the female dogs enrolled, 70 percent passed.
(2) Of the male dogs enrolled, 63 percent passed

Question Discussion.

Example 1.11 (Difficulty: 505)
In a certain classroom, there are 80 books, of which 24 are fiction and 23 are written in Spanish. How many of the fiction books are written in Spanish?

(1) Of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish.
(2) Of the books written in Spanish, there are 5 more nonfiction books than fiction books.

Question Discussion.

Example 1.12 (Difficulty: 505)
For what percent of those tested for a certain infection was the test accurate; that is, positive for those who had the infection and negative for those who did not have the infection?

(1) Of those who tested positive for the infection, 1/8 did not have the infection.
(2) Of those who tested for the infection, 90 percent tested negative.

Question Discussion.

Example 1.13 (Difficulty: 705)
When a certain drug was administered to a group of patients, some patients showed improvement and some patients experienced side effects. If at least 10 patients experienced side effects and 60 percent of the patients who showed improvements also experienced side effects, did more patients in the group show improvement than experience side effects?

(1) 90 percent of the patients in the group showed improvement.
(2) 90 percent of the patients in the group who experienced side effects also showed improvement.

Question Discussion.

Example 1.14 (Difficulty: 705)
The table above shows the results of a survey of 100 voters who each responded "Favorable" or "Unfavorable" or "Not Sure" when asked about their impressions of Candidate M and of Candidate N. What was the number of voters who responded "Favorable" for both candidates?

(1) The number of voters who did not respond "Favorable" for either candidate was 40.
(2) The number of voters who responded "Unfavorable" for both candidates was 10.

Question Discussion.

1.2 Conditional Overlap (Subgroup Ratios) - Overlaps defined as fractions/percentages of subgroups

Example 1.15 (Difficulty: 505)
Of the employees at a certain company, 60 percent participate in the company's retirement plan; and of those employees, 20 percent also participate in the company's health plan. If 48 percent of the company's employees participate in its health plan, what fraction of those employees also participate in the company's retirement plan?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4

Question Discussion.

Example 1.16 (Difficulty: 505)
Each of 42 people purchased one bouquet of carnations from a flower shop, and each bouquet had red carnations only, white carnations only, or a combination of red and white carnations. Of the bouquets purchased, if 28 contained red carnations and 24 contained white carnations, how many contained red carnations only?

A. 4
B. 10
C. 18
D. 24
E. 28

Question Discussion.

Example 1.17 (Difficulty: 605)
In Diana's stamp collection, \(\frac{4}{5}\) of the stamps are Canadian, and \(\frac{3}{7}\) of the Canadian stamps were issued before 1940. If 192 stamps in Diana's collection are Canadian stamps that were issued in 1940 or later, how many stamps in her collection are not Canadian?

A. 84
B. 88
C. 96
D. 104
E. 112

Question Discussion.

Example 1.18 (Difficulty: 605)
In a certain school, \(\frac{1}{5}\) of the students own an older computer, made more than three years ago; \(\frac{1}{30}\) of the students own a newer computer, made in the last three years; and \(\frac{7}{9}\) of the students own no computer. What fraction of the students own both an older and a newer computer?

A. \(\frac{1}{180}\)
B. \(\frac{1}{90}\)
C. \(\frac{1}{60}\)
D. \(\frac{1}{45}\)
E. \(\frac{2}{9}\)

Question Discussion.

Example 1.19 (Difficulty: 605)
Of the vehicles (cars and trucks) for sale at a certain dealership, 48 percent are cars that have 4 doors. If 80 percent of the cars for sale at the dealership have 4 doors, what percent of the vehicles are cars that do not have 4 doors?

A. 12%
B. 20%
C. 24%
D. 32%
E. 40%

Question Discussion.

Example 1.20 (Difficulty: 605)
In a certain nursery, 15 percent of the plants developed a fungus and died. If 40 percent of the plants that developed the fungus did not die, what percent of the plants in the nursery developed the fungus?

A. 25%
B. 27%
C. 35%
D. 40%
E. 55%

Question Discussion.

Example 1.21 (Difficulty: 605)
Of the vehicles (cars and trucks) for sale at a certain dealership, 48 percent are cars that have 4 doors. If 80 percent of the cars for sale at the dealership have 4 doors, what percent of the vehicles are cars that do not have 4 doors?

A. 12%
B. 20%
C. 24%
D. 32%
E. 40%

Question Discussion.

Example 1.22 (Difficulty: 605)
At a certain corporation, the planning committee has 23 members and the finance committee has 20 members. If a total of 21 people are members of only one of the two committees, how many are members of both committees?

A. 3
B. 11
C. 15
D. 20
E. 22

Question Discussion.

Example 1.23 (Difficulty: 505)
Of the cans of peaches inspected yesterday at a certain plant, 1.5 percent failed to pass inspection. Of the cans that failed inspection, five sixths were incorrectly labeled and the rest were dented. If all of the cans that were incorrectly labeled or dented failed inspection, how many of the cans of peaches inspected yesterday at the plant were dented?

(1) 450 of the cans of peaches inspected yesterday at the plant failed to pass inspection.
(2) 29,550 of the cans of peaches inspected yesterday at the plant passed inspection.

Question Discussion.

Example 1.24 (Difficulty: 605)
In a survey of retailers, what percent had purchased computers for business purposes?

(1) 85 percent of the retailers surveyed who owned their own store had purchased computers for business purposes.
(2) 40 percent of the retailers surveyed owned their own store.

Question Discussion.

Example 1.25 (Difficulty: 605)
In a survey of 200 college graduates, 30 percent said they had received student loans during their college careers, and 40 percent said they had received scholarships. What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers?

(1) 25 percent of those surveyed said that they had received scholarships but no loans.
(2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.

Question Discussion.

Example 1.26 (Difficulty: 705)
Last year in Country X, 15% of all students who received an undergraduate degree received their degree from a Type 1 university, while 30% of all students who received an undergraduate degree in a STEM (science, technology, engineering, or mathematics) major received their degree from a Type 1 university. Last year in Country X, what percent of all students who received an undergraduate degree from a Type 1 university received an undergraduate degree in a STEM major?

(1) Last year in Country X, 30% of all students who received an undergraduate degree received an undergraduate degree in a STEM major.
(2) Last year in Country X, a total of 9,000 students received a degree in a STEM major from a Type 1 university.

Question Discussion.

1.3 "Exactly One" or "At Least One" - Focus on exclusive counts (only A/only B) vs. any overlap

Example 1.27 (Difficulty: 505)
Of the 32 students in a certain class, 15 are in a music club and 20 are in a dance club. If 10 of the students are not in either club, how many of the students are in only one of the two clubs?

A. 5
B. 9
C. 10
D. 17
E. 25

Question Discussion.

Example 1.28 (Difficulty: 505)
Of the employees at a certain company, 60 percent participate in the company retirement’s plan; and of those employees, 20 percent also participate in the company’s health plan. If 48 percent of the company’s employees in its health plan, what fraction of those employees also participate in the company’s retirement plan?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4

Question Discussion.

Example 1.29 (Difficulty: 505)
Of w people in a room, x people had a sandwich for lunch and y people had soup for lunch. Of the people who had a sandwich or soup, z people had both a sandwich and soup. In terms of w, x, y, and z, what fraction of the people in the room had neither a sandwich nor soup?

A.

B.

C.

D.

E.

Question Discussion.

Example 1.30 (Difficulty: 605)
At a certain college last year, a total of 260 students took College Algebra in either the fall semester or the spring semester or both. If 40 students took College Algebra in both semesters and twice as many students took College Algebra in the fall semester as in the spring semester, how many students took College Algebra in the fall semester last year?

A. 73
B. 100
C. 142
D. 160
E. 200

Question Discussion.

Example 1.31 (Difficulty: 505)
Of the 40 researchers at a certain company, 60 percent work on project P and 30 percent work on project Q. How many of the researchers at the company work on project P but not on project Q?

(1) 4 of the researchers at the company work on project Q but not on project P.
(2) 12 of the researchers at the company do not work on either project P or project Q.

Question Discussion.

Example 1.32 (Difficulty: 505)
Of the 240 toy-import businesses in an association, 80 percent imported toys from Asia and 25 percent imported toys from Europe. How many of these businesses imported toys from Europe but not from Asia?

(1) Of the businesses in the association that imported toys from Asia, 25 percent also imported toys from Europe.
(2) Of the businesses in the association, 15 percent neither imported toys from Asia nor imported toys from Europe.

Question Discussion.

Example 1.33 (Difficulty: 505)
Is the number of members of Club X greater than the number of members of Club Y?

(1) Of the members of Club X, 20 percent are also members of Club Y.
(2) Of the members of Club Y, 30 percent are also members of Club X.

Question Discussion.

Example 1.34 (Difficulty: 705)
At a certain banquet, meals were served in only 2 locations. In the main dining room, 40 servings of eggplant parmesan and 60 servings of risotto were served. In the annex, 60 servings of eggplant parmesan and 30 servings of risotto were served. If each guest at the banquet was served exactly 1 serving of eggplant parmesan only, exactly 1 serving of risotto only, or exactly 2 servings, one of eggplant parmesan and the other of risotto, how many guests were at the banquet?

(1) In the main dining room, each of exactly 5 guests were served both eggplant parmesan and risotto; and in the annex, exactly 5 guests were served risotto only.
(2) The number of guests that were served eggplant parmesan only was the same in the 2 locations.

Question Discussion.

Example 1.35 (Difficulty: 705)
When a certain drug was administered to a group of patients, some patients showed improvement and some patients experienced side effects. If at least 10 patients experienced side effects and 60 percent of the patients who showed improvements also experienced side effects, did more patients in the group show improvement than experience side effects?

(1) 90 percent of the patients in the group showed improvement.
(2) 90 percent of the patients in the group who experienced side effects also showed improvement.

Question Discussion.

Pattern #2: Three-Set Overlap
- Problems with three groups (A, B, and C) and overlaps between them; solved using a three-set Venn diagram or standard formulas.

Example 2.1 (Difficulty: 605)
A certain firm has 28 lawyers in three states. These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z. How many lawyers have license only in state Z?

A. 3
B. 6
C. 10
D. 11
E. 13

Question Discussion.

Example 2.2 (Difficulty: 605)
A certain law firm has 28 lawyers and has offices in three states, X, Y, and Z. Every lawyer in the firm is either licensed in all three states or is licensed in only one of the three states. If 10 lawyers are licensed in State X, 11 are licensed in State Y, and 13 are licensed in State Z, how many lawyers are licensed only in State Z?

A. 3
B. 7
C. 8
D. 10
E. 25

Question Discussion.

Example 2.3 (Difficulty: 705)
Of the 45 households in a certain neighborhood, 28 subscribe to Newspaper Q, 17 subscribe to Newspaper R, 12 subscribe to Newspaper S, 7 subscribe to both Q and R, 8 subscribe to both Q and S, and 9 subscribe to both R and S. The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers. If 39 of the households subscribe to at least one of the three newspapers, how many households subscribe to only one of the newspapers?

A. 15
B. 21
C. 27
D. 33
E. 46

Question Discussion.

Example 2.4 (Difficulty: 705)
To get a job at Company X, an applicant must be recommended by three interviewers. Out of 30 applicants, 15 were recommended by the first interviewer, 17 by the second and 20 by the third interviewer. What is the least number of applicants who would have to have been recommended by all three interviewers?

A. 0
B. 2
C. 3
D. 5
E. 10

Question Discussion.

Example 2.5 (Difficulty: 605)
Of the students at a certain school, 40 took French, 30 took Latin, and 20 took Spanish. How many students at the school took only two of the three languages?

(1) 5 students at the school took all three of the languages.
(2) 52 students at the school took at least one of the languages.

Question Discussion.

Example 2.6 (Difficulty: 705)
There are 210 households in a certain residential complex. All households in the complex that have more than one dog also have at least one cat. All households that have at least one cat have a pet rodent. How many households in the complex have a pet rodent?

(1) 18 households in the complex have two or more dogs.
(2) 90% of households in the complex that have a pet rodent have at least one cat.

Question Discussion.

Pattern #3: Minimum/Maximum Overlap
- Questions asking for the smallest or largest possible overlap between groups.

Example 3.1 (Difficulty: 605)
Of the 190 bicycle stores in a certain region, 90 percent repair bicycles and 40 percent rent bicycles. If the integer n denotes the number of bicycle stores in the region that both repair and rent bicycles, which of the following indicates all possible values of n?

A. 38 ≤ n ≤ 76
B. 57 ≤ n ≤ 76
C. 57 ≤ n ≤ 171
D. 76 ≤ n ≤ 95
E. 76 ≤ n ≤ 171

Question Discussion.

Example 3.2 (Difficulty: 605)
All 15 members of a foreign language club speak one or more of three languages—Spanish, French, and German. If 1/3 of the members speak Spanish, 2/5 of the members speak French, 2/3 of the members speak German, and 1 member speaks all three of the languages, how many members speak exactly two of the languages?

A. 0
B. 1
C. 2
D. 3
E. 4

Question Discussion.

Example 3.3 (Difficulty: 605)
Among the N books in Estelle's library, 40 are fiction and the rest are nonfiction. Fifty of the N books are hardcovers and the rest are paperbacks. If 30 of the N books are hardcover fiction books, what is the least possible value of N?

A. 50
B. 60
C. 70
D. 80
E. 90

Question Discussion.

Example 3.4 (Difficulty: 705)

The table gives three factors to be considered when choosing an Internet service provider and the percent of the 1,200 respondents to a survey who cited that factor as important. If 30 percent of the respondents cited both "user-friendly" and "fast response time", what is the maximum possible number of respondents who cited "bargain prices", but neither "user-friendly" nor "fast response time"?

A. 312
B. 336
C. 360
D. 384
E. 420

Question Discussion.

Theory:

Questions:

Problem Solving

Data Sufficiency

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SwethaReddyL · Answer

Hi  Bunuel , discussion mapped on question 1.11 takes to some other question and not the one that is mentioned. Could you please look into it?

Bunuel · Answer

Fixed the link. Thank you!

Bunuel · Answer

GMAT Question Patterns:

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  GMAT Question Patterns: Remainders  
  GMAT Question Patterns: Algebra  
  GMAT Question Patterns: Overlapping Sets

rak08 · Answer

same issue with 2.3 Q

Bunuel · Answer

Fixed the link. Thank you!

Bunuel · Answer

New one:  GMAT Question Patterns: Sequences

Bunuel · Answer

The new pattern topic:  GMAT Question Patterns: Functions and Custom Characters .

GMAT Question Patterns: Overlapping Sets

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