### All GMAT Math Resources

## Example Questions

### Example Question #11 : Problem Solving Questions

Let universal set represent the set of all restaurants in the city of Bancroft. Let represent the set of all restaurants in Bancroft that accept the America's Best credit card. Let represent the set of all restaurants in Bancroft that accept the Bank Zero credit card. Let represent the set of all restaurants that accept the Chargitall credit card.

Of the following five restaurants in Bancroft:

Ally's accepts America's Best and Bank Zero, but not Chargitall.

Benny's accepts Chargitall, but not American's Best or Bank Zero.

Carleton's accepts America's Best, but neither Bank Zero nor Chargitall.

Danny's accepts Bank Zero, but neither America's Best nor Chargitall.

Eddie's does not accept any of the three cards.

Which of the following restaurants would fall in the set ?

**Possible Answers:**

Benny's only

Carleton's and Danny's only

Benny's and Eddie's only

Ally's and Eddy's only

Ally's only

**Correct answer:**

Benny's only

To fall in the set , a restaurant would have to be in the union of three sets - that is, any one or more. The three sets are:

, the set of all restaurants in Bancroft that accept the America's Best credit card;

, the set of all restaurants in Bancroft that accept the Bank Zero credit card; and

, the complement of the set of all restaurants in Bancroft that accept the Chargitall credit card, or, equivalently, set of all restaurants in Bancroft that do *not* accept the Chargitall credit card.

The complement of this set, , is the set of all restaurants that belong to *none* of , so the correct choice must be a restaurant that belongs to *and *. The restaurant must accept Chargitall, and must *not* accept the other two cards.

The only one of the five restaurants fitting this description is Benny's.

### Example Question #11 : Word Problems

Let be the universal set of all people. Let be the set of all people who like cola. Let be the set of all people who like milk.

Let represent Victor. Which of the following English-language sentences is the opposite of ?

**Possible Answers:**

Victor likes cola or milk, but not both.

Victor likes both cola and milk.

Victor likes cola, milk, or both.

Victor likes neither cola nor milk.

None of the other responses is correct.

**Correct answer:**

Victor likes cola, milk, or both.

The opposite of the statement is the statement

,

which, by DeMorgan's Law, is equivalent to

, or

Victor belongs to either or both of , the set of all people who like cola, and , the set of all people who like milk. Therefore, Victor likes cola, milk, or both.

### Example Question #11 : Sets

Let be the universal set of all people. Let be the set of all people who like Frank Sinatra. Let be the set of all people who like The Beatles. Let be the set of all people who like Elvis Presley. Let stand for Tommy.

Let . Which of the following *must* be false?

**Possible Answers:**

Tommy likes Frank Sinatra and The Beatles, but not Elvis Presley.

Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles.

Tommy likes Frank Sinatra, but neither The Beatles nor Elvis Presley.

Tommy likes Frank Sinatra, The Beatles, and Elvis Presley.

Tommy does not like Frank Sinatra, The Beatles, or Elvis Presley.

**Correct answer:**

Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles.

We can best answer this question using a Venn diagram. We want to examine the union of two sets:

, the intersection of and , which is the region the two circles have in common, and

, the complement , which is the region outside the circle representing this set.

Since we are examing the union, we want to shade in everything in one or both of these two sets. This is shown below:

If Tommy likes Frank Sinatra, The Beatles, and Elvis Presley, falls in all three circles, placing him in Region (i).

If Tommy does not like Frank Sinatra, The Beatles, or Elvis Presley, falls in none of the three circles, placing him in Region (ii).

If Tommy likes Frank Sinatra, but neither The Beatles nor Elvis Presley, falls in circle A, but neither B nor C, placing him in Region (iii).

If Tommy likes Frank Sinatra and The Beatles, but not Elvis Presley, falls in circles A and B, but not C, placing him in Region (iv).

If Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles, falls in circles A and C, but not B, placing him in Region (v).

In all of the first four cases, is a true statement, since the region is part of the grayed area. In the fifth case, is false. Therefore, the only impossible statement among the choices is "Tommy likes Frank Sinatra and Elvis Presley, but not The Beatles."

### Example Question #1 : Union & Intersect

Let universal set be the set of all people. Let represent Brittany.

Let be the set of people who like James Blunt, , the set of people who like John Legend, and , the set of people who like Pharrell Williams.

True or false: Brittany likes James Blunt, John Legend, and Pharrell Williams.

Statement 1:

Statement 2:

**Possible Answers:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Assume Statement 1 alone. is the intersection of the sets , , and . Since Brittany falls in this intersection, she falls in all three sets, and it follows that she likes all three of James Blunt, John Legend, and Pharrell Williams.

Assume Statement 2 alone. is the union of the three sets. Since Brittany falls in this union, she falls in one, two, or all three sets, which means that she likes at least one of James Blunt, John Legend, and Pharrell Williams. However, more information is needed before it can be established whether or not she likes all three.

### Example Question #11 : Gmat Quantitative Reasoning

Let be the universal set of all people. Let be the set of all people who are dancers and be the set of all people that are singers.

Let represent Jeremy. Which of these statements states the *opposite* of "Jeremy is not a dancer, but he is a singer" in set notation?

**Possible Answers:**

**Correct answer:**

In set notation, "Jeremy is not a dancer, but he is a singer" can be stated as

That is, Jeremy falls in the intersection of the complement of the set of dancers and the set of singers .

The opposite of this is that is in the complement of this set:

By DeMorgan's law, this is equivalent to saying

, or

.

### Example Question #11 : Word Problems

Every Moog is a Noog. Every Noog is a Poog. Every Poog is a Roog.

True or False: Stumpy is a Moog.

Statement 1: Stumpy is not a Poog.

Statement 2: Stumpy is not a Roog.

**Possible Answers:**

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

**Correct answer:**

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Let be the sets of Moogs, Noogs, Poogs, and Roogs, respectively, From the premise of the problem, we have that

.

It follows by transitivity that and .

Assume Statement 1 alone. If Stumpy is a Moog, then

.

, makng Stumpy a Poog. But since Stumpy is NOT a Poog, then it follows, contrapositively, that Stumpy is not a Moog.

Assume Statement 2 alone. If Stmpy is a Moog, then

,

making Stumpy a Roog. Since Stumpy is NOT a Roog, then it follows, contrapositively, that Stumpy is not a Moog.

### Example Question #11 : Gmat Quantitative Reasoning

Let universal set be the set of all people. Let be the set of people who like The Rolling Stones. Let be the set of people who like The Who. Let be the set of people who like Lynyrd Skynyrd.

Let stand for Nancy.

True or false:

Statement 1: Nancy does not like The Who.

Statement 2: Nancy likes Lynyrd Skynyrd.

**Possible Answers:**

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

**Correct answer:**

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

If , then falls in the intersection of two sets. One is , the union of , the set of people who like The Rolling Stones, and the complement of , the set of people not in - the set of all people who do not like The Who. The other set is the complement of , the set of all people not in - the set of people who don't like Lynyrd Skynyrd.

Therefore, if , *both* of the following must happen: Nancy must like The Rolling Stones or not like The Who, or both, *and* Nancy must not like Lynyrd Skynyrd.

From Statement 1 alone, Nancy does not like The Who, but it is not clear who else she likes. She is in , but it is not clear whether she is in the set of people who don't like Lynyrd Skynyrd, so the question of whether she falls in is open.

From Statement alone, since Nancy likes Lynyrd Skynyrd, she cannot be in . This makes false.

### Example Question #11 : Word Problems

Let be the universal set of all people. Let be the set of all people who can swim. Let be the set of all people who can ride a bicycle.

Let represent Kim. Which of the following English-language sentences is the opposite of ?

**Possible Answers:**

Either Kim cannot swim or she can ride a bicycle, but not both.

Kim cannot swim, but she can ride a bicycle.

Either Kim can swim or she cannot ride a bicycle, or both.

Kim can swim, but she cannot ride a bicycle.

Either Kim cannot swim or she can ride a bicycle, or both.

**Correct answer:**

Kim can swim, but she cannot ride a bicycle.

The opposite of the statement is the statement

,

which, by DeMorgan's Law, is equivalent to

or

This means that Kim is in the intersection of and , or, equivalently, she is in *both *, the set of people who can swim, and , the complement of the set of people who can ride a bicycle. Therefore, Kim can swim, but she cannot ride a bicycle.

### Example Question #1 : Relations

Let be the universal set of all people. Let be the set of all people who like Band A. Let be the set of all people who like Band B. Let be the set of all people who like Band C. Let stand for Julianna.

Let . Which of the following *could* be true?

**Possible Answers:**

Julianna likes Band A and Band B, but not Band C.

Julianna likes Band C, but not Band A or Band B.

Julianna likes Band B, but not Band A or Band C.

Julianna likes Band B and Band C, but not Band A.

Julianna likes Band A and Band C, but not Band B.

**Correct answer:**

Julianna likes Band A and Band C, but not Band B.

, which is the intersection of and . It follows that and .

, so it follows that Julianna likes Band A. The three choices that state that she does not can be eliminated.

. This is the union of , the complement of - that is, the set of people *not* in - and . It follows that either Julianna does not like Band B, does like Band C, or both. Therefore, it is not true that she likes Band B *and* does not like Band C. This can be eliminated.

The only possible choice is that Julianna likes Band A and Band C, but not Band B.

### Example Question #11 : Gmat Quantitative Reasoning

Let be the universal set of all people. Let be the set of all people who like catfish. Let be the set of all people who like flounder.

Let represent Wendy. Which of these statements states the *opposite* of "Wendy likes neither catfish nor flounder" in set notation?

**Possible Answers:**

**Correct answer:**

In set theory, "Wendy likes neither catfish nor flounder" can be stated as

.

The opposite of this is that is in the complement of this set:

.

By DeMorgan's law, this is equivalent to saying

, or

.

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