### All GMAT Math Resources

## Example Questions

### Example Question #11 : Data Sufficiency Questions

In the above Venn diagram, universal set represents the residents of a city. The sets represent the set of all teenagers, even birth month, and males, respectively.

Jamie is a resident of this city. Does Jamie have an even birth month?

Statement 1: Jamie is not a male.

Statement 2: Jamie is not a teenager.

**Possible Answers:**

EITHER statement ALONE is sufficient 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.

BOTH statements TOGETHER are insufficient to answer the question.

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

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The question is whether or not Jamie is an element of .

Assume both statements to be true. Then Jamie is an element of the set , shaded in this Venn diagram:

Some elements of are elements of , but some are not, making the two statements together insufficient to answer the question of whether Jamie is an element of . Whether Jamie was born in an even month or not cannot be determined.

### Example Question #11 : Data Sufficiency Questions

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

Let be the set of all people who like cheddar cheese. Let be the set of all people who like Swiss cheese.

True or false: .

Statement 1: Gina likes cheddar cheese.

Statement 2: Gina does not like Swiss cheese.

**Possible Answers:**

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

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

is the intersection of , the complement of the set of people who like cheddar cheese - that is, the set of people who *don't* like cheddar cheese - and , the set of people who like Swiss cheese. For to be true, it must hold that Gina doesn't like cheddar cheese *and* Gina likes Swiss cheese. Either statement alone makes this false.

### Example Question #11 : Data Sufficiency Questions

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

Let be the set of people who like The Eagles, , the set of people who like Bruce Springsteen, and , the set of people who like Crosby, Stills, and Nash.

True or false: Holly likes The Eagles.

Statement 1:

Statement 2:

**Possible Answers:**

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

EITHER statement ALONE is sufficient to answer the question.

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

BOTH statements TOGETHER are insufficient to answer the question.

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

**Correct answer:**

Assume Statement 1 alone. is the intersection of two sets - and , the complements of, respectively, the set of people who like The Eagles and the set of people who like Bruce Springsteen. Equivalently, this is the intersection of the set of people who do not like The Eagles and the set of people who do not like Bruce Springsteen. Since , Holly falls in both sets. Specifically, she does not like The Eagles.

Assume Statement 2 alone. is the union of two sets - , the set of people who do not like The Eagles and , the set of people who like Crosby, Stills, and Nash. If , Holly falls in either or both sets, so *either* she doesn't like The Eagles, *or* she likes Crosby, Stills, and Nash,* or both*. It is not resolved whether she likes The Eagles or not.

### Example Question #11 : Word Problems

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

Let be the set of people who like Pearl Jam, , the set of people who like Nirvana, and , the set of people who like Soundgarden.

True or false: Veronica likes at least *two* of Pearl Jam, Nirvana, and Soundgarden.

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements to be true.

is the union of the sets , , and ; Veronica falls in this union, so she falls in at least *one* of the three sets, so she likes one, two, or all three of Pearl Jam, Nirvana, and Soundgarden.

Similarly, from Statement 2, Veronica falls in at least *one* of the three sets , , and - the complements of the three sets - meaning that she *doesn't* like at least one of the three. Therefore, Veronica likes none, one, or two of the artists.

From the two statements together, it follows that Veronica likes one or two of the artists. Without further information, however, it cannot be determined whether she likes two of the artists.

### Example Question #11 : Sets

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

Let be the set of all people who like tea. Let be the set of all people who like coffee.

True or false: .

Statement 1: Frank likes tea.

Statement 2: Frank likes coffee.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

is the union of the set , the set of all people who like coffee, and , the complement of the set of all people who like tea - that is, the set of all people who do not like tea. if *either* Frank likes coffee *or* Frank does not like tea *or both*.

Statement 1 alone, that Frank likes tea, does not prove or disprove this to be true true; if Frank likes coffee, then this is true, and if he does not like coffee, then this is not true. Statement 2 alone, however, proves this statement true.

### Example Question #12 : Sets

Every Bee is a Cee. Every Dee is a Cee. Every Cee is a Fee.

True or false: Bumpy is a Fee.

Statement 1: Bumpy is not a Dee.

Statement 2: Bumpy is not a Bee.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

and .

This can be respresented in a Venn diagram as follows:

Assume both statements to be true. If we let represent Bumpy, then from the diagram, it can be seen that if Bumpy is neither a Bee or a Dee, it is possible for Bumpy to be be a Fee or to not be a Fee.

### Example Question #2121 : Gmat Quantitative Reasoning

No Fee is a Fi. No Fi is a Fo. No Fo is a Fum.

True or false: Jack is a Fum.

Statement 1: Jack is not a Fo.

Statement 2: Jack is not a Fi.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Let be the sets of Fe's, Fi's, Fo's and Fum's, respectively. According to the premise of the problem, the sets and are disjoint (they have no elements in common); the sets and are disjoint; and the sets and are disjoint. No other relationships are known. A Venn diagram of these relationships is below:

Assume both statements to be true. The question is whether or not Jack falls in set . If Jack is neither a Fi nor a Fo, then Jack falls in neither set nor set . As evidenced by the two X's in the diagram, it is possible for Jack to fall in set or to not fall in set . Therefore, the question of whether Jack is a Fum is unresolved.

### Example Question #12 : Data Sufficiency Questions

Everyone in Jonesville likes exactly three of the following - apples, bananas, cranberries, dates, figs, and guavas. No one in Jonesville who likes apples also likes guavas. No one in Jonesville who likes bananas also likes figs. No one in Jonesville who likes cranberries also likes dates.

Smith lives in Jonesville. True or false: Smith likes figs.

Statement 1: Smith doesn't like bananas.

Statement 2: Smith doesn't like cranberries.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Let represent the sets of people who like apples, bananas, etc. , respectively. Let represent Smith.

Assume Statement 1 alone.

If , then . Contrapositively, if , then . Therefore, cannot be in both and ; is in one, the other, or neither. By similar reasoning, is in at most one of and and at most one of and .

Suppose and . Then is in three of , and either falls in both and or both and ; since both are impossible, it follows that or , but not both. Therefore, Smith likes exactly one of bananas and figs, but not both. Since, from Statement 1, Smith doesn't like bananas, he must like figs.

Assume Statement 2. By similar reasoning, Smith likes cranberries or dates, but not both. Since he does not like cranberries, he likes dates. However, without further information, it cannot be determined what else he likes or dislikes.

### Example Question #11 : Data Sufficiency Questions

Every Beep is a Deep. Every Deep is a Meep. No Veep is a Deep.

Is Harpy a Veep?

Statement 1: Harpy is a Meep.

Statement 2: Harpy is a Beep.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. Let represent the sets of Beeps, Deeps, Meeps, and Veeps. Then and, since all Veeps are not Deeps, . This is represented by the Venn diagram below:

Note that the set (Meeps) need not fall completely inside the set (Veeps). Therefore, Harpy being a Meep does not necessarily make him a Veep or not a Veep.

Assume Statement 2 alone. Every Beep is a Deep, but no Veep is a Deep. If Harpy is a Beep, then he is a Deep. If Harpy were a Veep, then he cannot be a Deep, so, by contradiction, Harpy cannot be a Veep.

### Example Question #20 : Word Problems

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

Let be the set of people who like Neil Young, , the set of people who like Prince, and , the set of people who like Marvin Gaye.

True or false: Phillip does not like Neil Young, Prince, or Marvin Gaye.

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. is the intersection of the complements of all three of , , and - the sets of people who do not like Neil Young, people who do not like Prince, and people who do not like Marvin Gaye. Any person who is in this intersection does not like any of these three. , so Phillip does not like Neil Young, Prince, or Marvin Gaye.

Assume Statement 2 alone. is the union of all three , , and . Any person who is in this union likes any one, two, or three of these musicians. However, , which is the complement of this union - the set of people who like *none* of Neil Young, Prince, or Marvin Gaye. Phil is in this set.

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