GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #18 : Sets

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:

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

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.

Correct answer:

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

Explanation:

Let \(\displaystyle A, B, C, D, F, G\) represent the sets of people who like apples, bananas, etc. , respectively. Let \(\displaystyle s\) represent Smith.

Assume Statement 1 alone. 

If \(\displaystyle s \in B\), then \(\displaystyle s \notin F\). Contrapositively, if  \(\displaystyle s \in F\), then \(\displaystyle s \notin B\). Therefore, \(\displaystyle s\) cannot be in both \(\displaystyle B\) and \(\displaystyle F\)\(\displaystyle s\) is in one, the other, or neither. By similar reasoning, \(\displaystyle s\) is in at most one of \(\displaystyle A\) and \(\displaystyle G\) and at most one of \(\displaystyle C\) and \(\displaystyle D\)

Suppose \(\displaystyle s \notin B\) and \(\displaystyle s \notin F\). Then \(\displaystyle s\) is in three of \(\displaystyle A, C, D, G\), and either \(\displaystyle s\) falls in both \(\displaystyle A\) and \(\displaystyle G\) or both \(\displaystyle C\) and \(\displaystyle D\); since both are impossible, it follows that \(\displaystyle s \in B\) or \(\displaystyle s \in F\), 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 #19 : Sets

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.

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

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

Assume Statement 1 alone. Let \(\displaystyle B,D,M,V\) represent the sets of Beeps, Deeps, Meeps, and Veeps. Then \(\displaystyle B \subseteq D\subseteq M\) and, since all Veeps are not Deeps, \(\displaystyle V \subseteq D'\). This is represented by the Venn diagram below:

Untitled

Note that the set \(\displaystyle M\) (Meeps) need not fall completely inside the set \(\displaystyle V\) (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 : Sets

Let universal set \(\displaystyle U\) be the set of all people. Let \(\displaystyle p\) represent Phillip. 

Let \(\displaystyle A\) be the set of people who like Neil Young, \(\displaystyle B\), the set of people who like Prince, and \(\displaystyle C\), the set of people who like Marvin Gaye.

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

Statement 1: \(\displaystyle p \in A' \cap B' \cap C'\)

Statement 2: \(\displaystyle p \in (A \cup B \cup C)'\)

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. \(\displaystyle A' \cap B' \cap C'\) is the intersection of the complements of all three of \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) - 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. \(\displaystyle p \in A' \cap B' \cap C'\), so Phillip does not like Neil Young, Prince, or Marvin Gaye.

Assume Statement 2 alone. \(\displaystyle A \cup B \cup C\) is the union of all three \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\). Any person who is in this union likes any one, two, or three of these musicians. However, \(\displaystyle p \in (A \cup B \cup C)'\), 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.

Example Question #1 : Work Problems

Clara spent \(\displaystyle \frac{1}{3}\) of her salary on rent and \(\displaystyle \frac{1}{5}\) of the rest on clothes. How much did she have left after paying the rent and buying the clothes?

(1) Her salary was $3000

(2) She spent $1400 on rent and clothes

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

D. EACH statement ALONE is sufficient.

Correct answer:

D. EACH statement ALONE is sufficient.

Explanation:

For statement (1), she spent

 \(\displaystyle \frac{1}{3}\cdot 3000=\$1000\)

on rent, and

\(\displaystyle \frac{1}{5}\cdot \left ( 3000-1000 \right )=\$400\) 

on clothes.

So she spent \(\displaystyle \$1000+\$400=\$1400\) in total.

Therefore, she has \(\displaystyle \$3000-\$1400=\$1600\) left.

For statement (2), we can set Clara’s salary to be \(\displaystyle x\), then we have

\(\displaystyle \frac{1}{3}x+\left ( 1-\frac{1}{3} \right )\cdot \frac{1}{5}x=\$1400\) 

Then simplify the equation:

\(\displaystyle \frac{1}{3}x+\frac{2}{15}x=\$1400\) 

Therefore, \(\displaystyle x=\$3000\).

Now we can just follow what we did for the first statement to calculate the money he had left.

Example Question #2 : Work Problems

Steve can paint his greenhouse in 4 hours 40 minutes minutes, working alone; his brother Phil can do the same job in 6 hours, working alone. If they work together, to the nearest minute, how long will it take them? 

Possible Answers:

2 hours 54 minutes

2 hours 42 miutes

3 hours 

2 hours 38 minutes

2 hours 22 minutes

Correct answer:

2 hours 38 minutes

Explanation:

Think of this in terms of "greenhouses per minute", not "minutes per greenhouse". Converting hours and minutes to just minutes, Steve can paint \(\displaystyle \frac{1}{280}\) greenhouses per minute; Phil can paint \(\displaystyle \frac{1}{360}\) greenhouses per minute. 

If we let \(\displaystyle t\) be the time in minutes that it takes to paint the greenhouse, then Steve and Phil paint \(\displaystyle \frac{1}{280} t\)  and \(\displaystyle \frac{1}{360} t\) greenhouses, respectively; since one greenhouse total is painted, then we can add the labor and set up this equation:

\(\displaystyle \frac{1}{280} t + \frac{1}{360} t = 1\)

Simplify and solve:

\(\displaystyle \left (\frac{1}{280} + \frac{1}{360} \right ) t = 1\)

\(\displaystyle \left (\frac{9}{280 \cdot 9} + \frac{7}{360\cdot 7} \right ) t = 1\)

\(\displaystyle \left (\frac{9}{2,520} + \frac{7}{2,520} \right ) t = 1\)

\(\displaystyle \frac{16}{2,520} t = 1\)

\(\displaystyle \frac{2}{315} t = 1\)

\(\displaystyle \frac{315}{2} \cdot \frac{2}{315} t = \frac{315}{2} \cdot 1\)

\(\displaystyle t = \frac{315}{2} = 315 \div 2 = 157 \frac{1}{2} \approx 158\)

or 2 hours, 38 minutes.

Example Question #3 : Dsq: Understanding Work Problems

A large water tower can be emptied by opening one or both of two drains of different sizes. On one occasion, both drains were opened at the same time. How long did it take to empty the water tower?

Statement 1: Alone, the larger drain can empty the tower in three hours.

Statement 2: The smaller drain can empty water at 75% of the rate at which the larger drain does.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

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

EITHER statement ALONE is sufficient 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

A work problem is actually a rate problem in disguise.

If you know that an object working alone can do a job in \(\displaystyle M\) hours, then you know that the object works at a rate of \(\displaystyle \frac{1}{M}\) jobs per hour. After \(\displaystyle t\) hours, the object accomplishes \(\displaystyle \frac{1}{M} t\) of a job. Similarly, the other object working alone does a job in \(\displaystyle N\) hours, and therefore does \(\displaystyle \frac{1}{N} t\) of a job. Together, the objects do one whole job, so solve this equation 

\(\displaystyle \frac{1}{M} t + \frac{1}{N}t = 1\)

for \(\displaystyle t\).

Statement 1 alone gives us half the picture; \(\displaystyle M = 3\), but \(\displaystyle N\) is unknown.

Statement 2 alone tells us that \(\displaystyle \frac{1}{N}\) is 75% of \(\displaystyle \frac{1}{M}\). But \(\displaystyle M\) is unknown.

From Statement 1 and 2 together, we know

\(\displaystyle \frac{1}{N}\) is 75% of \(\displaystyle \frac{1}{3}\) - this allows us to calculate \(\displaystyle N\):

75% of \(\displaystyle \frac{1}{3}\) is \(\displaystyle \frac{3}{4} \times \frac{1}{3} = \frac{1}{4}\), so \(\displaystyle N = 4\). Since we have both \(\displaystyle M\) and \(\displaystyle N\), we have the complete equation

\(\displaystyle \frac{1}{3} t + \frac{1}{4}t = 1\)

and we can calculate \(\displaystyle t\).

Example Question #3 : Work Problems

A large water tower can be emptied by opening one or both of two drains of different sizes. On one occasion, both drains were opened at the same time. How long did it take to empty the water tower?

Statement 1: Alone, the smaller drain can empty the tower in three hours.

Statement 2: Alone, the larger drain can empty the tower in two hours.

Possible Answers:

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

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. 

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:

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

Explanation:

A work problem is actually a rate problem in disguise.

If you know that an object working alone can do a job in \(\displaystyle M\) hours, then you know that the object works at a rate of \(\displaystyle \frac{1}{M}\) jobs per hour. After \(\displaystyle t\) hours, the object accomplishes \(\displaystyle \frac{1}{M} t\) of a job. Similarly, the other object working alone does a job in \(\displaystyle N\) hours, and therefore does \(\displaystyle \frac{1}{N} t\) of a job. Together, the objects do one whole job, so solve this equation 

\(\displaystyle \frac{1}{M} t + \frac{1}{N}t = 1\)

for \(\displaystyle t\).

Statement 1 alone tells us that \(\displaystyle M = 3\), and Statement 2 alone tells us that \(\displaystyle N = 2\).

Therefore, each statement alone gives us only half the picture, but together, they give us the equation

\(\displaystyle \frac{1}{3} t + \frac{1}{2}t = 1\),

which can be solved to yield the answer. 

Example Question #4 : Dsq: Understanding Work Problems

Three brothers - David, Eddie, and Floyd - mow a lawn together, starting at the same time. How long will it take them to finish?

Statement 1: Working alone, David can mow the lawn in four hours.

Statement 2: Working together, but without David, Eddie and Floyd can mow the lawn in three hours.

Possible Answers:

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.

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

Correct answer:

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

Explanation:

A work problem is actually a rate problem in disguise.

If you know that David working alone can do a job in \(\displaystyle M\) hours, then you know that the object works at a rate of \(\displaystyle \frac{1}{M}\) jobs per hour. After \(\displaystyle t\) hours, the object accomplishes \(\displaystyle \frac{1}{M} t\) of a job. Similarly, Eddie and Floyd, working without David, do a job in \(\displaystyle N\) hours, and therefore does \(\displaystyle \frac{1}{N} t\) of a job. Together, the objects do one whole job, so solve this equation 

\(\displaystyle \frac{1}{M} t + \frac{1}{N}t = 1\)

for \(\displaystyle t\).

Statement 1 alone tells us that \(\displaystyle M = 4\), and Statement 2 alone tells us that \(\displaystyle N = 3\); each one alone leaves the other value unknown. However, if both statements are given, the equation 

\(\displaystyle \frac{1}{4} t + \frac{1}{3}t = 1\)

can be solved for \(\displaystyle t\) to yield the correct answer.

Example Question #1 : Work Problems

Last week, Mrs. Smith, Mrs. Edwards, and Mrs. Hume were able to write \(\displaystyle 400\) invitations to a party in two hours. 

Today, Mrs. Smith is sick and cannot help, so Mrs. Edwards, and Mrs. Hume have to work without her. They must write \(\displaystyle 300\) more invitations to the same party. How long should they take, working together?

Statement 1: Working alone, Mrs. Smith can write \(\displaystyle 100\) invitations in one and one-half hours.

Statement 2: Working alone, Mrs. Hume can write \(\displaystyle 100\) invitations in one hour.

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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

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

Explanation:

Since the group is now without the help of Mrs. Smith, we look at Mrs. Smith's contribution to the work as a whole, and the sum of the other two ladies' contribution as a whole; Statement 2, which deals with Mrs. Hume alone, is irrelevant and unhelpful.

Since the three ladies together wrote 400 invitations in 120 minutes, we can infer that they would have spent three-fourths of this time, or 90 minutes, writing 300 invitations.

If Statement 1 is true, then Mrs. Smith, who can write 100 invitations in 90 minutes, would take three times this, or 270 minutes, to write 300 invitations. 

A work problem is a rate problem in disguise. Think in terms of "jobs per hour", and take the reciprocal of each "hours per job". The three ladies together would have done \(\displaystyle \frac{1}{90}\) job in one hour, and Mrs. Smith alone would have done \(\displaystyle \frac{1}{270}\) job in one hour.

Therefore, in one hour today, the two ladies will do

\(\displaystyle \frac{1}{90} - \frac{1}{270}\) 

jobs, and in \(\displaystyle t\) hours today, they will do

\(\displaystyle \left (\frac{1}{90} - \frac{1}{270} \right )t = 1\)

job.

Solve for \(\displaystyle t\) in this equation. 

This proves that Statement 1 alone allows us to find the answer - but not Statement 2.

Example Question #1 : Rate Problems

Data sufficiency question- do not actually solve the question

How many hours did it take to drive from city A to city B without stopping?

1. The drive started at 10 am.

2. The average speed during the trip is 65 miles/hour.

Possible Answers:

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question

Both statements taken together are sufficient to answer the question but neither statement alone is sufficient

Statement 1 is sufficient, but statement 2 is not sufficient to answer the question

Statement 2 is sufficient, but statement 1 is not sufficient to answer the question

Each statement alone is sufficient to answer the question

Correct answer:

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question

Explanation:

The total time is calculated by the equation \small Time=\frac{distance}{rate}\(\displaystyle \small Time=\frac{distance}{rate}\). Statement 2 provides the rate, but we have no information regarding distance, therefore, the quesiton is impossible to solve without more information.

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