GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #19 : Dsq: Understanding Real Numbers

is a real number. True or false: is a rational number.

Statement 1: A square whose side has length has area $\frac{1}{4}$ .

Statement 2: $N \div 6$ is a rational number.

Possible Answers:

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 do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides 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.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

From Statement 1 alone, since a side of a square with area $\frac{1}{4}$ has as its sidelength

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$ ,

$N= \frac{1}{2} = 1 \div 2$ .

This is the quotient of two integers; by definition, this is rational.

From Statement 2 alone, if we let $M = N \div 6$ , then $N = 6 \cdot M$ . is rational, and so is 6, so their product, , is also rational.

Report an Error

Example Question #11 : Real Numbers

Evaluate $A \cdot (B + C)$ .

Statement 1: A = 0

Statement 2: B = 0

Possible Answers:

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 do NOT provide sufficient information to answer the question.

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.

Correct answer:

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

Explanation:

Assume Statement 1 alone.

$A \cdot (B + C) = 0 \cdot (B + C) = 0$ , since 0 multiplied by any number yields a product of 0.

Assume Statement 2 alone.

$A \cdot (B + C) = A \cdot (0 + C) = A \cdot C$ , since 0 added to any number yields the sum 0. However, without knowing and , or their product, it is impossible to determine the result.

Report an Error

Example Question #21 : Real Numbers

Evaluate AB + AC .

Statement 1: is the multiplicative inverse of .

Statement 2: is the additive inverse of .

Possible Answers:

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 do 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.

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

Explanation:

Assume Statement 1 alone. The product of a number and its multiplicative inverse is 1, so AB = 1 , and AB + AC = 1+ AC . But without knowing anything else, the expression cannot be evaluated.

Assume Statement 2 alone. The sum of a number and its additive inverse is 0, so B + C = 0 . By the distributive property,

$AB + AC = A (B + C ) = A \cdot 0 = 0$ .

Report an Error

Example Question #22 : Real Numbers

is a real number. True or false: is a rational number.

Statement 1: $N^{2} - 1$ is a rational number.

Statement 2: N + 1 is a rational number.

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.

EITHER STATEMENT ALONE provides 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.

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:

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

Explanation:

Statement 1 provides insufficient information to determine whether is rational or not.

If N = 2 , which, being an integer, is rational, then $N^{2} - 1 = 2^{2} - 1 = 4 - 1 = 3$ , which, being an integer, is rational.

If $N = \sqrt{2}$ . which is irrational, then $N^{2} - 1 =\left ( \sqrt{2} \right )^{2} - 1 = 2 - 1 = 1$ , which, being an integer, is rational.

Statement 2 is, however, sufficient. If N + 1 = M for some rational number , then N = M - 1 . 1, being an integer, is rational, and as the difference of rational numbers, is rational.

Report an Error

Example Question #23 : Real Numbers

is a real number. True or false: is a rational number.

Statement 1: $N + \frac{6}{11}$ is an irrational number.

Statement 2: $\frac{5}{7}N$ is an irrational number.

Possible Answers:

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

EITHER STATEMENT ALONE provides 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.

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 do NOT provide sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. $\frac{6}{11}$ , the quotient of integers, is rational. The sum of any two rational numbers is rational, so if is rational, then $N + \frac{6}{11}$ is a rational number. However, $N + \frac{6}{11}$ is irrational, so, contrapositively, is irrational.

Assume Statement 2 alone. $\frac{5}{7}$ , the quotient of integers, is rational. The product of any two rational numbers is rational, so if is rational, then $\frac{5}{7}N$ is a rational number. However, $\frac{5}{7}N$ is irrational, so, contrapositively, is irrational.

Report an Error

Example Question #24 : Real Numbers

Evaluate AB + CD .

Statement 1: is the additive inverse of .

Statement 2: is the additive inverse of .

Possible Answers:

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 do 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.

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

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

Explanation:

Assume both statements to be true. The additive inverse of a number is the number that can be added to that number to yield sum 0. We show that the value of AB + CD cannot be determined from these two statements by examining two cases:

Case 1: A = 1, B = -1, C = 2, D = -2

A + B = 1 + (-1) = 0 and C+ D = 2 + (-2) = 0 , so is the additive inverse of and is the additive inverse of .

AB + CD = 1(-1)+ 2 (-2) = -1 +(-4) = -5 .

Case 2: A = 1, B = -1, C = 3, D = -3

A + B = 1 + (-1) = 0 and C+ D = 3 + (-3) = 0 , so is the additive inverse of and is the additive inverse of .

AB + CD = 1(-1)+ 3(-3) = -1 +(-9) = -10

In both scenarios, the conditions of both statements are met, but AB + CD assumes different values. The two statements together are insufficient to answer the question.

Report an Error

Example Question #21 : Real Numbers

True or false: is an integer.

Statement 1: The multiplicative inverse of is not an integer.

Statement 2: The additive inverse of is an integer.

Possible Answers:

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.

EITHER STATEMENT ALONE provides 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.

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

Explanation:

Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of is not an integer, it is possible for to be an integer or to not be one, as is shown in these examples:

If N = 2 , which is an integer, then, since $2 \cdot \frac{1}{2} = 1$ , the multiplicative inverse of is $\frac{1}{2}$ , which is not an integer, If $N = \frac{2}{3}$ , which is not an integer, then, since $\frac{2}{3} \cdot \frac{3} {2} = 1$ , the multiplicative inverse of is $\frac{3} {2}$ , which is not an integer.

Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If is the additive inverse of the number , then

M+N = 0 , or

$N = -M = -1 \cdot M$

by Statement 2, is an integer; ., the product of integers, is itself an integer.

Report an Error

Example Question #26 : Real Numbers

Define an operation $\bigstar$ on the real numbers as follows:

If both and are whole numbers, then $a \bigstar b = a+b$ .

If and are not both whole numbers, then $a \bigstar b = 0$ .

Evaluate $M \bigstar N$ .

Statement 1: is not an integer.

Statement 2: N = 0 .

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 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 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:

Assume Statement 1 alone. Every whole number (0, 1, 2, 3,...) is an integer, so if is not an integer, it cannot be a whole number. Therefore, since and are not whole numbers, the second defintion of $\bigstar$ is used, and $M \bigstar N= 0$ .

Assume Statement 2 alone. N = 0 , which is a whole number, so it is not clear what definition of $\bigstar$ is used. If is not a whole number, the second defintion of $\bigstar$ is used, and $M \bigstar N= 0$ . If is also a whole number, then the first defintion is used:

$M \bigstar N = M +N = M + 0 = M$ .

However, we do not know the value of .

Overall, Statement 2 provides insufficient information to answer the question.

Report an Error

Example Question #27 : Real Numbers

Define an operation $\bigstar$ on the positive integers as follows:

If and are both prime integers, then $a \bigstar b = 1$ .

If and are not both prime integers, then $a \bigstar b = 2$ .

Evaluate $M \bigstar N$ .

Statement 1: M= 1 .

Statement 2: is a factor of .

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.

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 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:

Assume Statement 1 alone. A prime number is an integer with exaclty two factors - 1 and the number itself. 1 is considered to not be prime, since it has one factor. Therefore, if M = 1 , and are not both prime, and the second defintion of $a \bigstar b$ is used. $M \bigstar N = 1$ .

Assume Statement 2 alone.

Case 1: M = 1 .

In this case, Statement 1 is true, and as demonstrated before, $M \bigstar N = 1$ .

Case 2: M = 2, N = 2 .

In this case, is a factor of , since any integer is a factor of itself. Both integers are prime, so the first defintion of $a \bigstar b$ is used. $M \bigstar N = 1$ .

Report an Error

Example Question #28 : Real Numbers

Define an operation $\bigstar$ on the real numbers as follows:

If both and are both positive, then $a \bigstar b = a-b$ .

If both and are not both positive, then $a \bigstar b = a+b$ .

Evaluate $M \bigstar N$ .

Statement 1: M = -N .

Statement 2: M = 0 .

Possible Answers:

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.

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 1 ALONE is sufficient to answer the question, but Statement 2 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:

Assume Statement 1 alone. M = -N , so and are each other's opposites. Either one is positive and one is negative, or both are equal to 0; either way, the definition of $a \bigstar b$ for and not both positive is used, and

$M \bigstar N = M+ N = M + (-M) = 0$ .

Assume Statement 2 alone. Again, the definition of $a \bigstar b$ for and not both positive is used, and

$0 \bigstar N = 0+ N =N$ .

However, we do not know the value of .

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Los Angeles, GMAT Tutors in Boston, GMAT Tutors in New York City, Chemistry Tutors in New York City, GMAT Tutors in Atlanta, Computer Science Tutors in Los Angeles, Reading Tutors in San Diego, Statistics Tutors in Los Angeles, ISEE Tutors in Denver, ACT Tutors in Washington DC

Popular Courses & Classes

MCAT Courses & Classes in Denver, LSAT Courses & Classes in Seattle, MCAT Courses & Classes in Chicago, MCAT Courses & Classes in Los Angeles, GMAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Seattle, LSAT Courses & Classes in Houston, GRE Courses & Classes in Atlanta, ISEE Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Phoenix

Popular Test Prep

SAT Test Prep in Houston, ISEE Test Prep in Philadelphia, ACT Test Prep in Seattle, GRE Test Prep in Boston, GRE Test Prep in Atlanta, SAT Test Prep in Seattle, ACT Test Prep in Chicago, ACT Test Prep in Houston, GRE Test Prep in Philadelphia, LSAT Test Prep in Philadelphia

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #19 : Dsq: Understanding Real Numbers

Example Question #11 : Real Numbers

Example Question #21 : Real Numbers

Example Question #22 : Real Numbers

Example Question #23 : Real Numbers

Example Question #24 : Real Numbers

Example Question #21 : Real Numbers

Example Question #26 : Real Numbers

Example Question #27 : Real Numbers

Example Question #28 : Real Numbers

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details