GMAT Math : Real Numbers

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 #1 : Dsq: Understanding Real Numbers

Is z a negative number?

(1) $\dpi{100} \small 13z>14z$

(2) $\dpi{100} \small 4+z$ is positive

Possible Answers:

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

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

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

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

EACH statement ALONE is sufficient.

Correct answer:

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

Explanation:

(1) Subtracting $\dpi{100} \small 13z$ from both sides of $\dpi{100} \small 13z>14z$ shows that $\dpi{100} \small 0>z$ , so $\dpi{100} \small z$ must be negative $\dpi{100} \small \rightarrow$ SUFFICIENT

(2) Subtracting 4 from both sides of $\dpi{100} \small 4+z>0$ gives $\dpi{100} \small z>-4$ . Therefore, for this statement, $\dpi{100} \small z$ can be negative but is not necessarily negative. $\dpi{100} \small \rightarrow$ NOT SUFFICIENT.

Report an Error

Example Question #2 : Dsq: Understanding Real Numbers

If , , and are points on the number line, what is the distance between and ?

(1) The distance between and is 7.

(2) The distance between and is 12.

Possible Answers:

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

D. EACH statement ALONE is sufficient.

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

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

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

Correct answer:

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

Explanation:

Looking at statements (1) and (2) separately, there is no information about the distance between and . Looking at statements (1) and (2) together, there are two possibilities: 1) and are on different sides of , and the distance between them is 19; 2) and are on the same side of , and the distance between them is 5. Therefore, we cannot get the only possible answer to this question based on the two statements.

Report an Error

Example Question #3 : Dsq: Understanding Real Numbers

Data sufficiency question- do not actually solve the question

How many children are attending a party?

1. If 6 more children attend the party, there will be 24 children there.

2. If 10 children leave the party, there will be less than 12 children in attendance.

Possible Answers:

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

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

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

Each statement alone is sufficient

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

Correct answer:

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

Explanation:

Statement 1 precisely tells how many children are attending the party while Statement 2 only provides a range.

Report an Error

Example Question #4 : Dsq: Understanding Real Numbers

A sandwich shop offers a discount in which customers pay $2.99 for each additional sandwich after purchasing their first sandwich. Emily bought her first sandwich for $4.95. If she purchased 5 additional sandwiches to take home, then the total amount she paid is equivalent to which of the following?

Possible Answers:

5(3.00)+5.00

5(3.00-0.01) + 4.90

5(3.00) + 4.95

5(3.00) + 4.90

5(3.00-.05)+4.95

Correct answer:

5(3.00) + 4.90

Explanation:

The price of 6 sandwiches can be expressed as 5(2.99)+4.95 . Regrouping is needed since this is not in the answer choices. If $3.00 is used instead of $2.99, each of the five additional sandwiches is .01 too high and the total is .05 too high. Therefore, this amount needs to be subtracted from the $4.95.

Report an Error

Example Question #5 : Dsq: Understanding Real Numbers

Given that x > 0 and y > 0 , is x - y positive, negative, or zero?

1) 3x = 2y

2) x + y = 20

Possible Answers:

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

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

BOTH statements TOGETHER are 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 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.

Explanation:

If it is known that 3x = 2y , it can be determined that $x = \frac{2}{3}y$ .

Then $x - y = \frac{2}{3} y - y = -\frac{1}{3} y$

Since y > 0 , $x - y = -\frac{1}{3} y < 0$ , and the question is answered.

But if we only know that x + y = 20 , it is not sufficient.

Case 1:

x = 10, y = 10 yields x - y = 0

Case 2:

x = 11, y = 9 yields x - y = 2 > 0

The answer is that Statement 1 is sufficient, but not Statement 2.

Report an Error

Example Question #6 : Dsq: Understanding Real Numbers

Which of the following is the decimal representation of an irrational number? (Assume any observed patterns continue)

Possible Answers:

0.22222222...

3.1415926

0.123451234512345...

0.101001000100001...

0.333333333

Correct answer:

0.101001000100001...

Explanation:

A number is rational if and only if its decimal representation is terminating or repeating.

0.333333333 and 3.1415926 are terminating decimals, which are rational (note: the latter number is not equal to $\pi$ ). Both can be eliminated.

0.22222222... has a repeating digit and 0.123451234512345... has a repeating group of five digits, and are therefore repeating decimals, which are also rational. Both can be eliminated.

0.101001000100001... has an infinite pattern - the number of zeroes that precede each one is increasing by one with each group - but since the pattern is not repeating, it is neither a terminating decimal nor a repeating decimal. It is irrational.

Report an Error

Example Question #7 : Real Numbers

Jenkins wanted to find all the real solutions of the following expression. Find them for him.

f(x)=4x^2-6x+c

I) f(x) has a vertex at the point (6,7) .

II) f(x) is never undefined.

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Correct answer:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Explanation:

To find the real solutions, we need to know what c is. An easy way to do that would be if we had a point on the function.

I) gives us such a point. Therefore, we are able to substitute in our values and solve for c using algebraic operations.

f(x)=4x^2-6x+c

7=4(6)^2-6(6)+c

$7=144-36+c \rightarrow 7=108+c$

-101=c

Thus our equation becomes:

f(x)=4x^2-6x-101 .

From here we can try to factor to find the real solutions or you can use the quadratic formula.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Plugging in our values we get the solution x=5.83, -4.33 .

II) Is not really relevant, we are dealing with a parabola, so it shouldn't be undefined anyway.

Report an Error

Example Question #8 : Real Numbers

$\small a + 8 = b+12$

$\small c+ 5 = d + 11$

$\small a + 4 = d + 3$

Order $\small a,b, c,d$ from least to greatest.

Possible Answers:

$\small \small a,b,d,c$

$\small a,b ,c,d$

$\small b,a,d,c$

$\small \small b,a,c,d$

$\small a,c,b,d$

Correct answer:

$\small b,a,d,c$

Explanation:

We can find each in terms of .

$\small a + 8 = b+12$

b+12 -12 = a+8 -12

b = a-4

$\small a + 4 = d + 3$

d + 3 -3 = a + 4 -3

d = a+1

$\small \small c+ 5 = d + 11$

c+ 5 = a+1 + 11

c+ 5 = a+12

c+ 5 -5 = a+12 -5

c = a+7

In ascending order, the four values are:

b = a-4

d = a+1

c = a+7

The correct choice is $\small b,a,d,c$ .

Report an Error

Example Question #9 : Real Numbers

is a real number. True or false: is an integer.

Statement 1: $N^{3} - 4N^{2}+ 4N -16 = 0$

Statement 2: $N^{3} - 4N^{2}- 4N +16 = 0$

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.

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.

EITHER STATEMENT ALONE provides 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. The polynomial expression in

$N^{3} - 4N^{2}+ 4N -16 = 0$

can be factored as follows:

$N^{2} (N - 4)+ 4 (N - 4) = 0$

$\LARGE (N^{2} + 4) (N - 4) = 0$

Either $\normal { N - 4 = 0 }$ , in which case N = 4 , or

$N^{2}+ 4 = 0$ - which is impossible for any real value of , since

$N^{2} \ge 0$ , and

$N^{2}+ 4 \ge 4 > 0$

Therefore, is an integer.

Assume Statement 2 alone. Again, we factor:

$N^{3} - 4N^{2}- 4N +16 = 0$

$N^{2} (N - 4)- 4 (N - 4) = 0$

$(N^{2} - 4)\: (N - 4) = 0$

(N+2)(N-2)(N-4) = 0

Similarly to the polynomial in Statement 1, N = -2 , N = 2 , or N = 4 . All three possible values of are integers.

Report an Error

Example Question #10 : Real Numbers

Evaluate $A \cdot B + C \cdot D$ .

Statement 1: $A \cdot B + D \cdot C = 100$

Statement 2: $A \cdot B = C \cdot D$

Possible Answers:

EITHER 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.

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 Statmemt 1 alone. By the commutative property of multiplication:

$C \cdot D = D \cdot C$

By the addition property of equality,

$A \cdot B + C \cdot D = A \cdot B + D \cdot C$

By substitution,

$A \cdot B + C \cdot D = 100$

Statement 2 alone, however, does not answer the question. If $A \cdot B = C \cdot D$ , then by substitution,

$A \cdot B + C \cdot D = A \cdot B + A \cdot B = 2 \cdot A \cdot B$ ;

however, without further information, we cannot evaluate this expression.

Report an Error

All GMAT Math Resources

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

Popular Subjects

French Tutors in Miami, Math Tutors in Miami, Algebra Tutors in San Diego, Computer Science Tutors in Phoenix, Reading Tutors in New York City, Math Tutors in Seattle, LSAT Tutors in Washington DC, GMAT Tutors in Phoenix, Biology Tutors in Seattle, Math Tutors in New York City

Popular Courses & Classes

Spanish Courses & Classes in Miami, LSAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Seattle, MCAT Courses & Classes in San Diego, Spanish Courses & Classes in Phoenix, SAT Courses & Classes in New York City, SSAT Courses & Classes in Washington DC, LSAT Courses & Classes in New York City, ISEE Courses & Classes in Chicago, GRE Courses & Classes in Philadelphia

Popular Test Prep

GMAT Test Prep in Denver, ISEE Test Prep in San Francisco-Bay Area, ACT Test Prep in Denver, LSAT Test Prep in Los Angeles, GRE Test Prep in Philadelphia, GRE Test Prep in New York City, MCAT Test Prep in New York City, SAT Test Prep in New York City, SAT Test Prep in Atlanta, SSAT Test Prep in Boston

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 : Real Numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Understanding Real Numbers

Example Question #2 : Dsq: Understanding Real Numbers

Example Question #3 : Dsq: Understanding Real Numbers

Example Question #4 : Dsq: Understanding Real Numbers

Example Question #5 : Dsq: Understanding Real Numbers

Example Question #6 : Dsq: Understanding Real Numbers

Example Question #7 : Real Numbers

Example Question #8 : Real Numbers

Example Question #9 : Real Numbers

Example Question #10 : Real Numbers

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details