GMAT Math : Real Numbers

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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:

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

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

EACH statement ALONE is sufficient.

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

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

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Example Question #1 : 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:

D. EACH statement ALONE is sufficient.

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

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Example Question #1 : 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:

Each statement alone is sufficient

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

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

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.

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Example Question #4 : 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) + 4.95

5(3.00)+5.00

5(3.00-0.01) + 4.90

5(3.00-.05)+4.95

5(3.00) + 4.90

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.

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Example Question #5 : 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 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient to answer the question.

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

EITHER Statement 1 or Statement 2 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.

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Example Question #6 : 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.

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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:

Either statement is 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.

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

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

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

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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.

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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.

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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 #1 : Dsq: Understanding Real Numbers

Example Question #1 : Dsq: Understanding Real Numbers

Example Question #4 : Real Numbers

Example Question #5 : Real Numbers

Example Question #6 : Real Numbers

Example Question #7 : Real Numbers

Example Question #8 : Real Numbers

Example Question #9 : Real Numbers

Example Question #10 : Real Numbers

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