GMAT Math : Arithmetic

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Example Questions

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.

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.

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 #1232 : Data Sufficiency Questions

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

Possible Answers:

0.101001000100001...

3.1415926

0.22222222...

0.123451234512345...

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:

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.

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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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Example Question #11 : Dsq: Understanding Real Numbers

Evaluate AB + CD .

Statement 1: is the multiplicative inverse of .

Statement 2: is the multiplicative 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

The multiplicative inverse of a number is the number that can be multiplied by that number to yield product 1.

Assume Statement 1 alone. is the multiplicative inverse of , so AB = 1 , and

AB + CD = 1+ CD .

However, we know nothing about or , so we cannot evaluate the expression. For similar reasons, Statement 2 alone is also insufficient.

Now assume both statements to be true. By Statement 1, AB = 1 , and by Statement 2, CD = 1 , so

AB + CD = 1+ 1= 2 .

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Example Question #252 : Arithmetic

is a real number, with $N^{2}$ a positive integer. True or false: is a rational number.

Statement 1: $N^{2}$ is a multiple of 9.

Statement 2: $N^{2}$ is a multiple of 12.

Possible Answers:

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.x

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.

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:

The two statements together provide insufficient information to answer the question. For example, 36 and 72 are both multiples of both 9 and 12. However, the positive square root of 36 is 6, making an integer and, consequently, rational. However, the positive square root of 72 is not an integer, since 72 falls between consecutive perfect squares 64 and 81 (the squares of 8 and 9, respectively), and any square root of an integer that is not itself an integer must be irrational.

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Example Question #11 : Dsq: Understanding Real Numbers

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

Statement 1: $C \cdot D + A \cdot B = 66$

Statement 2: $B \cdot A + D \cdot C= 66$

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.

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.

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. By the commutative property of addition,

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

By substitution,

$A \cdot B + C \cdot D = 66$ .

Assume Statement 2 alone. By the commutative property of multiplication,

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

By two substitutions,

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

$A \cdot B + C \cdot D = 66$

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Example Question #254 : Arithmetic

Evaluate AB+C .

Statement 1: is the multiplicative inverse of .

Statement 2: is the additive inverse of .

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.

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.

Correct answer:

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

Explanation:

Assume both statements to be true. The multiplicative inverse of a number is the number that can be added to that number to yield product 1; the additive inverse of a number is the number that can be added to that number to yield sum 0. Therefore,

AB = 1 , or, equivalently, $A = \frac{1}{B}$ , and

B + C = 0 , or, equivalently, C = -B .

The expression can be restated in terms of :

$AB+C = \frac{1}{B} \cdot B + (-B) = 1 - B$ .

However, we do not know , and we have no way of finding it out, so we cannot evaluate the expression. Therefore, the question cannot be answered.

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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Understanding Real Numbers

Example Question #1232 : Data Sufficiency Questions

Example Question #7 : Real Numbers

Example Question #8 : Real Numbers

Example Question #9 : Real Numbers

Example Question #10 : Real Numbers

Example Question #11 : Dsq: Understanding Real Numbers

Example Question #252 : Arithmetic

Example Question #11 : Dsq: Understanding Real Numbers

Example Question #254 : Arithmetic

All GMAT Math Resources

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