# GMAT Math : DSQ: Understanding real numbers

## Example Questions

### Example Question #11 : Real Numbers

Evaluate .

Statement 1:  is the multiplicative inverse of .

Statement 2:  is the multiplicative inverse of .

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 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.

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

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 , and

.

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, , and by Statement 2, , so

.

### Example Question #1241 : Data Sufficiency Questions

is a real number, with a positive integer. True or false:  is a rational number.

Statement 1:  is a multiple of 9.

Statement 2:  is a multiple of 12.

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.x

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.

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.

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.

### Example Question #1242 : Data Sufficiency Questions

Evaluate .

Statement 1:

Statement 2:

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

Explanation:

Assume Statement 1 alone. By the commutative property of addition,

.

By substitution,

.

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

and

By two substitutions,

### Example Question #1243 : Data Sufficiency Questions

Evaluate .

Statement 1:  is the multiplicative inverse of .

Statement 2:  is the additive inverse of .

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.

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.

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,

, or, equivalently, , and

, or, equivalently, .

The expression can be restated in terms of :

.

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.

### Example Question #1244 : Data Sufficiency Questions

Evaluate .

Statement 1:

Statement 2:  is the additive inverse of .

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.

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

Explanation:

Assume Statement 1 alone. Since , then by substitution,

.

But without further information, the expression cannot be evaluated.

Assume Statement 2 alone.  is the additive inverse of , so, by definition, . Applying the commutative, then associative, properties,

But without further information, the expression cannot be evaluated.

Now assume both statements to be true. From Statement 2 and Statement 1 combined,

.

### Example Question #1245 : Data Sufficiency Questions

Evaluate .

Statement 1:

Statement 2:

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.

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.

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

Explanation:

By the distributive property of multiplication over addition,

Assume Statement 1 alone. , so, by substitution,

.

But, without knowing , or their product, it is impossible to determine the value of the expression. Statement 2 provides insufficient information, for similar reasons.

Now assume both statements to be true. Then

Since  and , by substitution,

.

### Example Question #1246 : Data Sufficiency Questions

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

Statement 1:

Statement 2:

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.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 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.

Explanation:

Assume Statement 1 alone. The polynomial expression in

can be factored as follows:

Either , in which case , which, as an integer is also rational, or

, in which case:

or either  or , both irrational.

Therefore, Statement 1 provides insufficient information.

Assume Statement 2 alone. If , then , making  a fourth root of 4. A rational root of an integer must itself be an integer, and there is no integer which, when taken to the fourth power, is equal to 4. Therefore,  must be irrational.

### Example Question #1247 : Data Sufficiency Questions

Evaluate .

Statement 1:

Statement 2:

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.

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.

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

Explanation:

From Statement 1 alone, since 1 raised to the power of any real number is equal to 1, we know that

.

However, we cannot determine the value of knowing only that . If is a nonzero number, then . However, if , then

,

which is an undefined expression.

### Example Question #1248 : Data Sufficiency Questions

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

Statement 1: A square whose side has length  has area .

Statement 2:  is a rational number.

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.

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.

Explanation:

From Statement 1 alone, since a side of a square with area  has as its sidelength

,

.

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

From Statement 2 alone, if we let , then  is rational, and so is 6, so their product, , is also rational.

### Example Question #1249 : Data Sufficiency Questions

Evaluate .

Statement 1:

Statement 2:

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.

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 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.

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.

, since 0 multiplied by any number yields a product of 0.

Assume Statement 2 alone.

, since 0 added to any number yields the sum 0. However, without knowing  and , or their product, it is impossible to determine the result.

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