**INTRODUCTION DATA SUFFICIENCY**

Most people have much more difficulty with the Data Sufficiency problems than with the Standard GMAT Math problems. However, the mathematical knowledge and skill required to solve Data Sufficiency problems is no greater than that required to solve standard math problems. What makes Data Sufficiency problems appear harder at first is the complicated directions. But once you become familiar with the directions, you’ll find these problems no harder than standard math problems. In fact, people usually become proficient more quickly on Data Sufficiency problems.

**THE DIRECTIONS**

The directions for GMAT Data Sufficiency questions are rather complicated. Before reading any further, take some time to learn the directions cold. Some of the wording in the directions below has been changed from the GMAT to make it clearer. You should never have to look at the instructions during the GMAT.

Directions: Each of the following Data Sufficiency problems contains a question followed by two statements, numbered (1) and (2). You need not solve the problem; rather you must decide whether the information given is sufficient to solve the problem.

The correct answer to a question is

**A** if statement (1) ALONE is sufficient to answer the question but statement (2) alone is not sufficient;

**B** if statement (2) ALONE is sufficient to answer the question but statement (1) alone is not sufficient;

**C** if the two statements TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient;

**D** if EACH statement ALONE is sufficient to answer the question;

**E** if the two statements TAKEN TOGETHER are still NOT sufficient to answer the question.

Numbers: Only real numbers are used. That is, there are no complex numbers.

Drawings: The drawings are drawn to scale according to the information given in the question, but may conflict with the information given in statements (1) and (2).

You can assume that a line that appears straight is straight and that angle measures cannot be zero.

You can assume that the relative positions of points, angles, and objects are as shown.

All drawings lie in a plane unless stated otherwise.

**Example:** If *x* is both the cube of an integer and between 2 and 200, what is the value of *x*?

(1) *x* is odd.

(2) *x* is the square of an integer.

Since *x* is both a cube and between 2 and 200, we are looking at the integers:

2^{3}, 3^{3}, 4^{3}, 5^{3}

which reduce to

8, 27, 64, 125

Since there are two odd integers in this set, (1) is not sufficient to uniquely determine the value of *x*. This eliminates choices A and D.

Next, there is only one perfect square, 64 = 8^{2}, in the set. Hence, (2) is sufficient to determine the value of *x*. The answer is B.

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**ELIMINATION**

GMAT Data Sufficiency questions provide fertile ground for elimination. In fact, it is rare that you won’t be able to eliminate some answer-choices. Remember, if you can eliminate at least one answer choice, the odds of gaining points by guessing are in your favor.

The following table summarizes how elimination functions with Data Sufficiency problems.

Statement |
Choices Eliminated |

(1) is sufficient | B, C, E |

(1) is not sufficient | A, D |

(2) is sufficient | A, C, E |

(2) is not sufficient | B, D |

(1) is not sufficient and (2) is not sufficient | A, B, D |

**Example 1:** What is the 1st term in sequence S?

(1) The 3rd term of S is 4.

(2) The 2nd term of S is three times the 1st, and the 3rd term is four times the 2nd.

(1) is no help in finding the first term of S. For example, the following sequences each have 4 as their third term, yet they have different first terms:

0, 2, 4

-4, 0, 4

This eliminates choices A and D. Now, even if we are unable to solve this problem, we have significantly increased our chances of guessing correctly–from 1 in 5 to 1 in 3.

Turning to (2), we completely ignore the information in (1). Although (2) contains a lot of information, it also is not sufficient. For example, the following sequences each satisfy (2), yet they have different first terms:

1, 3, 12

3, 9, 36

This eliminates B, and our chances of guessing correctly have increased to 1 in 2.

Next, we consider (1) and (2) together. From (1), we know “the 3rd term of S is 4.” From (2), we know “the 3rd term is four times the 2nd.” This is equivalent to saying the 2nd term is 1/4 the 3rd term: (1/4)4 = 1. Further, from (2), we know “the 2nd term is three times the 1st.” This is equivalent to saying the 1st term is 1/3 the 2nd term: (1/3)1 = 1/3. Hence, the first term of the sequence is fully determined: 1/3, 1, 4. The answer is C.

**Example 2:** What is the value of *x* – *y*?

(1) *x* + *y* = 3*y* – *x*

(2) *x* + *y* = *x*^{3} + *y*^{3}

Start with (1): *x* + *y* = 3*y* – *x*

Subtract 3*y* and add *x* to both sides of the equation: 2*x* – 2*y* = 0

Divide by 2: *x* – *y* = 0

Hence, (1) is sufficient to determine the value of *x* – *y*, and therefore the answer is either A or D.

Turning to (2), we suspect there is not enough information since there are no like terms that can be combined as in (1). So use substitution to look for a counterexample. Let *x* = *y* = 0. Then 0 + 0 = 0^{3} + 0^{3} and *x* – *y* = 0. However, if *x* = 1 and *y* = 0, then 1 + 0 = 1^{3} + 0^{3} and *x* – *y* = 1. This shows that there are different pairs of numbers that satisfy (2) yet yield different values for *x* – *y*. Hence, (2) is not sufficient to determine the value of *x* – *y*. The answer is A. (Note, for the information to be sufficient, it is not enough to find *a* value of *x* – *y*; there must be a *unique* value.)

**Example 3:** Is *p* < *q* ?

(1) *p*/3 < *q*/3

(2) –*p* + *x* > –*q* + *x*

Multiplying both sides of *p*/3 < *q*/3 by 3 yields *p* < *q*.

Hence, (1) is sufficient. As to (2), subtract *x* from both sides of –*p* + *x* > –*q* + *x*, which yields

–*p* > –*q*

Multiplying both sides of this inequality by -1, and recalling that multiplying both sides of an inequality by a negative number reverses the inequality, yields *p* < *q*.

Hence, (2) is also sufficient. The answer is D.