Most people have much more difficulty with the Data Sufficiency problems than with the Standard 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 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 test.

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:

In triangle ABC to the right, what is the value of y? (1) AB = AC (2) x = 30

Explanation: By statement (1), triangle ABC is isosceles. Hence, its base angles are equal: y = z. Since the angle sum of a triangle is 180 degrees, we get x + y + z = 180. Replacing z with y in this equation and then simplifying yields x + 2y = 180. Since statement (1) does not give a value for x, we cannot determine the value of y from statement (1) alone. By statement (2), x = 30. Hence, x + y + z = 180 becomes 30 + y + z = 180, or y + z = 150. Since statement (2) does not give a value for z, we cannot determine the value of y from statement (2) alone. However, using both statements in combination, we can find both x and z and therefore y. Hence, the answer is C.

Notice in the above example that the triangle appears to be a right triangle. However, that cannot be assumed: angle A may be 89 degrees or 91 degrees, we can't tell from the drawing. You must be very careful not to assume any more than what is explicitly given in a Data Sufficiency problem.

ELIMINATION

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: In the figure to the right, what is the area of the triangle? (1) (2) x = 90

Recall that a triangle is a right triangle if and only if the square of the longest side is equal to the sum of the squares of the shorter sides (Pythagorean Theorem). Hence, (1) implies that the triangle is a right triangle. So the area of the triangle is (6)(8)/2. Note, there is no need to calculate the area--we just need to know that the area can be calculated. Hence, the answer is either A or D.

Turning to (2), we see immediately that we have a right triangle. Hence, again the area can be calculated. The answer is D.

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.

Example 4: 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:

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, in the set. Hence, (2) is sufficient to determine the value of x. The answer is B.

Example 5: Is CAB a code word in language Q?

(1) ABC is the base word.
(2) If C immediately follows B, then C can be moved to the front of the code word to generate another word.

From (1), we cannot determine whether CAB is a code word since (1) gives no rule for generating another word from the base word. This eliminates A and D.

Turning to (2), we still cannot determine whether CAB is a code word since now we have no word to apply this rule to. This eliminates B.

However, if we consider (1) and (2) together, then we can determine whether CAB is a code word:

From (1), ABC is a code word.

From (2), the C in the code word ABC can be moved to the front of the word: CAB.

Hence, CAB is a code word and the answer is C.

UNWARRANTED ASSUMPTIONS

Be extra careful not to read any more into a statement than what is given.

• The main purpose of some difficult problems is to lure you into making an unwarranted assumption.

If you avoid the temptation, these problems can become routine.

Example 6: Did Incumbent I get over 50% of the vote?

(1) Challenger C got 49% of the vote.
(2) Incumbent I got 25,000 of the 100,000 votes cast.

If you did not make any unwarranted assumptions, you probably did not find this to be a hard problem. What makes a problem difficult is not necessarily its underlying complexity; rather a problem is classified as difficult if many people miss it. A problem may be simple yet contain a psychological trap that causes people to answer it incorrectly.

The above problem is difficult because many people subconsciously assume that there are only two candidates. They then figure that since the challenger received 49% of the vote the incumbent received 51% of the vote. This would be a valid deduction if C were the only challenger (You might ask, "What if some people voted for none-of-the-above?" But don't get carried away with finding exceptions. The writers of the GMAT would not set a trap that subtle). But we cannot assume that. There may be two or more challengers. Hence, (1) is insufficient.

Now, consider (2) alone. Since Incumbent I received 25,000 of the 100,000 votes cast, I necessarily received 25% of the vote. Hence, the answer to the question is "No, the incumbent did not receive over 50% of the vote." Therefore, (2) is sufficient to answer the question. The answer is B.

Note, some people have trouble with (2) because they feel that the question asks for a "yes" answer. But on Data Sufficiency questions, a "no" answer is just as valid as a "yes" answer. What we're looking for is a definite answer.

CHECKING EXTREME CASES

• When drawing a geometric figure or checking a given one, be sure to include drawings of extreme cases as well as ordinary ones.

Example 1: In the figure to the right, AC is a chord and B is a point on the circle. What is the measure of angle x?

Although in the drawing AC looks to be a diameter, that cannot be assumed. All we know is that AC is a chord. Hence, numerous cases are possible, three of which are illustrated below:

In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in Case III, x is less than 45 degrees. Hence, the given information is not sufficient to answer the question.

Example 2: Three rays emanate from a common point and form three angles with measures p, q, and r. What is the measure of q + r ?

It is natural to make the drawing symmetric as follows:

In this case, p = q = r = 120, so q + r = 240. However, there are other drawings possible. For example:

In this case, q + r = 180. Hence, the given information is not sufficient to answer the question.

Problems:

1. Suppose 3p + 4q = 11. Then what is the value of q?

(1) p is prime.
(2) q = -2p

Solution: (1) is insufficient. For example, if p = 3 and q = 1/2, then 3p + 4q = 3(3) + 4(1/2) = 11. However, if p = 5 and q = -1, then 3p + 4q = 3(5) + 4(-1) = 11. Since the value of q is not unique, (1) is insufficient.

Turning to (2), we now have a system of two equations in two unknowns. Hence, the system can be solved to determine the value of q. Thus, (2) is sufficient, and the answer is B.

2. What is the perimeter of triangle ABC above?

(1) The ratio of DE to BF is 1: 3.
(2) D and E are midpoints of sides AB and CB, respectively.

Solution: Since we do not even know whether BF is an altitude, nothing can be determined from (1). More importantly, there is no information telling us the absolute size of the triangle.

As to (2), although from geometry we know that DE = AC/2, this relationship holds for any size triangle. Hence, (2) is also insufficient.

Together, (1) and (2) are also insufficient since we still don't have information about the size of the triangle, so we can't determine the perimeter. The answer is E.

3. A dress was initially listed at a price that would have given the store a profit of 20 percent of the wholesale cost. What was the wholesale cost of the dress?

(1) After reducing the asking price by 10 percent, the dress sold for a net profit of 10 dollars.
(2) The dress sold for 50 dollars.

Solution: Consider just the question setup. Since the store would have made a profit of 20 percent on the wholesale cost, the original price P of the dress was 120 percent of the cost: P = 1.2C. Now, translating (1) into an equation yields:

P - .1P = C + 10

Simplifying gives

.9P = C + 10

Solving for P yields

P = (C + 10)/.9

Plugging this expression for P into P = 1.2C gives

(C + 10)/.9 = 1.2C

Since we now have only one equation involving the cost, we can determine the cost by solving for C. Hence, the answer is A or D.

(2) is insufficient since it does not relate the selling price to any other information. Note, the phrase "initially listed" implies that there was more than one asking price. If it wasn't for that phrase, (2) would be sufficient. The answer is A.

4. What is the value of the two-digit number x?

(1) The sum of its digits is 4.
(2) The difference of its digits is 4.

Solution: Considering (1) only, x must be 13, 22, 31, or 40. Hence, (1) is not sufficient to determine the value of x.

Considering (2) only, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 59. Hence, (2) is not sufficient to determine the value of x.

Considering (1) and (2) together, we see that 40 and only 40 is common to the two sets of choices for x. Hence, x must be 40. Thus, together (1) and (2) are sufficient to uniquely determine the value of x. The answer is C.

5. If x and y do not equal 0, is x/y an integer?

(1) x is prime.
(2) y is even.

Solution: (1) is not sufficient since we don't know the value of y. Similarly, (2) is not sufficient. Furthermore, (1) and (2) together are still insufficient since there is an even prime number--2. For example, let x be the prime number 2, and let y be the even number 2 (don't forget that different variables can stand for the same number). Then x/y = 2/2 = 1, which is an integer. For all other values of x and y, x/y is not an integer. (Plug in a few values to verify this.) The answer is E.

6. Is 500 the average (arithmetic mean) score on the GMAT?

(1) Half of the people who take the GMAT score above 500 and half of the people score below 500.
(2) The highest GMAT score is 800 and the lowest score is 200.

Solution: Many students mistakenly think that (1) implies the average is 500. Suppose just 2 people take the test and one scores 700 (above 500) and the other scores 400 (below 500). Clearly, the average score for the two test-takers is not 500. (2) is less tempting. Knowing the highest and lowest scores tells us nothing about the other scores. Finally, (1) and (2) together do not determine the average since together they still don't tell us the distribution of most of the scores. The answer is E.

7. The set S of numbers has the following properties:

I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.

Is 3 in S?

(1) 1/3 is in S.
(2) 1 is in S.

Solution: Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3) = 3 is in S. Hence, (1) is sufficient.

Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2 (Note, nothing in Property II prevents x and y from standing for the same number. In this case both stand for 1.) is in S. Applying Property II again shows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.

8. What is the area of the triangle above?

(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.

Solution: From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3x = 6x = 180

Dividing the last equation by 6 gives: x = 30

Hence, a = 30, b = 60, and c = 90. However, different size triangles can have these angle measures, as the diagram below illustrates:

Hence, (1) is not sufficient to determine the area of the triangle.

Turning to (2), be careful not to assume that c is a right angle. Although from the diagram c appears to be a right angle, it could be 91 degrees or 89 degrees--we can't tell. Hence, (2) is not sufficient to determine the area of the triangle.

However, with both (1) and (2), c is a right angle and the area of the triangle is (1/2)(base)(height) = (1/2)(4)(3). The answer is C.

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