ตัวอย่างข้อสอบ gmat
EXAM - GMAT TEST Data Sufficiency
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 |  |
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) |  |
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 <>
(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.
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