BMM10236 Chapter 2 - DS Strategies.pdf

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    How the stem helps

    The Data Sufficiency questions are broken into the stem (the top question and two statements). You answer the question by

    determining if the information in the two statements is sufficient to answer the question.

    Lets us look at an example to clarify this.

    (stem) What is the sum of a + b?

    (statement) (1) A = 5

    (statement) (2) B = 10

    Explanation: Statement (1) tells you that A is 5. This is not enough information to answer the question. Statement (2) alone

    is also not enough to answer the question. However, if you combine the two statements, knowing that A = 5 and B = 10, then

    you can determine the solution to the question

    Let us dwell into details of strategies for solving Data Sufficiency questions:

    Strategy 1. Memorize the Data Sufficiency answer choices

    The directions and answer choices for Data Sufficiency questions never change. Memorize them so that you have no

    problems on test day. There is no excuse for walking into test day without these five answer choices perfectly memorized!

    A. Statement (1) by itself is sufficient to answer the question, but statement (2) by itself is not.

    B. Statement (2) by itself is sufficient to answer the question, but statement (1) by itself is not.

    C. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is

    sufficient.

    D. Either statement by itself is sufficient to answer the question.E. Statements (1) and (2) taken together are not sufficient to answer the question, requiring more data pertaining to the

    problem.

    What does it mean that a statement is sufficient?

    Sufficient does not mean that a statement is right or true, just that you can use the statements to derive an answer. Many

    beginning students err and think a statement is not sufficient if it proves a statement false.

    Strategy 2. Methodically progress through the two statements

    It takes mental discipline to progress through the Data Sufficiency questions. The test writers deliberately build tricks to

    each question. There are two basic questions that you must ask yourself on every Data Sufficiency question:

    Step 1: Can you answer the question using the information from statement (1) only?

    Step 2: Can you answer the question using the information from statement (2) only?

    Step 3: If the answer to both of these questions is no, then you ask yourself a third question: can you answer the

    question if you combine the information from both statements?

    Data Sufficiency Strategies

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    Example:Does 3 + x = 1?

    (1) x is positive

    (2) x is an odd number

    Solution

    (1) alone is sufficient, because it proves that 3 + x cannot equal 1. 3 plus a positive number cannot equal 1. Thus, statement

    (1) is sufficient because it establishes that the statement is false. (2) Statement (2) is also sufficient, because it proves 3 +

    x cannot equal 1. 3 plus an odd number cannot equal 1. Therefore, it is sufficient. Since both statements are sufficient,the answer must be D.

    Strategy 3. Analyze questions in terms of sufficiency

    Do not think in terms of what is the exact value, is this true or false? Instead, review questions in terms of one question

    is there enough information to answer the question? Look at each statement and ask yourself if it provides enough

    information to arrive at a conclusion.

    Test-Taking Tactics - Having discussed the strategies, we will discuss some test taking tactics for DS section:

    1. Make sure you understand the directions.

    Make sure you know what is being asked. If you have never seen this type of question before, make sure you do the practice

    problems that follow later. At first, these questions may seem difficult, but once you have worked through several examples,

    you will start to feel comfortable with them.

    The answer choices are summarized below as you will see them on the CAT exam. They can be of the following two variants.

    Also, according to the latest trends, five options can be there in questions as stated in strategy 1.

    1. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

    2. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

    3. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

    4. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the

    statements.

    1. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

    2. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

    3. Each statement alone is sufficient to answer the question.

    4. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the

    statements.

    2. Dont waste time figuring out the exact answer.

    Always keep in mind that you are never asked to supply an answer for the problem; you are only asked to determine if there

    is sufficient data available to find the answer. Once you know whether or not it is possible to find the answer from the given

    information, you are done. If you waste time figuring out the exact answer, you may not be able to finish the entire section.

    The CAT testers use the data sufficiency questions to test your ability to reason quantitatively. This is opposed to theproblem solving questions which are supposed to test how well you manipulate numbers. As a result, if you find yourself

    doing a lot of numbers crunching in the data sufficiency section, you are most likely doing something wrong.

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    Example: How much were the profits of the XYZ company in 1988?

    (1) XYZ company had revenues of $112,234 in 1988. (2) Costs of XYZ company were $102,479 in 1988.

    The information given states that to find the profit you need to know both the revenues and the costs. So it is easy to see

    that both (1) and (2) are needed and that the profit could be determined using (1) and (2). So the answer is (C). Do not

    compute the profit. If you perform the subtraction needed to compute the profit, you are just wasting time that could be spent

    on other problems.

    3. Draw a picture whenever possible.

    Make a copy of any diagrams on your paper and mark them up. If a diagram is not supplied, draw one on your paper. Pictures

    can be especially helpful in any question that involves geometry.

    4. Dont make extra assumptions.

    You are only allowed to use the information given and facts that are always true (such as the number of hours in a day) to

    answer these questions. Do not make assumptions about things such as prices rising every year. If you are given a diagram

    dont assume two lines that appear to be perpendicular are perpendicular unless you are given specific information that says

    the lines are perpendicular. If an angle looks like a 45 angle dont assume it is 45 unless you are given that fact.

    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 1: 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 thenfigure 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 dont get

    carried away with finding exceptions. The writers of the CAT 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.

    Please note that 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 were looking for is a definite answer.

    Example:

    In triangle ABC to the right, what is the value of y?

    (1) AB = AC(2) x = 30

    x z

    y

    B

    CA

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    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 maybe 89 degrees or 91 degrees, we cant tell from the drawing. You must be very careful not to assume any more than what is

    explicitly given in a Data Sufficiency problem.

    5. Use a system to work through the questions.

    Try to adopt a consistent approach to these types of problems. The system that follows will help you to answer the

    questions and also let you guess intelligently, if you cant complete the problem. You will have to invest some time to

    understand the method, but once you have done so, you should be much better prepared for these types of questions.

    Systematic Method for Data Sufficiency Questions

    A systematic analysis can improve your score on Data Sufficiency sections. By answering three questions, you will always

    arrive at the correct choice. In addition, if you can answer any one of the three questions, you can eliminate at least one of

    the possible choices so that you can make an intelligent guess. The three questions are:

    I. Is the first statement alone sufficient to solve the problem?II. Is the second statement alone sufficient to solve the problem?

    III. Are both statements together sufficient to solve the problem?

    As a general rule try to answer the questions in order I, II, III, since in many cases you will not have to answer all three to get

    the correct choice.

    Here is how to use the three questions:

    If the answer to I is YES, then the only possible choices are A or D. Now, if the answer to II is YES, the choice must be B, and

    if the answer to II is NO, the choice must be A. If the answer to I is NO then the only possible choices are B, C, or E. Now, if

    the answer to II is YES, then the choice must be B, and if the answer to II is NO, the only possible choices are C or E. So,

    finally, if the answer to III is YES, the choice is C, and if the answer to III is NO, the choice is E. A good way to see this is to

    use a decision tree.

    Is first statement sufficient? (I)

    YES NO

    Is second statement sufficient?(II)

    Choice is (B), (C)or (E)Choice is (A)or (D)

    YES

    Choice is (D).

    NO

    Choice is (A).

    YES

    Choice is (B).

    NO

    Choice is (C)or (E).

    YES

    Choice is (C).

    NO

    Choice is (E).

    Are both statements together

    sufficient? (III)

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    To use the tree simply start at the top and by answering YES or NO move down the tree until you arrive at the correct choice.

    For example, if the answer to I is YES and the answer to II is NO, then the correct choice is A. (Notice that in this case you

    dont need to answer III to find the correct choice.) The decision tree can also help you make intelligent guesses. If you can

    only answer one of the three questions, then you can eliminate the choices that follow from the wrong answer to the

    question.

    Practice this system to improve your ability to solve as well as make educated guesses for Data Sufficiency problems.

    ELIMINATION

    Data Sufficiency questions provide fertile ground for elimination. In fact, it is rare that you wont 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 favour.

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

    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) First statement 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 correctlyfrom 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.

    6. In many cases you can use simple values to check quickly whether a statement follows from a given statement.

    This can be especially useful in deciding that a statement does not follow from a given statement.

    Ex. Is k a multiple of 6?

    (1) k is a multiple of 3. (2) k is a multiple of 12.

    Write out some simple multiples of 3 (3 1 = 3, 3 2 = 6, etc.). Since 3 is not a multiple of 6, k is a multiple of 6 does not follow

    from k is a multiple of 3. So statement (1) is not sufficient, and the only possible choices are B, C, or E. Write some multiples

    of 12 (for example, 12, 24, 36, 48, . . .). All these are multiples of 6, since 12 is 2 6. So statement (2) is sufficient, and the correct

    answer is B.

    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

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    7. Remember if there is sufficient information to show that the answer to the question is NO, that means that there is sufficient

    information to answer the question

    Example: Is n an even integer?

    (1) n = 3k, for some integerk (2) n = 2j + 1, for some integer j.

    The first statement is not sufficient, since 3 2 is 6, which is even, but 3 3 is 9, which is odd. The second statement issufficient, since it means that n is odd. This means that the answer to the main question is no, and therefore B is the correct

    choice.

    Long-Term Strategy for Data Sufficiency Questions : Practice, Practice just go on Practicing Working Data Sufficiency

    Questions.

    Most students have not had much experience with these types of questions. The more examples you work out the better you

    will perform on this section of the test. By the time you have finished the Practice Exercises, you should feel confident about

    your ability to answer Data Sufficiency questions.

    A Quick Recap On the Test taking Tactics

    (1) Study the questions carefully. The questions generally ask one of 3 things: 1) a specific value, 2) a range of numbers, or 3)

    a true/false value. Make sure you know what the question is asking.

    (2) Determine what information is needed to solve the problem. This will obviously vary depending on what type of question

    is asked. To determine the area of a circle, you will need to know either the circles diameter, radius, or circumference.

    (3) Look at each of the two statements independently of each other. Follow the process of elimination rules covered above when

    considering each statement individually.

    (4) If step 3 did not produce an answer, then combine the two statements. If the two statements combined can answer the

    question, then the answer choice is C. Otherwise, E.