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Data Sufficiency
Option #1:
Print out Probability section to use offline (9 pages)
Option #2
View the Data Sufficiency online. Browse through the sections of this chapter >>This section is broken into 4 parts:
I. Introduction
II. Strategies for Solving Data Sufficiency Questions
III. Data Sufficiency Trick Questions
IV. More practice questions
I. Introduction
In this chapter, we will review strategies for the Data Sufficiency questions and go over trick questions test designers write to fool you on these questions.
The Data Sufficiency questions (typically 1/3 of all the math questions) do not require the test taker to find a solution. Instead, the Data Sufficiency questions require the test takers only to find if each of the statements provides enough information for solving the question.
Data Sufficiency question instructions look like this:
Directions: In each of the problems, a question is followed by two statements containing certain data. You are to determine whether the data provided by the statements are sufficient to answer the question. Choose the correct answer based up on the statement's data, your knowledge of mathematics, and your familiarity with everyday facts (such as number of minutes in an hour or cents in a dollar). (international students: 100 cents to the dollar).
Choose choice
A) if statement (1) by itself is sufficient to answer the question, but statement (2) by itself is not;
B) if statement (2) by itself is sufficient to answer the question, but statement (1) by itself is not;
C) if statements (1) and (2) taken together are sufficient to answer the question, even though neither
statement by itself is sufficient;
D) If either statement by itself is sufficient to answer the question;
E) If statements (1) and (2) taken together are not sufficient to answer the question, requiring more data pertaining to the problem.Note: Diagrams accompanying problems agree with information given in the questions, but may not agree with additional information given in statements (1) and (2).
All numbers used are real numbers.
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 look at an example to clarify this.(stem) What is the sum of a + b?
(statement) (1) A = 5
(statement) (2) B = 10Explanation: 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
II. Strategies for Solving Data Sufficiency Questions
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.Some students confuse C and D. Read each answer choice carefully.
1. Note that A requires that B not be sufficient, and vice versa with B.
2. C stipulates that A and B cannot be able to answer the question alone. This means that although A and B together may be able to answer, the answer is not C if either one can answer the question alone.
3. The answer is D if both can answer the question independently, even if both can answer the question together.
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.
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?
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.
3. Data Sufficiency process of elimination strategies
In Step 2, as you progress through each statement, you may eliminate questions. Just solve for one of the statements and you are halfway done.Statement 1 is insufficient: automatically eliminate choices A and D, which require (1) to be sufficient.
Statement 1 is sufficient: automatically eliminate choices B, C and E, which require (1) to be insufficient.
Statement 2 is insufficient: automatically eliminate choices B and D, which require (1) to be sufficient.
Statement 2 is sufficient: automatically eliminate choices A, C and E, which require (1) to be insufficient.
4. 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.
III. Data Sufficiency Trick Questions
The tricks used in Data Sufficiency questions come from a narrow pool of tricks. Learn to identify each of the tricks, and you will be in a strong position to answer each question. In approaching each question, apply the three step method we discussed earlier:
Step (1) Look at the question stem.
Step (2) Look at each statement individually.
Step (3) Then look at both statements in combination.
It is important that you have the discipline to stick to this approach. The Data Sufficiency questions tend to be trick questions, particularly the difficult ones, and straying from this basic strategy will increase the chances of you being fooled.Remember that standardized tests are based on the premise that you can separate students into groups of ability. In order to do this, the less capable students must get questions wrong. To make sure less capable students get low scores, the tests are deliberately designed with trick questions specifically made to fool you.
Selected Trick Questions
AMNESIA TRICK
How many adults ride bicycles in city A if all adults in City A either ride bicycles or drive cars?
(1) 85% of the 10,000 adults in city A drive cars.
(2) 8500 adults in city A drive cars.
(A) Statement (1) is sufficient since if 8,500 drive cars then 1,500 ride bicycles. Statement (2) is not sufficient since we do not know the total population; it cannot be assumed from (1).
The trick here is that 1 alone can answer the question. Although 1 and 2 together may answer the question, the answer is still A. The unskilled reader will carry over the information from statement 1 when reading statement 2 and not catch the flaw with statement 2 (it does not tell you the population). Trick #2: note that the question doesn't tell you the total population of City A, but the total population is not relevant since the question only asks for "Adults".
This question shows how you must have discipline and stick to the 3 step process.
(1) Read the stem
(2) Read each statement individually
(3) If both statements cannot answer the question alone, then look at both statements together. Before you try to combine statement 1 and 2, make sure each answer can/cannot answer the question. When you first read statement 2, temporarily forget what you read in statement 1 so that you may evaluate if (2) alone is sufficient. Hence the name of the trick question: "Amnesia." Get temporary amnesia after reading statement 1 and don't use statement 1's information when you first evaluate statement 2 (because you need to see if statement 2 is sufficient alone.
DELAY TRICK
How much was a certain Babe Ruth baseball card worth in January 1991?
(1) In January 1997 the card was worth $100,000.
(2) Over the ten years 1987-1997, the card steadily increased in value by 10% each 12 months.
(1) alone is obviously insufficient. To use (2) you need to know what the card was worth at some time between 1987 and 1997. So (2) alone is insufficient, but by using (1) and (2) together you can figure out the worth of the baseball card in January 1991. The trick here is not to do the calculations. If you tried to actually calculate the value in January 1991, you have fallen into the trap. All that matters is that sufficient information is available.
The test designers make these questions to make you waste time so that you do not finish the test on time. This is called the DELAY trick because it causes you to be delayed and lose valuable time if you do unnecessary calculations.
BACKSOLVE
Is the two-digit integer, with digits r (first) and m(second), a multiple of 7?
(1) r + m = 13
(2) r is divisible by 3
With statements 1 and 2 we may determine that the two digit number is not a multiple of 7. Using statement (1) Try all the two digit numbers that sum 13: 94, 85, 76, 67, 58, 49. Of those, only 49 is divisible by 7. So, using statement 1, rm may or may not be a multiple of 7; it is insufficient. (2) Is not sufficient because there are many numbers with r that are divisible by 3 and that are multiples of 7 (35, 63, 98). Combined, there are NO possible numbers rm that are divisible by 7 and satisfy statements 1 and 2. The answer is NO, rm is not a multiple of 7. Using statements 1 and 2 we may deduce this.
Using statement 2, however, 49 is not a multiple of 3. So, combining the two statements proves that rm is not a multiple of 7. In other words, we've used the two statements to deduce that rm is not a multiple of 7.
This looks like a very intimidating question. As a rule, when you encounter a highly intimidating question such as this one, you should plug in possible answers. This question defies an algebraic solution, so it must be solved through backsolving.
RED HERRING TRICK
Billy sells twice as many $20 tickets as Tim, and Tim sells three times as many $10 tickets as Billy. How many tickets did Billy sell? Tickets are either $10 or $20.
(1) Tim sold a total of 35 tickets
(2) Together Billy and Tim sold 70 tickets for $1000
1 is not sufficient. Let x = the number of $20 tickets sold by Tim and y = the number of $10 tickets sold by Billy. Then
Billy sold 2x ($20 tickets) + y ($10 tickets)
Tim sold x ($20 tickets) + 3y ($10 tickets)
(2) implies 70 = x + 2x + y + 3y and 1000 = 20(x + 2x) + 10 (y + 3y)- divide this equation by 20 to simplify. Subtract these two equations
70 = 3x + 4y
-50 = -3x - 2y
20 = 2y may be solved for x and y and subsequently y = 10 and x = 10, Billy sold 2(10) + 10 = 30 tickets.
The trick here is that 1 is completely unnecessary and a distraction. The information in 1 may help answer the question, but it is unnecessary; 2 can do it alone.
International students: A "red herring" is an American/English phrase for something that is a distraction to the issue. In this case, the first statement is a distraction.
SUPER STATEMENT TRICK
What is the average (arithmetic mean) of 3x and 12z?(1) x + 4z = 20
(2) x + z = 8
Yes, combining A and B will solve the question, but A can do it alone. The trap is C. Students will know that the two statements together can solve the question. SUPER STATEMENT questions involve questions where together both statements can solve a question, but carefully examined, one statement may solve it alone.
The given information asks for the average of 3x and 12z, which is (3x + 12z) / 2, or 3(x + 4z)/2. Statement 1 tells us the value of x + 4z, (x + 4z)/3. So you can solve the average formula directly without using the second statement. x + 4z = 20, so 3x + 12z = 60, meaning that the average = 30. You may use statement 2 to solve the problem, but statement 1 can do it itself (thus disqualifying choice C, which requires both 1 and 2 to be insufficient).
HINT: on difficult Data Sufficiency questions, the statements usually have more value than it appears at first glance.
IV. More practice questions
Example 1
Is x > 4?1) x squared = 9
2) x squared =25
Solution
(1) implies that x = +/- 3 (+/- means positive or negative). Both +3 and -3 are less than 4, so the answer is "NO" and (1) is sufficient, that is NO, x is not greater than 4. A "NO" answer is equally acceptable as a "YES" answer. It is only necessary that there is sufficient information to answer the question. (2) implies x = positive or negative 5. -5 is less than 4 and + 5 is greater than 4, so the question cannot be answered with the information given in (2). The correct response is A.
Example 2
What is x - y?
1) x + y = 8
2) x - 2y = 2
Solution
(1) is not sufficient since (x - y) is the quantity desired. Likewise, (2) is not sufficient. But (1) and (2) together provide us with 2 equations and two unknowns from which x - y can be determined. The correct response is C. (We may solve the problem by subtracting (2) from (1): 3y = 6, therefore y = 2 and x = 6, so that x - y = 6 - 2 = 4. This calculation is, however, unnecessary.)
Example 3
How old is Gloria?1) Gloria's age is four times Alex's age plus Becky's age.
2) Becky was Alex's age fifteen years ago.
Solution
(1) is obviously not sufficient as is (2). Can the question be answered with (1) and (2)? Let x be Gloria's age, y be Alex's age, and z be Becky's age. (1) states that x = 4y + z. (2) states that z - 15 = y. These two equations contain three unknowns; consequently, we cannot determine x. More information is needed and the correct response is E.
Example 4
A student group sold only donuts and GMAT books to raise funds. How many GMAT books were sold?
1) 30% of the 90 items sold were GMAT books.
2) 63 donuts were sold.
Solution
(1) is sufficient since 30% of 90 is 0.3 x 90 = 27. (2) is not sufficient since we do not know the total number of items sold. So the correct response is A. A note of caution: Never let information in (1) influence your decision regarding the information in (2). In this example we cannot assume that 90 items were sold when deciding if (2) provides sufficient information. This is the Amnesia trick that undisciplined test takers will always fall into. Remember to look at each statement individually before comparing the two.