To begin understanding Data Sufficiency, it’s helpful to look at the official directions and answer choices for these problems, courtesy of the Graduate Management Admissions Council.
Directions
This Data Sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of
mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked;
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
You should see from the answer choices that the name of the game is “is the data sufficient?”, so let’s spend some time discussing what constitutes sufficient information.
More important than those solutions, however, are these takeaways:
• A statement is sufficient when it guarantees exactly one (and only one) answer to the question.
• This means that in a Yes/No question, you have sufficient information if the answer is “Definitely Yes” or if the answer is “Definitely No”. You do not have sufficient information when the answer is “Sometimes Yes but Sometimes No” (or “Maybe”).
• This means that in a “What is the Value?” question, you have sufficient information when you can pin down exactly one value for the question, but you do not have sufficient information when more than one value is possible.
• Data Sufficiency questions require attention to detail – the drills on the previous page came in pairs, and to the untrained eye each pair might have seemed the same. But subtle differences in what was given or asked – variable squared vs. variable cubed; cube vs. rectangular box; 2A + 3P vs. 3A + 2P – can make all the difference.
As tricky as Data Sufficiency can look at first, the way in which they are constructed ensures that you can attack them systematically. As you know, the answer choices are always the same. And you should also know that there are only two types of question stems that can ask:
1. Yes or No Questions
• A statement gives multiple solutions, but they all give the same answer.
• A statement provides a no answer instead of a yes answer.
2. What Is the Value? Questions
• A statement appears to be giving one value because you have assumed properties of the number that were not actually given (positive, integers, etc.).
• Restrictions were placed in the problem that you did not properly leverage (for instance, the problem is asking for the number of children, which must be an integer and cannot be negative).
Avoid assumptions. Every time you approach a Data Sufficiency problem, you must actively consider any assumptions that you may have been baited into making. Avoiding assumptions is perhaps the most important skill in all with Data Sufficiency.
1. Yes or No Question
Statement:Is x > 3?
Arguments:(1) The sum of x and the square of x is 12.
(2) square of x > 9
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution. A
Question Type: Yes/No. This question asks: “Is x > 3?”
Given information in question stem or diagram: No important information is given in the question stem.
Statement 1: The first step in this statement is to translate the wording into the following equation: x + x2 = 12. Since this is a quadratic equation, you should set everything equal to zero so that x2 + x – 12 = 0. Factoring this, you see that (x + 4)(x - 3) = 0 and x would be -4 or 3. The difficulty in this statement is that many people assume that this information is not sufficient because there are two values, one negative and one positive. However, remember that to prove sufficiency in a yes or no question requires only a definitive answer, not one value. Since each of these values (-4 and 3) gives a “no” answer to the question, this statement is sufficient. The answer is either A or D.
Statement 2: x2 > 9. If x2 > 9 then either x > 3, which gives you a “yes” answer, or x < 3, which gives you a “no” answer. For example x could be -5 (which when squared is > 9) or 5 (which when squared is also > 9). This statement is thus not sufficient, and the correct answer is A.
Note: This question is created to prey on two common mistakes, one relating to Data Sufficiency itself and one relating to algebra:
1.) People (even those who have done lots of data sufficiency) tend to forget to look for the “no” answer inYes/No questions and they often make mistakes about what is really requiredfor sufficiency on Yes/No questions.
2.) People forget about the negative possibilities when dealing with squared variables in inequalities.
2. What Is the Value? Question
Statement: If a, b, and c are distinct positive integers where a < b < c and √abc = c, what is
the value of a?
Arguments:(1) c = 8
(2) The average of a, b, and c is 14/3
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution:D
Question Type: What Is the Value? This question asks for the specific value of
positive integer a.
Given information in the question stem or diagram: There is a lot of information SOLUTIONS to leverage from this question stem. a, b, and c are distinct positive integers; a< b < c; and the square root of abc = c. You should first manipulate the last one algebraically by squaring both sides to see that abc = c2. Divide both sides by c (you can do this because you know that c cannot be 0 from the question stem) and the equation becomes ab = c. So before you even go to the statements you know that ab = c and all of the variables are different positive integers.Statement 1: c = 8. Combined with what you learned from the question stem, this means that ab = 8. Since a and b are distinct positive integers and a < b, the only possibility is a = 2 and b= 4. You might consider a = 1 and b = 8 but since
the integers must be distinct, you cannot have b = 8 since c = 8. This is sufficient but you will only see that if you properly leverage every piece of information given in the question stem. Remember: When you are given even a small piece of information in the question stem it is usually very important. The correct answer is A or D.
Statement 2: The average of a, b, and c is 14/3
. This means that the total of a+ b + c = 14. This statement is even trickier than the last but requires a similar leveraging of all available information. It may seem at first glance that there are many possibilities for the values of a, b, and c. However, the only way that ab = c and a + b + c = 14 is for a = 2, b= 4, and c = 8. There is no other way to have three distinct numbers add up to 14 and have ab = c. This statement is also sufficient and the correct answer is D. This question provides an excellent example of a phenomenon you will see often in Data Sufficiency: When a lot of information
is given in the question stem, statements are usually sufficient with much less information than you might first think.
Directions
This Data Sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of
mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked;
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
You should see from the answer choices that the name of the game is “is the data sufficient?”, so let’s spend some time discussing what constitutes sufficient information.
More important than those solutions, however, are these takeaways:
• A statement is sufficient when it guarantees exactly one (and only one) answer to the question.
• This means that in a Yes/No question, you have sufficient information if the answer is “Definitely Yes” or if the answer is “Definitely No”. You do not have sufficient information when the answer is “Sometimes Yes but Sometimes No” (or “Maybe”).
• This means that in a “What is the Value?” question, you have sufficient information when you can pin down exactly one value for the question, but you do not have sufficient information when more than one value is possible.
• Data Sufficiency questions require attention to detail – the drills on the previous page came in pairs, and to the untrained eye each pair might have seemed the same. But subtle differences in what was given or asked – variable squared vs. variable cubed; cube vs. rectangular box; 2A + 3P vs. 3A + 2P – can make all the difference.
As tricky as Data Sufficiency can look at first, the way in which they are constructed ensures that you can attack them systematically. As you know, the answer choices are always the same. And you should also know that there are only two types of question stems that can ask:
1. Yes or No Questions
• A statement gives multiple solutions, but they all give the same answer.
• A statement provides a no answer instead of a yes answer.
2. What Is the Value? Questions
• A statement appears to be giving one value because you have assumed properties of the number that were not actually given (positive, integers, etc.).
• Restrictions were placed in the problem that you did not properly leverage (for instance, the problem is asking for the number of children, which must be an integer and cannot be negative).
Avoid assumptions. Every time you approach a Data Sufficiency problem, you must actively consider any assumptions that you may have been baited into making. Avoiding assumptions is perhaps the most important skill in all with Data Sufficiency.
1. Yes or No Question
Statement:Is x > 3?
Arguments:(1) The sum of x and the square of x is 12.
(2) square of x > 9
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution. A
Question Type: Yes/No. This question asks: “Is x > 3?”
Given information in question stem or diagram: No important information is given in the question stem.
Statement 1: The first step in this statement is to translate the wording into the following equation: x + x2 = 12. Since this is a quadratic equation, you should set everything equal to zero so that x2 + x – 12 = 0. Factoring this, you see that (x + 4)(x - 3) = 0 and x would be -4 or 3. The difficulty in this statement is that many people assume that this information is not sufficient because there are two values, one negative and one positive. However, remember that to prove sufficiency in a yes or no question requires only a definitive answer, not one value. Since each of these values (-4 and 3) gives a “no” answer to the question, this statement is sufficient. The answer is either A or D.
Statement 2: x2 > 9. If x2 > 9 then either x > 3, which gives you a “yes” answer, or x < 3, which gives you a “no” answer. For example x could be -5 (which when squared is > 9) or 5 (which when squared is also > 9). This statement is thus not sufficient, and the correct answer is A.
Note: This question is created to prey on two common mistakes, one relating to Data Sufficiency itself and one relating to algebra:
1.) People (even those who have done lots of data sufficiency) tend to forget to look for the “no” answer inYes/No questions and they often make mistakes about what is really requiredfor sufficiency on Yes/No questions.
2.) People forget about the negative possibilities when dealing with squared variables in inequalities.
2. What Is the Value? Question
Statement: If a, b, and c are distinct positive integers where a < b < c and √abc = c, what is
the value of a?
Arguments:(1) c = 8
(2) The average of a, b, and c is 14/3
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution:D
Question Type: What Is the Value? This question asks for the specific value of
positive integer a.
Given information in the question stem or diagram: There is a lot of information SOLUTIONS to leverage from this question stem. a, b, and c are distinct positive integers; a< b < c; and the square root of abc = c. You should first manipulate the last one algebraically by squaring both sides to see that abc = c2. Divide both sides by c (you can do this because you know that c cannot be 0 from the question stem) and the equation becomes ab = c. So before you even go to the statements you know that ab = c and all of the variables are different positive integers.Statement 1: c = 8. Combined with what you learned from the question stem, this means that ab = 8. Since a and b are distinct positive integers and a < b, the only possibility is a = 2 and b= 4. You might consider a = 1 and b = 8 but since
the integers must be distinct, you cannot have b = 8 since c = 8. This is sufficient but you will only see that if you properly leverage every piece of information given in the question stem. Remember: When you are given even a small piece of information in the question stem it is usually very important. The correct answer is A or D.
Statement 2: The average of a, b, and c is 14/3
. This means that the total of a+ b + c = 14. This statement is even trickier than the last but requires a similar leveraging of all available information. It may seem at first glance that there are many possibilities for the values of a, b, and c. However, the only way that ab = c and a + b + c = 14 is for a = 2, b= 4, and c = 8. There is no other way to have three distinct numbers add up to 14 and have ab = c. This statement is also sufficient and the correct answer is D. This question provides an excellent example of a phenomenon you will see often in Data Sufficiency: When a lot of information
is given in the question stem, statements are usually sufficient with much less information than you might first think.
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