Data Sufficiency - Platinum Techniques
While it is true that the only sure way to master the GMAT is to master the content tested on the GMAT, it is also true that knowing certain techniques can improve your score. Below, we have assembled a list of techniques useful for GMAT data sufficiency questions.
Note: Some of the example questions that follow are not entirely realistic and you should not expect to see questions like these on the GMAT. These example questions contain traits designed to make the illustration of the Platinum Technique easier to see and understand.
Work additional data sufficiency questions.
Determining Sufficiency Does Not Require Solving
One common mistake individuals make on data sufficiency questions is solving equations completely instead of simply determining whether sufficient data exist to answer the question. Remember that data sufficiency questions do not require you to find a value--they require you to determine definitively whether sufficient data exist to answer the question. In order to answer a data sufficiency question, you do not need to determine the solution to each equation (e.g., x = 7 and y = 6). Rather, you need to have a definitive answer (e.g., x is a single value). Consider the following example:
If a, b, and c are positive integers, what is the value of a9 + b8 + c6?
- a3 = 8, b = 18, and c = 28989
- a = 2, b8 = 11,019,960,576
Correct Answer: A
Try plugging in values for the variables and determining whether you can find a definitive value for the expressiona9 + b8 + c6.
- In evaluating Statement (1), we know definitive values for a, b, and c. Consequently, we have sufficient information to find a definitive value for a9 + b8 + c6.
- Note that we do not need to determine what the exact value of a9 + b8 + c6 is. We simply need to know definitively that one-and-only-one value for the expression a9 + b8 + c6 can exist given the information in Statement 1. Statement (1) is SUFFICIENT.
- In evaluating Statement (1), we can perform the following algebraic substitutions:
=(a)9 + b8 + c6
=(2)9 + 11,019,960,576 + c6
- Since we cannot definitively determine that there will only be one value for c6, we cannot definitively determine that there will be only one value for the entire expression a9 + b8 + c6. (Note that we do not need to determine what the exact value is.) Statement (2) is NOT SUFFICIENT.
Statements (1) and (2) Do Not Contradict
Statements (1) and (2) will NEVER contradict each other. Consequently, if you are simplifying or solving statements and the result is a situation where Statements (1) and (2) contradict each other, you made an error. For example, if after simplifying the information in the statements, you are left with Statement (1) x = 10 and Statement (2) x = 19, you must go back and re-do your calculations as you made an error.
In A Time Crunch, Evaluate the Easier Statement and Guess
If you are in a time crunch, look at the statements and see which one is easier to evaluate and act accordingly. By evaluating one statement, you improve your odds of guessing the correct answer significantly.
If x is an integer, is x3 > 0?
- x5 + x3 + x + 15 = 298
- x5 + 10 > 15
Correct Answer: D
Know that the question is asking whether x is positive or negative. Seek to determine quickly the answer sign.
- When you raise an integer to an odd number, it does not change the sign of the expression. In other words, if x is negative, x3 will be negative. Likewise, if x is positive, x3 will be positive.
- The question can be simplified to: "is x positive?"
- If time is short and Statement (1) looks complicated, move on and evaluate Statement (2) first and rule out clearly wrong answers.
- Evaluating Statement (2):
x5 + 10 > 15
x5 > 5
x must be positive. Statement (2) is SUFFICIENT.
- Since Statement (2) is sufficient, you can quickly rule out answer choices A, C, and E. You have now quickly improved your chances of choosing the correct answer from 20% to 50% (i.e., you are now choosing from 2 answer choices instead of from 5).
- In evaluating Statement (1), begin by simplifying:
x5 + x3 + x + 15 = 298
x5 + x3 + x = 283
- Since raising a number to an odd exponent does not change the sign of the number, the sign of every term in Statement (1) must be the same (i.e., x5, x3, and x all have the same sign.)
- Logically, x must be positive since it is impossible to add together only negative numbers and arrive at a sum that is a positive number. In other words, if you add any two negative numbers, you will have a negative number with a larger absolute value. Since x5 + x3 + x adds up to a positive number (i.e., 283), it is impossible for x to be negative (otherwise, x5 + x3 + x would be negative). x is positive and Statement (1) is SUFFICIENT.
- It turns out that x = 3.0227, meaning Statement (1) is SUFFICIENT because the information in it produces one-and-only-one value for the expression above.
- Note, however, that the test would never ask you to solve an equation such as the one in Statement (1). This complicated equation is used simply to elucidate the technique.
Symmetry Amongst Statements (1) and (2) --> D or E
Parallelism or symmetry in the two statements means that D or E is the correct answer. In other words, if you rephrase the statements and you discover they are saying the same thing, you can immediately rule out A, B, and C. This is best illustrated by an example:
What is the value of x + 15?
- x3 = 27
- x5 = 243
Correct Answer: D
Seek to determine if x is a unique value. If x is unique, x + 12 will be unique and the information will make the statement sufficient.
- Evaluate Statement (1).
x3 = 27
x = 3
- Evaluate Statement (2).
x5 = 243
x = 3
- Since Statements (1) and (2) provide the same information, answer choices A, B, and C cannot be correct. Consequently, answer choices D or E must be true.
- Statements (1) and (2) are each SUFFICIENT alone because x + 15 will always be 18.
Avoid Unwarranted Assumptions
In intermediate to difficult questions, the GMAT test-writers try to trap test-takers by getting them to make unwarranted assumptions. Consider the following examples of unwarranted assumptions:
- A jar contains 10 marbles. If there are 4 red marbles in the jar, how many blue marbles are in the jar? Unwarranted Assumption: There are 6 blue marbles in the jar. There could be 5 blue marbles and 1 yellow marble. Do not assume that there are only two colors of marbles in the jar.
- If x > 10 and x < 12, what is the value of x? Unwarranted Assumption: x is 11. You cannot assume that x is an integer. Nothing in the given information said that x must be an integer. Consequently, x could be 10.5 or 11.5.
- If Ms. Watson's 4th grade class has 20 students and 50% of these students have blonde hair, how many girls in Mrs. Watson's class have blonde hair? Unwarranted Assumption: There are 10 students with blonde hair so 50% of these must be girls. Consequently, 5 girls have blonde hair. You cannot assume that the ratio of the number of boys to girls is 1:1, or 50%.
Beware of Even Exponents
Whenever dealing with even exponents, you must be cognizant that an even exponent hides the sign of the base. In other words, if x2 = 4, x = 2 AND -2. Consider the following example where forgetting this would take you to the wrong answer.
What is the value of x + 1?
- x2 = 16
- x3 = 64
Correct Answer: B
Find out whether a definitive value can found for the expression in the question.
- In evaluating Statement (1), many beginning and intermediate-level test-takers automatically assume that since x = 4, Statement (1) is sufficient because 4 + 1 is always 5.
- However, Statement (1) is NOT SUFFICIENT because x also equals -4 since (-4)2 = 16. Consequently, x + 1 also equals (-4) + 1 = -3
- In evaluating Statement (2), we find that Statement (2) is SUFFICIENT because there is only one possible value for x, 4. Negative four is not a possible value given the information in Statement (2) since (-4)3 is not 64 but -64. Therefore, there is only one possible value for x + 1, 5.
- Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.
Statements Producing Two Values Are Not Automatically Insufficient
This is probably one of the dirtiest tricks the GMAT test-writers can pull on a medium-level test-taker. Just because a statement, when simplified, yields two values does not mean the statement is automatically insufficient. Those two values could produce the same value for the question, in which case the statement is sufficient. Consider the following example:
If x is an integer, what is the value of x2 + x - 12?
- x = 3
- x = 3, x = -4
Correct Answer: D
Try all possible values and seek to determine whether there is a single defintiive answer to the question. If yes, the statement is sufficient. If no, the statement is not sufficient.
- Since Statement (1) gives a single definitive value that can be plugged in to x2 + x - 12, Statement (1) is SUFFICIENT since we will be able to determine for sure the value of the expression.
- To see this more clearly, plug in x = 3 and see that it produces one definitive answer to the question:
x2 + x - 12
32 + 3 - 12 = 9 + 3 - 12 = 0
- Evaluate Statement (2). At this point, some are tempted to say that since Statement (2) provides two values for x, Statement (2) is not sufficient because two different values of x (i.e., 3 and -4) will produce two separate values of the equation in the question. However, this is not always true. Two separate values of x can produce the same value.
Evaluate x = 3
x2 + x - 12
32 + 3 - 12 = 9 + 3 - 12 = 0
Evaluate x = -4
x2 + x - 12
(-4)2 -4 - 12 = 16 -4 - 12 = 0
- Since the information in Statement (2) provides one definitive value for the expression in the question (i.e., 0), Statement (2) is SUFFICIENT.
- Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.
Two Equations With Two Variables Does Not Necessarily Mean Sufficiency
In high school math, most students learned that if two equations have two separate variables, then a definitive solution exists. Although this is technically true, the GMAT plays upon this and seeks to trick test-takers by providing two equations that look different but are actually the same. The GMAT test writers hope that test-takers will assume that a solution exists for the equations. However, under the GMAT's scheme, there is only one unique equation with two variables, so a solution does not need to exist. Consider the following example:
If x and y are integers, is x > 0 and y < 0?
- 4x + y = 8
- x - 2 = -y/4
Correct Answer: E
Minuplate the equations in Statements (1) and (2) and see whether a unique equation exists.
- Most test-takers realize that Statement (1) alone and Statement (2) alone are NOT SUFFICIENT.
- However, the tests' authors hope that test-takers, when evaluating answer choice C, will assume that there are two unique equations with two variables and, as a result, there is a unique solution for x and y and the question can be definitively answered.
- But, the above assumption does not hold in this question because there are not two unique equations. The equation in Statement (2) is the same as the equation in Statement (1).
Start with: 4x + y = 8
Subtract 8 from each side: 4x - 8 + y = 0
Subtract y from each side: 4x - 8 = -y
Divide each side by 4: x -2 = -y/4
So, there is only one unique equation with two variables. Consequently, it is impossible to solve for the value of x and y. Both Statements (1) and (2), even when taken together, are NOT SUFFICIENT.