Data Sufficiency and Quantitative Comparison for the DAT
/Learn key DAT concepts related to data sufficiency and quantitative comparison, plus practice questions and answers
Learn key DAT concepts related to data sufficiency and quantitative comparison
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Part 1: Introduction to data sufficiency and quantitative comparison
Many questions in the quantitative reasoning section are multiple choice and straightforward, just like you’re used to seeing in high school and college. However, the QR section also includes some multiple choice questions that are formatted differently than you’ve probably seen before. These questions relate to data sufficiency and quantitative reasoning, and can initially present a challenge for some students. However, these types of questions test the same concepts covered in other quantitative reasoning guides. This guide will teach you how to master data sufficiency and quantitative comparison questions. As you study this guide, practice the strategies using questions and answers at the end.
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Part 2: Data sufficiency
Data sufficiency questions can be some of the hardest problems in the QR section. There are a few reasons why these questions can be difficult, as they require you to think critically, can ask about any QR topic, and are uniquely formatted. Data sufficiency questions take this form:
How many boys are in a certain school?
Statement 1: The ratio of boys to girls at the school is 3:4
Statement 2: There are 120 students in the school
A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient
A question is asked, followed by two statements that offer additional information. Your task is to determine whether each statement on its own or combined has sufficient information to answer the question. The answer choices for data sufficiency questions are always the same. Choice A states that you can answer the question using only the information from statement 1, but not from statement 2, while choice B is the opposite. If both statements separately contain enough information to answer the question, choice D is correct. Choice C states that the information from both statements combined is sufficient to answer the question. If neither statement alone or combined is sufficient to answer the question, choose choice E. For the previous question, choice C is correct.
A few things should be noted in relation to these questions. The first is the question is asking about the statements and whether you could solve the problem, not what the answer to the question actually is. Second, each statement is true. Data sufficiency questions don’t try to trick you by having false statements; both statements contain true information related to the question. Third, data sufficiency questions can pull from any quantitative reasoning topic. You don’t have to study any new material to answer these questions, but you do need to know the other QR topics (algebra, graphing, geometry, etc.).
There are proven strategies you can use to excel at data sufficiency questions. The first strategy is to follow a decision tree. While you will not have this decision tree on the actual exam, using it will help you develop a way of approaching these questions.
FIGURE 1: DATA SUFFICIENCY DECISION TREE
The first decision on the tree asks “Is statement 1 sufficient?” After answering this question, the next decision to make is “Is statement 2 sufficient?” Depending on the answer to these questions, you may already be able to confidently solve the problem. If both answers to this point are no, then you need to determine if both statements together are sufficient.
This decision tree teaches an important principle for these questions, which is to evaluate each statement separately. Let’s apply this decision tree to a practice question.
Given the following equation, where x and y are variables, is x greater than y?
\(2x + 3y = 10\)
Statement 1: \(x=3\)
Statement 2: \(y=\frac\)
A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient
Start by determining if the first statement alone is sufficient. In this case, we can plug in 3 for x in the original equation and see if we can figure out if x is greater than y.
\[2(3)+3y=10\]
\[3y=4\]
\[y=\frac\]
With only statement 1, we can solve for y and determine whether or not it is less than x. Following the decision tree, our next step is to answer if statement 2 is sufficient. Without even plugging in the given value of y, however, we can see that it will be sufficient. The original equation only has two variables, x and y. For statement 1, we determined that we can answer the question if we are given one of the two variables. Since the answer to both questions is yes (meaning both statements alone are sufficient), D is the correct choice.
This way of evaluating statement 2 leads us to the next strategy, which is don’t answer the question. You’re probably thinking this strategy makes no sense, as you have to choose one of the multiple choice answers. However, that’s not the question this strategy is referring to. Rather, this strategy is to, where possible, not actually answer the first question. Remember, data sufficiency questions are asking whether or not you have enough information to answer the original question, and they don’t actually require you to answer that question. Here is an example.
Kevin received a raise of $3 per hour at his job. Last week, he worked 40 hours. How many less hours does he need to work this week in order to make the same amount of money as he did last week?
Statement 1: Kevin’s hourly pay rate is 30% higher than it was when he was hired
Statement 2: Last week, Kevin made $700
A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient
In order to answer this question, we need to know what Kevin’s hourly pay was before his raise. Then, we can determine what his new hourly wage is, and divide the total amount of money he earned last week by his new hourly wage. Looking at statement 1, we can see that his pay is 30% higher than when he was originally hired. This information, however, won’t help us determine his previous hourly wage.
Statement 2 says that Kevin earned $700 last week. From the question prompt, we also know that Kevin worked 40 hours last week. Using this information, we could calculate that Kevin’s wage prior to his raise was $700 ÷ 40 hrs = $17.5/hr. Then, we could add $3/hr to this wage and divide $700 by the new hourly wage. This would give us a value of about 34 hours, meaning Kevin would need to work 6 fewer hours to make the same wage he did before.
If this were a normal question, you would definitely do all these calculations. However, this is a data sufficiency question. Solving it after determining that statement 2 is sufficient is a waste of precious time and mental energy. Without actually doing all of the calculations, we can determine that choice B is the right answer.
How did we know that the information we needed to solve the problem was related to finding the hourly wage before the raise? This question leads to the next strategy, which is to think critically. To do well on data sufficiency questions, you need to critically evaluate each statement. Ask yourself questions such as
What steps would I take to solve the question in the prompt?
What information would I need to answer this question?
In what ways does this statement apply (or not apply) to the prompt?
By answering these questions, you can determine what information you need to answer the prompt and whether or not each statement provides this information. Again, let’s apply this strategy to a practice question.
Is the median of the following set of numbers greater than 15?
Statement 1: The range of the numbers is 20.
Statement 2: The mean of the numbers is 18.
A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient
One of the first questions we ask ourselves is “what information would we need to answer the prompt?” We are not given a set of numbers, so think about how we could determine the median without knowing what the numbers are. To hypothetically solve this problem, we would need to know how many numbers are in the data set. Additionally, information such as the frequency of each number, or the mode, or perhaps a combination of the mean with other information would be needed.
Now, look at statement one, which gives us the range of the data set. This information alone is not sufficient to determine the median, so evaluate statement 2. This statement gives us the mean of the data set, which again is not sufficient. Looking at the information from statements 1 and 2 together, we can see that we were not given much of the information that we previously determined that we needed (frequency, how many numbers, etc.). Because of this, we cannot answer the question with only the information from statements 1 and 2. Choice E is correct.
Another strategy is to carefully read the question. Test writers often try to trick you on data sufficiency questions by giving you statements that are related to the prompt but don’t quite answer it. Carefully read the prompt, and then critically think about each statement.
Some data sufficiency questions will require you to interpret a graph or figure. When answering questions with figures, take time to read and understand the information it provides. With these and all other data sufficiency questions, do not make assumptions. These questions are designed so that if you assume information that isn’t given, you will not get the right answer. Let’s go through a final example with a pie chart.
Is the number of students who study engineering and graduate in eight semesters greater than the number of students who study education and graduate in eight semesters?
Statement 1: 75% of engineering majors graduate in eight semesters
Statement 2: 95% of education majors graduate in eight semesters
A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient
After carefully reading the prompt and looking at the pie chart, we see that we are given percentages of enrollment at a college by area of study. The prompt asks about graduation numbers for education and engineering majors. We are not given any information about graduation, and the graph does not actually say how many people are in each major.
Looking at statement 1, we are given the percentage of engineering majors that graduate in eight semesters. This says nothing about education majors, so statement 1 alone is insufficient. Statement 2 alone is also insufficient because it only mentions education majors. Together, though, we know what percentage of engineering and education majors graduate in eight semesters.
Remember that the pie chart gives percentages, not actual numbers. To theoretically calculate how many students graduate in eight semesters from these two majors, we would need to multiply the percentages together (engineering: 0.16*0.75=0.12; education: 0.12*0.95=0.114). Even though we do not know the actual number of students at this college, we know that a greater total percentage of engineering students graduate in eight semesters than education majors. Both statements together are sufficient (choice C).
