## Common DS Mistakes

### Transcript

Common data sufficiency mistakes. In this lesson, we will look at some common mistakes on the GMAT data sufficiency questions. The first concerns the confusion with the direction of logic. Well, what do I mean by that? The d, the data sufficiency is primarily a test of logic.

We have to be very careful in understanding the logical relationships. Consider this question. You can pause the video and think about this for a minute. Okay. Here's the mistake thinking.

Of course, if the prompt were really true, if JKLM were really a square, then each one of these statements would also be true. It's absolutely true that if we were guaranteed that, that shape is a square, then JM would equal KL and that angle would equal 90 degrees. Yes. That is geometrically correct, but that's not the task at hand in the data sufficiency.

Don't get caught in backwards logic. Would the prompt make the statements true? That's not what this is about. In the data sufficiency, the task is always to figure out whether one or both of the statements make the prompt true. So, is JKLM a square?

That's really a question, that is something unknown. And suppose we know the statements, does that help us answer the question? For this question, we've already discussed how difficult it is to prove that a shape is a square. One right angle and two congruent sides guarantee nothing. The shape could look like this shape here.

Obviously, that is not even close to a square. So both statements together are not sufficient. The answer has to be E. The next mistake we have already discussed in the context of picking numbers. Don't automatically assume that all numbers are nice, neat positive integers, the ordinary grade-school counting numbers.

People see the word number and they think, one, two, three, four, five, six. They think the numbers they can count on their fingers. The GMAT loves to throw out a variable or a general term such as number and let people incorrectly assume that the only possibilities are these numbers that you can count on your fingers. Here's the practice question.

Pause the video and then we'll talk about this. Okay. If the square root of K is not an integer, then is K a prime number? Notice that in order to be a prime number, K would have to be a positive integer. Nothing in the question guarantees that K is a positive integer.

If K were a positive integer, then it's absolutely true, especially with that second statement. The positive integers less than five are one, two, three, four. And the square root of 1 is an integer and the square root of 4 is an integer, so the square root of 2 and 3, those would be the two that aren't integers. And indeed, two and three are both prime numbers, but we're not guaranteed that K is an integer.

For example, 2.5 or pi would satisfy the prompt and both statements. Those are two numbers that less than ten, less than five and when we take the square root is not an integer. But those numbers are not integers at all, so they're certainly not prime numbers. So, one and two are both insufficient. Okay? Could be a prime number, doesn't have to be a prime number.

And so the answer is E. The next broad category of mistakes concerns exceptions, mathematical exceptions. Many areas of math, particularly with integers, there are certain general properties that are true for a certain category of numbers, but there's an exception for one number or some small category of numbers.

For example, all prime numbers are odd except for two. The only even prime number. That's an important exception to know. All positive integers are prime or composite, except for one, which is neither. One is not a prime number and it has no factors of them in itself.

All negative numbers, all numbers on the number line are either positive or negative, except for zero. Zero is the only number that is neither positive, nor negative. For most numbers, positive and negative, x squared is greater than x. But for x equals 0 and x equals 1, x squared actually equals x and for fractions between 0 and 1, x squared is less than x.

We can't take the square root of a negative, but we can take the cube root of a negative. These are some examples of mathematical exceptions. So, I can't list every possible mathematical exception right here. But you'll notice that these are precisely the things we were discussing in the math lessons.

The GMAT loves to design data sufficiency questions that examine one particular exception, perhaps an easy to overlook exception that many students will forget. If one remembers the exception, then the information leads to one answer, the correct answer. But if one does not remember the exception, one, the student will be led to a very different answer.

Here's a practice problem. Pause the video and then we'll talk about this. Okay. F is a positive factor of 105, is F a prime number? Well, positive factor of 105.

Let's think about this. So we'll find the prime factorization first. We're gonna have to do a little simplification, but we get down to 3 times 5 times 7. That's the prime factorization. Well, for statement number one, if F is not divisible by 3, well, F could be 5.

That gives us a yes answer or F could be 35, that's a no answer. State number one is insufficient. Statement number two, what are the factors less then ten? Well, three, five, and seven, those are all prime numbers, but here's the exception. One is a factor and one is not a prime number.

So one is a factor of every number. So, it's a factor of 105 and 1 is not a prime number. So we have some primes, but we have one, which is not prime. So this is also insufficient. And then we combine the statements, then yes, we could have five and seven, which could be yes answers, but one gives a no answer.

So, so we can still get a yes or a no answer to the prompt questions, even with the statements combined. Everything is insufficient and the answer is E. That problem was simultaneously testing two special cases. Did the student know that one is a factor of 105 and of every positive integer? And also, did the student know that one is not a prime number?

Forgetting either one of these important distinctions or both of them, would have led to a wrong answer. Make sure you know the mathematical rules in detail. The final very common mistake is something we discussed way back in the introductory video about data sufficiency. Of course, the first task in a data sufficiency problem is to examine each statement on its own separate from the other.

It does not matter in which order we examine the two separate statements. Many people will do one first, then two, but there's no rule that you have to do one versus the other first. The obvious mistake is to carry information from one into your considerations of statement two by itself. You're supposed to treat statement two by itself first without com, any of the information from statement one.

But if we spend a lot of time thinking about statement number one, it's more natural that you would bring some of that information to statement two. This is particularly likely if statement number one is long and complicated and the GMAT likes doing this. They like giving a long, beefy, complicated statement number one. A relatively short statement number two.

And of course, what most people are gonna do, they're gonna read that statement number one, they're gonna think a lot about it. And inevitably, they're gonna remember some of that information, drag some of that information to statement two and make a mistake. If statement number two is much shorter or easier than number one. You can avoid a lot of trouble simply by examining statement number two by itself before you examine statement number one by itself.

So, very easy. If statement number one looks long, statement number two looks short and easy, do statement number two first. Very simple. Here's a practice problem. Pause the video and then we'll talk about this.

Okay. Triangle T has three angles that we don't know. Is angle A less than 45 degrees? Well, statement number two is very short, statement number two must be insufficient, because it give gives the measure of angle B, tells us nothing about angle A or angle C.

So we have no way to draw a conclusion about angle A. So statement number two, insufficient. So now that we know that, let's look at statement number one by itself. One by itself. Well, we can pick some numbers here. If we pick B equals 50, since that's something that we're gonna have to deal with anyway.

We picked that number and then we, then it looks like angle C has to be bigger than 100. So let's say, it's a 101. Then if we solve for A, we have two of the angles, 50. We have 101. Subtract from 180, we get 29 degrees.

So that would give us a yes answer. A is less than, than 45 degrees. Okay? But if we pick B equals 1 degree, a small angle, then C has to be greater than two. Say, C equals 3. Well, then if we subtract those from 180, we get angle A equals 176, which is much bigger than 45 degrees.

That gives us a no answer. So this statement by itself, one by itself is insufficient. So, each statement by itself is insufficient. Well, now what we're gonna do is look at the two statements combined. We already found a yes answer with a picking number approach. So that implies, we're probably done picking numbers.

We have to think about logic to demonstrate that something is sufficient. So we know that B is 50. We know that C is greater than 100. Well, if we add those two, we know that B plus C is greater than 150. Well, those two angles take up more than 150 of the angles in the triangle, whereas the three angles together have to add up to 180.

That leaves less than 30 degrees for angle A, so A would have to be less than 30. And if a is less than 30, it absolutely has to be less than 45. That gives us a definitive yes answer with the statements combined we're led inexpli, we're inescapably to a definitive answer. That means that it's sufficient together and the answer is C. So this lesson has covered some of the more common mistakes on the GMAT data sufficiency.

When you make mistake or have an oversight in GMAT math, never dismiss it as simply a stupid mistake. Always get curious about the nature of this mistake and what it would take to learn deeply from it and never make it again. Remember, the mark of a truly excellent student is never making the same mistake twice and you may go back and look at that, look at that video in the general math strategies section about learning from your mistakes.

That, that's a discussion that might help you getting the most out of each time you make a mistake in math. In summary, remember, never to use backwards logic on the GMAT. Your job is to determine whether the statements support the prompt, not vice versa. Don't automatically assume that all numbers are integers or positive integers if that is not specified.

Beware of questions that target specific mathematical exceptions. All prime numbers are odd except for two. Something like that. Don't carry information from statement one to statement two, analyze them separately. Again, if statement number one is a big statement, a big complicated statement and statement number two is really short, do statement number two first.

And of course, always learn from your mistakes.

Read full transcriptWe have to be very careful in understanding the logical relationships. Consider this question. You can pause the video and think about this for a minute. Okay. Here's the mistake thinking.

Of course, if the prompt were really true, if JKLM were really a square, then each one of these statements would also be true. It's absolutely true that if we were guaranteed that, that shape is a square, then JM would equal KL and that angle would equal 90 degrees. Yes. That is geometrically correct, but that's not the task at hand in the data sufficiency.

Don't get caught in backwards logic. Would the prompt make the statements true? That's not what this is about. In the data sufficiency, the task is always to figure out whether one or both of the statements make the prompt true. So, is JKLM a square?

That's really a question, that is something unknown. And suppose we know the statements, does that help us answer the question? For this question, we've already discussed how difficult it is to prove that a shape is a square. One right angle and two congruent sides guarantee nothing. The shape could look like this shape here.

Obviously, that is not even close to a square. So both statements together are not sufficient. The answer has to be E. The next mistake we have already discussed in the context of picking numbers. Don't automatically assume that all numbers are nice, neat positive integers, the ordinary grade-school counting numbers.

People see the word number and they think, one, two, three, four, five, six. They think the numbers they can count on their fingers. The GMAT loves to throw out a variable or a general term such as number and let people incorrectly assume that the only possibilities are these numbers that you can count on your fingers. Here's the practice question.

Pause the video and then we'll talk about this. Okay. If the square root of K is not an integer, then is K a prime number? Notice that in order to be a prime number, K would have to be a positive integer. Nothing in the question guarantees that K is a positive integer.

If K were a positive integer, then it's absolutely true, especially with that second statement. The positive integers less than five are one, two, three, four. And the square root of 1 is an integer and the square root of 4 is an integer, so the square root of 2 and 3, those would be the two that aren't integers. And indeed, two and three are both prime numbers, but we're not guaranteed that K is an integer.

For example, 2.5 or pi would satisfy the prompt and both statements. Those are two numbers that less than ten, less than five and when we take the square root is not an integer. But those numbers are not integers at all, so they're certainly not prime numbers. So, one and two are both insufficient. Okay? Could be a prime number, doesn't have to be a prime number.

And so the answer is E. The next broad category of mistakes concerns exceptions, mathematical exceptions. Many areas of math, particularly with integers, there are certain general properties that are true for a certain category of numbers, but there's an exception for one number or some small category of numbers.

For example, all prime numbers are odd except for two. The only even prime number. That's an important exception to know. All positive integers are prime or composite, except for one, which is neither. One is not a prime number and it has no factors of them in itself.

All negative numbers, all numbers on the number line are either positive or negative, except for zero. Zero is the only number that is neither positive, nor negative. For most numbers, positive and negative, x squared is greater than x. But for x equals 0 and x equals 1, x squared actually equals x and for fractions between 0 and 1, x squared is less than x.

We can't take the square root of a negative, but we can take the cube root of a negative. These are some examples of mathematical exceptions. So, I can't list every possible mathematical exception right here. But you'll notice that these are precisely the things we were discussing in the math lessons.

The GMAT loves to design data sufficiency questions that examine one particular exception, perhaps an easy to overlook exception that many students will forget. If one remembers the exception, then the information leads to one answer, the correct answer. But if one does not remember the exception, one, the student will be led to a very different answer.

Here's a practice problem. Pause the video and then we'll talk about this. Okay. F is a positive factor of 105, is F a prime number? Well, positive factor of 105.

Let's think about this. So we'll find the prime factorization first. We're gonna have to do a little simplification, but we get down to 3 times 5 times 7. That's the prime factorization. Well, for statement number one, if F is not divisible by 3, well, F could be 5.

That gives us a yes answer or F could be 35, that's a no answer. State number one is insufficient. Statement number two, what are the factors less then ten? Well, three, five, and seven, those are all prime numbers, but here's the exception. One is a factor and one is not a prime number.

So one is a factor of every number. So, it's a factor of 105 and 1 is not a prime number. So we have some primes, but we have one, which is not prime. So this is also insufficient. And then we combine the statements, then yes, we could have five and seven, which could be yes answers, but one gives a no answer.

So, so we can still get a yes or a no answer to the prompt questions, even with the statements combined. Everything is insufficient and the answer is E. That problem was simultaneously testing two special cases. Did the student know that one is a factor of 105 and of every positive integer? And also, did the student know that one is not a prime number?

Forgetting either one of these important distinctions or both of them, would have led to a wrong answer. Make sure you know the mathematical rules in detail. The final very common mistake is something we discussed way back in the introductory video about data sufficiency. Of course, the first task in a data sufficiency problem is to examine each statement on its own separate from the other.

It does not matter in which order we examine the two separate statements. Many people will do one first, then two, but there's no rule that you have to do one versus the other first. The obvious mistake is to carry information from one into your considerations of statement two by itself. You're supposed to treat statement two by itself first without com, any of the information from statement one.

But if we spend a lot of time thinking about statement number one, it's more natural that you would bring some of that information to statement two. This is particularly likely if statement number one is long and complicated and the GMAT likes doing this. They like giving a long, beefy, complicated statement number one. A relatively short statement number two.

And of course, what most people are gonna do, they're gonna read that statement number one, they're gonna think a lot about it. And inevitably, they're gonna remember some of that information, drag some of that information to statement two and make a mistake. If statement number two is much shorter or easier than number one. You can avoid a lot of trouble simply by examining statement number two by itself before you examine statement number one by itself.

So, very easy. If statement number one looks long, statement number two looks short and easy, do statement number two first. Very simple. Here's a practice problem. Pause the video and then we'll talk about this.

Okay. Triangle T has three angles that we don't know. Is angle A less than 45 degrees? Well, statement number two is very short, statement number two must be insufficient, because it give gives the measure of angle B, tells us nothing about angle A or angle C.

So we have no way to draw a conclusion about angle A. So statement number two, insufficient. So now that we know that, let's look at statement number one by itself. One by itself. Well, we can pick some numbers here. If we pick B equals 50, since that's something that we're gonna have to deal with anyway.

We picked that number and then we, then it looks like angle C has to be bigger than 100. So let's say, it's a 101. Then if we solve for A, we have two of the angles, 50. We have 101. Subtract from 180, we get 29 degrees.

So that would give us a yes answer. A is less than, than 45 degrees. Okay? But if we pick B equals 1 degree, a small angle, then C has to be greater than two. Say, C equals 3. Well, then if we subtract those from 180, we get angle A equals 176, which is much bigger than 45 degrees.

That gives us a no answer. So this statement by itself, one by itself is insufficient. So, each statement by itself is insufficient. Well, now what we're gonna do is look at the two statements combined. We already found a yes answer with a picking number approach. So that implies, we're probably done picking numbers.

We have to think about logic to demonstrate that something is sufficient. So we know that B is 50. We know that C is greater than 100. Well, if we add those two, we know that B plus C is greater than 150. Well, those two angles take up more than 150 of the angles in the triangle, whereas the three angles together have to add up to 180.

That leaves less than 30 degrees for angle A, so A would have to be less than 30. And if a is less than 30, it absolutely has to be less than 45. That gives us a definitive yes answer with the statements combined we're led inexpli, we're inescapably to a definitive answer. That means that it's sufficient together and the answer is C. So this lesson has covered some of the more common mistakes on the GMAT data sufficiency.

When you make mistake or have an oversight in GMAT math, never dismiss it as simply a stupid mistake. Always get curious about the nature of this mistake and what it would take to learn deeply from it and never make it again. Remember, the mark of a truly excellent student is never making the same mistake twice and you may go back and look at that, look at that video in the general math strategies section about learning from your mistakes.

That, that's a discussion that might help you getting the most out of each time you make a mistake in math. In summary, remember, never to use backwards logic on the GMAT. Your job is to determine whether the statements support the prompt, not vice versa. Don't automatically assume that all numbers are integers or positive integers if that is not specified.

Beware of questions that target specific mathematical exceptions. All prime numbers are odd except for two. Something like that. Don't carry information from statement one to statement two, analyze them separately. Again, if statement number one is a big statement, a big complicated statement and statement number two is really short, do statement number two first.

And of course, always learn from your mistakes.