Number and Percentage (#%) Problem Tips

[Dave Killoran]

Hi Rosie,

I'm going to try to help you out with these problems by first starting with a question for you. You mentioned that you were still getting #% problems wrong in LR. Was that from taking practice tests? Or was it from missing the questions in that chapter? Or was it more a general sense of unease you had with the concepts? The more you can tell me about the problems that you are having, the more accurate advice I can give you So, just let me know anything you think would be useful, and and then I'll add more to this response.

Ok, those questions aside, #% problems are going to appear at least a few times on your test, so you are right to want to make sure you have these ideas locked down. And, they can be among the most difficult type of problems, so the fact that they are giving you some trouble isn't unusual. Lots of people have difficulty with these questions!

The first tip I always give about these questions is:


1. Make sure that the #s and %s that are used in the problem are the focus of the problem.

The people who make the test know that these problems bother a lot of people, so one of their tricks is to toss in numerical or percentage info, but not have it be what the problem is really about. so, make sure that if they do throw in some number or percentage info, that it's not a red herring and that it is central to the problem.


Next, the advice I give in the LRB about #% problems is fairly general on the whole, and that is intentional. There are so many different scenarios that can occur that getting too specific would mean that the advice wouldn't apply in many situations. But, %# questions tend to revolve around a few basic errors:

* confusing a number with a percentage;
* confusing a percentage with a number;
* failing to tell you what happened with the total.


So, the second piece of advice is:


2. Identify the exact number or percentage that is being discussed. And if an overall total is given, make note of it as well.

Many student assume that they know what was said without making sure of it, and the makers of the LSAT know that will occur, and create wrong answers that then look good. So, make sure you know what was talked about in detail.

Numbers and percentage problems on the LSAT often hinge on the interplay between three elements: total, a number within the total, and percentage within the total. If you are given just a percentage, that does not prove anything about the number. If you are given just a number, that does not tell you anything about the percentage. But if you are given two out of those three elements, then you can deduce the third. For example, if you are given the percentage and the total, you can determine the number.


So, whereas in point #2 above I advised making sure you know exactly what was said about each number, percentage, or total, here my advice is to look on the other side as well:


3. Look for what's missing. The LSAT often gives you information on 1 or 2 elements, but leaves other parts out. The questions often hinge on what's been left out, so make sure you know what it was.

For example, if you have just a number and the total, but nothing on the percentage, there's a reasonable chance that the answer will revolve around that percentage idea. More on this shortly.


If you are given static, fixed numbers (for example, you have 100 marbles, and 40 of them are blue) then it's easy to figure out what's going on (40% of them are blue; 60% of them are not blue). The LSAT doesn't want to make your life easy, and so often one or more of the numbers they give you is rising or falling. That means the next rule is very important:


4. Watch for movement in any number, percentage, or total, and make note of whether it is rising or falling.

When these numbers and percentages begin rising or falling, change is occurring and it becomes trickier to draw solid conclusions. This is especially the case because of the next tip:


5. Often, #% arguments contain an error of reasoning. So, you are frequently looking at conclusions that are unsupported.

As my colleague Adam Tyson wrote: "A classic numbers and percentages flaw is that the authors attempt to use a percentage to prove a number or a number to prove a percentage, without providing a crucial piece of information. In order to prove a percentage, you need to have a total and a subtotal - that is, if I know I have 100 marbles and 40 of them are green, I can prove that 40% are green and that 60% are not green. What if I only say that later I get more green marbles - can I prove that my percentage increased? No, not without knowing the new total number of marbles. Maybe I got ten more green marbles, but I also got 90 more that were not green, bringing my total marbles to 200, and reducing the green percentage to 25% (50 out of 200)."

In this tip, I specifically point out that the arguments often have errors. This means that the presence of an argument is key, and as we know from earlier chapters, to have an argument you need a conclusion. So:


6. If you have a conclusion, that's when you see an error that is usually based on confusing the number and percentage idea. If there's no conclusion (meaning you are looking at a Fact Set), then often a piece of information is missing and you have to determine what that is.

The presence of an argument often tells you what direction you might be going in, which also means that the question type can help tell you what's happening. If you have a Must Be True question, often you've read a Fact Set, and the correct answer supplies a missing piece of info. But if you have a Strengthen or Weaken question, you've most probably read something with a flaw in it, and it's up to you to have identified that flaw.

To help highlight this idea, I'll use an example one of my colleagues cited recently:

• Let's say I told you there are more violent crimes in Gainsville this year than ever before, and concluded that the average resident in Gainsville is now more likely than ever to fall victim to a violent crime. What do you think of this argument?
  
  Hopefully, you can see this as a terrible argument: clearly, we don't know how the Gainsville population has changed over time. The conclusion is about likelihood (i.e. a percentage idea), whereas the premise describes an increase (a numerical idea). If the total population of Gainsville has tripled over the years, an increase in the total number of crimes wouldn't necessarily mean that the average person is more likely to become a victim. In fact, the crime rate - usually measured by the number of crimes per 100,000 people - could easily have gone down!
  
  So, to weaken the argument, we can propose that the Gainsville's total population has drastically increased over the years. And, inversely, to strengthen the argument we can say that its population has decreased: more crime, spread across fewer people, would guarantee that the crime rate is now higher, and that your risk of being a victim has increased. If they ask you to identify an assumption upon which the argument depends, it would be that Gainsville's population has not substantially increased over the years. (Another assumption would be that we haven't changed the definition of what constitutes a "violent crime".) Does that make sense?

That brings us to our last point for now:

7. You don't have to do real math.

The LSAT isn't a math test, and so you never really have to compute any of these problems, or find an actual mathematical solution. What you do have to do, however, is understand the range of possibilities under the scenario they describe. such as with the Gainsville example above, I don't need to know the exact number of residents, just that the total number of residents has an effect on how we view the argument. So, don't worry about doing real math, and instead worry about how the numbers work together in general.


Ok, those tips hopefully are helpful, and if you are looking to work on more #% problems, we do offer LSAT Logical Reasoning Training Type books that contain sections specifically devoted to these types of problems (and many others). For more info on those books, check out:


PowerScore's LSAT Logical Reasoning: Question Type Training

PowerScore's LSAT Logical Reasoning: Question Type Training Volume 2


Alright, that's it for now. Please let me know if that helps!

[Arindom]

Hi Dave,

Thank you for your excellent tips. I am yet to encounter number and percentage issues on the tests I have done so and I have done only about 5 or 6. However, I did encounter this issue once and chose the correct answer because a) I knew the flaw and b) none of the other answers were even close to this flaw. But I did have any issue trying to fully and lucidly explain what the answer meant. The question I am referring to is PrepTest 4, October 1992, LR section 3 where the question was on declines in the automobile industry revenues. If you wouldn't mind could you explain ans. choice B?

In general, could you give me a few examples of how this number-percentage issue might be expressed in terms of the correct answer choice?

Many thanks.

- Arindom

[Nikki Siclunov]

Hi Arindom,

This is Question 24 from Section 3 (LR) in the October 1992 LSAT. Check out the explanation I've posted for this question here:

lsat/viewtopic.php?f=693&t=9399

Note that we discuss arguments and fact patterns involving numerical evidence in Lesson 9 of the Full-Length and Live Online courses. There is also an extensive discussion of these concepts in the Logical Reasoning Bible and Workbook.

Good luck!

[jbrown1104]

Hi PS!

Quick question, I went over the 6 misconceptions and understand that you have to have at least two pieces of information when determining if the author's use of numbers and percentages were valid. But to be clear, just having the numbers and percentages does not suffice as the two necessary pieces of information. Or could having just two be enough?

How I am interpreting this is of the information provided by an author providing a conclusion based solely off of percentages and numbers isn't sufficient. You need either number and total OR numbers and percent. Let me provide an example: "the apple farmer had 50 more apples this year from apple trees so apple trees are increasing in the percentage of apples they yield". This would be wrong because if there was only 5 trees last year with 50 apples and 10 trees this year with 50 apples, the percentage would be smaller per tree".

So, having numbers and percentages within an argument where the author concludes something about their relationship is NOT enough because you need the total amount. Meaning out of the two elements needed, total will ALWAYS be one of them? Am I interpreting this correctly?


Best,
~JB

[Alex Bodaken]

JB,

Thanks for the question! What Dave alludes to above, and what I believe you are getting at in this post, is that the LSAT testmakers will often provide insufficient information as a basis for a conclusion - the example with the apple farmer you lay out is a good one. The key is to figure out what information is given, what the conclusion is, and if the conclusion can properly be drawn from the information. If I tell you I have 25% of the apples and I have 4 apples, you can properly draw the conclusion that there are 16 total apples - so in that example, I didn't give a total but you were able to work it out. But if I say that I had 25% of the apples and now I have 26%, and therefore, I only got a couple more apples - now we have a problem because I've only told you the percentage, and without a total you can't know how many more apples I got with my extra 1% (maybe 1! maybe 1000!). So to sum up, I wouldn't worry too much about the total always needing to be included in the stimulus, but instead about what can be properly worked out with the information given, and then once you've figured that out, whether or not the conclusion matches that.

Hope that helps!
Alex

[jbrown1104]

Hi Alex!

Thank you for taking the time to decode my example! I get what you're saying! The problem arises when you try to make a comparison of numbers with only the percentages and vice versa. The example with, having 25% of apples and you have 4, the logical inference has to be there was a total of 16. However, you cannot make a logical comparison (of numbers and percentages) when giving only one piece of information that isn't the total. Let me know if I dissected this correctly please.

Thank you so much!
~JB

[Adam Tyson]

Looks like you've got it, JB, but I want to clarify one thing, and that is that the total may or may not be included but is never enough by itself.

If I know there are 50 apples (the total) and I have 10 of them (the subtotal), I can figure out a percentage (20%)

If I know there are 50 apples (the total) and I have 20% of the apples (a percentage), I can figure out the subtotal that I have (10)

If I know I have 10 apples (the subtotal) and that those represent 20% of the apples (a percentage), I can calculate the total (50 apples)

With any two pieces, I can figure out the third, but with only one of the three I cannot prove anything about the other two. So, having 10 apples last week and 20 apples now tells me nothing about the percentage or the total then or now. I can say that I have more apples now, and that I have twice as many as I had before, but I cannot say that I have a higher or lower percentage of all apples, nor can I say that there must be a higher or lower total number of apples. I only know about my subtotal.

Always ask yourself if you have sufficient information to make a numerical or percentage conclusion, and then use that info to answer all kinds of different questions!

[jbrown1104]

Thank you Adam! I get it know, I feel more prepared to tackle # and % problems because I now understand the importance the conclusion plays and what is sufficient information to make particular conclusions.

Best,
JB!