Why keep a GMAT error log?

What usually happens when you encounter a difficult GMAT math problem for a second time that you struggled to solve initially? Do you cruise right through the solution? Or do you hem and haw and spin around in the same frustrating circles you did the first time you saw the problem? If the latter, it’s time to ask yourself a serious question: if you can’t even solve difficult problems that you’ve already seen, what chance do you have of solving the difficult problems you’ll see for the first time when you take the real GMAT? If you’re not retaining the things that you learn when you do practice problems, you’re wasting your time.

Retaining a newly learned mathematical concept is a bit like trying to keep a non-helium balloon up in the air: you have to keep tapping it to keep it afloat. If you only tap it once and then forget about it for weeks, don’t be shocked when you come back and discover that it’s on the ground. As a private tutor, the most frustrating thing that I experience (professionally) is when I’m able to successfully help a student really understand a new concept, and then a week later he or she has totally forgotten how to use it. Remember that the only thing that a tutor or GMAT book or a generous guy on a forum can do is help you get that balloon off the ground. After that it’s entirely on you to keep it afloat. And I don’t know of any better way to do that than by keeping a detailed error log and reviewing it constantly.

So what should go in your error log? Each entry should include the following things: the problem, the nature of your error or difficulty, what you tried the first time and why it worked or didn’t work, a complete solution, and any points you want to take away from the problem. An entry might look something like this:

The good news is that very few math problems are unique. Any concept that you learn from any specific GMAT problem will most likely be applicable to some other problem. But first you have to understand the concept well and retain it. If you review the above entry in your error log two or three times per week, your chances of forgetting any of the key concepts will be very low. What’s more, you’ll be more likely to think of using them if you encounter a similar problem.

-GMAT Math Pro
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What weird old tip did an unemployed single mom use to score an 800 on the GMAT?

It’s been a banner year for unemployed single moms. Their waistlines are shrinking, their monthly incomes from working at home are skyrocketing, and their teeth are whiter than freshly fallen snow. And it’s all thanks to weird old tips! Naturally, I began to wonder if members of this blessed demographic ever stumble across any weird old GMAT preparation tips. Not knowing any single, unemployed moms myself, I asked my girlfriend if she’d be willing to quit her job, get pregnant, and break up with me in order to assist me in this crucial research. Alas, she was not. So, the question posed in the blog title will remain unresolved…….for now. In the meantime, here’s the weirdest tip I could think of:

You’re probably familiar with the idea that a great way to test your understanding of a concept is to try to teach it. If you’ve ever had to teach anything, you know it’s true. The gulf between looking at a problem as a student and saying “Oh, yeah, I get it,” and being able to explain the problem confidently and coherently can be huge. Ideally, we would all just get jobs as professional GMAT tutors while we prepare for, master, and ultimately score in the 99th percentile of the test. But those jobs can be difficult to get before you master the test. So what’s to be done? Family and friends usually have limited patience for listening to you stammer your way through an explanation of the differences between combinations and permutations. And dogs, while willing to listen, are generally frustrating students. So if you want to apply this weird tip, you’re going to have to get creative. Here’s how:

1. Participate in a GMAT discussion forum

Many of you may already be doing this, but make sure you’re making the most of this opportunity. Don’t just go there to post questions for experts to answer; help other users, too. Force yourself to write clear, comprehensive explanations. It’s a great way to reinforce knowledge and reveal any holes in your understanding that you may not have realized were there. Don’t worry about telling someone the wrong thing. Just do your best, and someone will correct you if necessary.

2. Make instructional GMAT videos

I discovered this trick when I was making instructional GMAT videos. The benefits are similar to those you get from teaching, except you don’t have to worry about the hardest part: finding a student. Pick a relatively challenging but not impossible problem and try to record a clean, polished explanation. Try to make sure it would be clear to someone whose level of understanding is below yours. Your first take will probably be pretty bad. Keep trying until you get something decent. If you can’t do it, it probably means you don’t understand the concept as well as you should. Write down exactly what you’re struggling with and discuss it with your private tutor or the experts in your online GMAT community.

3. Find a study partner

If you don’t have any friends who are studying for the GMAT, find someone through an online community who is at approximately the same skill level that you are and would be willing to meet with you once a week via Skype. Then do something like the following: select ten challenging problems and divide them into two sets of five. On one group you’ll be the teacher. On the other group you’ll be the student. Over the next week, solve the ones that you’ll be teaching. Be ready not only to explain the problems but also to field any questions the other person may have about them. Consult experts if necessary. For the other five questions, read them and try to solve them. Don’t go to extraordinary lengths to solve them, but think about them enough so that you’ll have some good questions to ask the other person when they teach them to you the next time you meet.

GMAT Math Pro speed tip: Multiplying by 11 or 15

To multiply a two-digit number, n, by 11 quickly, do the following: Add the digits of n and insert this sum between the digits of n. For example, 11*36=396. Calculate 3+6=9 and insert 9 between the 3 and 6 of 36 to get 396. If the sum of the digits is a two-digit number, add one to the ten’s digit of n, and insert the unit’s digit between the new two-digit number. For example, 11*87=957. Calculate 8+7=15. This is a two-digit number, so add one to the ten’s digit of 87 to get 97. Now, insert the unit’s digit, 5, between the 9 and 7 to get 957. Try this trick a few times and verify your answers with a calculator.

Multiplying any number by 15 can be made easier by thinking of 15 in different forms. For example, 15 is the same as 1.5*10. If we want to multiply 28*15, this is the same as multiplying 28*1.5*10. But why would we want to do this? The trick is realizing that 28*1.5 is the same as increasing 28 by 50%. In other words, all you have to do is take half of 28, 14, and add it to 28 to get 42. Then multiply 42 by 10 to get the answer, 420. So, this method works well with even numbers because it is easy to see what half of an even number is. With odd numbers it is slightly more difficult but still very doable. Let’s say we want to calculate 13*15. You could do it the same way: 13*15=13*1.5*10=19.5*10=195. Or, you could think of 15 as being the same as 3/2 times 10. Now 13*15=13*(3/2)*10. Now you could multiply 3*13 to get 39, and divide 39 by 2 to get 19.5 and then multiply by 10 to get 195. Use whichever method makes you more comfortable.

Despite the fact that 11 and 15 are only 2 out of infinitely many numbers, these tricks are applicable fairly often on the GMAT. Any new method takes some getting used to, so try a few practice problems until you are convinced that these tricks work. If you’re too paranoid to try new things like this on an actual GMAT, you can at least use them as a way to check answers you get from more traditional multiplication methods.

GMAT Math Pro speed tip: Which fraction is greater?

Whenever someone asks me which of two fractions is greater, and I am, for whatever reason, obliged to take their query seriously, I’m always hopeful that the two fractions will turn out to be something like 999/1000 and 1/1000000. This is rarely the case, but I’m almost as grateful to get something like 2/21 and 3/19. In this case I can reason, without tedious calculation, that \frac{3}{19}, with its bigger numerator and smaller denominator, certainly has the greater value. But what if the fractions turn out to be something like \frac{2}{29} and \frac{3}{43}. Now it is not quite so easy. \frac{3}{43} has the bigger numerator, but it also has a bigger denominator. Is the increase in the denominator enough to offset the increase in the numerator? It’s difficult to say. In the old days your only option would be to get out your abacus, learn how to use an abacus, and compute the equivalent form of each fraction over a common denominator. \frac{2}{29} becomes \frac{86}{1297} and \frac{3}{43} becomes \frac{87}{1247}. So \frac{3}{43} wins by a nose. The increased speed comes in realizing that there is no point in computing the actual value of the denominator. You KNOW the denominators will be the same because that’s the way you’re constructing them. The whole comparison depends on the numerators. So why not skip straight to them?

The fastest way to do it is to think of it as being akin to cross-multiplication. You’re comparing \frac{2}{29} to \frac{3}{43}. Start with the bottom right number, 43, and draw a diagonal to the top left number, 2. Multiply along this diagonal to get 86, and write this to the left of \frac{2}{29}. Then start with the bottom left number, 29, and draw a diagonal up to the top right number, 3, and multiply along this diagonal to get 87. Write this number to the right of \frac{3}{43}. 87 is the bigger number, and it is next to \frac{3}{43}, so \frac{3}{43} is the bigger fraction.

To summarize, if you want to compare the the sizes of two fractions, \frac{a}{b} and \frac{c}{d}, if ad>bc, then \frac{a}{b} is bigger and if bc>ad then \frac{c}{d} is bigger. But don’t think of it this abstractly. If it comes up, your eyes should immediately be moving along those diagonals from bottom to top to see which side has the bigger product.

Now if someone asks you to compare the sizes of two fractions, whether it be a friend, neighbor, or the computerized testing agent of an organization that controls your future, you can confidently answer in a matter of seconds.

–GMAT Math Pro
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Don’t multiply yet!

Suppose you are trying to solve the following problem:

If p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28, what is the value of \displaystyle \frac{p}{q}?

Here is one (bad) option for solving the problem:

p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28

2184p=17472q (Multiply everything together)

\displaystyle p=\frac{17472q}{2184} (Divide both sides by 2184)

\displaystyle \frac{p}{q}=\frac{17472}{2184} (Divide both sides by q)

\displaystyle \frac{p}{q}=8 (Simplify.)

This method produces the correct answer, 8, but it requires a lot of complex arithmetic, which wastes time and increases the likelihood of mistakes.

Here is a better way to solve it:

\displaystyle p \cdot 12 \cdot 13 \cdot 14=q \cdot 24 \cdot 26 \cdot 28

\displaystyle p=\frac{q \cdot 24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14} (Divide both sides by 12 \cdot 13 \cdot 14}

\displaystyle \frac{p}{q}=\frac{24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14} (Divide both sides by q)

\displaystyle \frac{p}{q}=\frac{24 \cdot 26 \cdot 28}{12 \cdot 13 \cdot 14}=\frac{2 \cdot 2 \cdot 2}{1 \cdot 1 \cdot 1}=8 (Simplify.)

Notice that in the last line of this solution, I divided the bottom numbers into the top numbers first instead of multiplying everything together. Dividing 24 by 12, 26 by 13, and 28 by 14 is a whole lot easier than multiplying 12 by 13 by 14 and 24 by 26 by 28 and then dividing 17,472 by 2,184.  The principle that we are exploiting is simple: smaller numbers are easier to work with.  And, generally, division produces smaller numbers than multiplication.  Hence, the title of this entry.  Don’t be in a rush to multiply everything together. Keep the numbers in factored form, and simplify anything you can with division first.  Don’t multiply until the end.


Writing Neatly

Note: This is the fourth entry of a series on preventing careless mistakes.  To start at the beginning, scroll down to the first entry, “Preventing Careless Mistakes”.

Get in the habit of writing your work in a clear, legible, organized fashion.

Those of you who already write your work clearly and legibly can probably skip this entry. For those of you who don’t, I’m going to try as hard as I can to sell you on it. The deceptive thing about writing your work sloppily is that it works out fine most of the time, so people think it doesn’t matter. But any chaotic element you introduce into a situation necessarily increases the likelihood of an error. It’s inevitable. The effect may be small, but its cumulative effect over several problems can be significant. Suppose you have a 97% chance of completing a given problem without making a careless error if you write your work sloppily, but you could increase that probability to 99% if you write your work neatly. That’s only a difference of 2%, but small differences can add up. In the former case, you’d have about a 32% chance of making it through an entire GMAT math section without making a careless mistake. In the latter case, your likelihood increases to about 69%. That’s a significant increase. Now, this is somewhat over-simplified, but it does illustrate the potential for small differences to accumulate to something significant. If you’re serious about maximizing your GMAT score, you should always take advantage of opportunities like these.

One obvious benefit to writing your work legibly is that it will reduce arithmetic errors. If you write a 3 that looks like an 8 and you forget that it’s supposed to be a 3, you’ve got a serious problem on your hands. Best-case scenario, you waste time trying to figure out your mistake. Worst-case scenario, you get the question wrong. One thing about the GMAT that you may not know if you’re just starting your preparation is that you’re not allowed to use regular paper and pencil when you do your work. Instead, they give you a black marker and books of laminated graph paper that are commonly called noteboards. They’re not awful, but they do take some getting used to. The tip of the marker is a lot thicker than the tip of a pencil, so it is more difficult to write neatly and precisely. Some companies sell replicas of these books and markers. If you’re nervous about dealing with this on test day or if your handwriting is naturally atrocious, I would recommend buying some of these so you can practice writing legibly on them.Writing your work in an organized fashion is important because it makes it easier to check for errors. There’s nothing quite so exasperating as finishing up a lengthy calculation, discovering that your answer is not one of the choices, and then having to dig through several lines of sloppy, jumbled, randomly placed work to find your error. Make sure you have plenty of space to finish a problem when you decide where to start it on your noteboard. Don’t worry about running out of pages. Remember that you can request more noteboards if you need them.

The final benefit of writing your work neatly comes when you are taking practice tests. One of the best things you can do to improve your math score is to thoroughly analyze the mistakes you make when you practice. When I started taking practice GMAT exams years ago, I would just grab whatever paper I saw lying around to use as my scratch paper. I would use envelopes, receipts from the grocery store, whatever I saw that had any blank white space. However, if you’re trying to review your work to see where you went wrong and half of it is written on the back of your gas bill and the other half is scrawled between the nutritional information on a Subway napkin, it can be very difficult to follow. I ended up buying a notebook specifically for GMAT math problems and writing my work in a way that I could still follow it weeks later. You should do this, too.

Categorizing Your Mistakes

Note: This is the third entry of a series on preventing careless mistakes.  To start at the beginning, scroll down to the first entry, “Preventing Careless Mistakes”.

When you make a careless mistake, determine its nature so that you can be alert for it next time.

Careless mistakes come in a variety of forms, so it is important to identify the types that you are most susceptible to. Once you are aware of the things you fall for, you can maintain a heightened alertness for them. For instance, the first time I moved into an apartment with a door that locked automatically, I was constantly locking myself out of my apartment. Then I would have to sit in the hallway for hours, like an idiot, while I waited for the maintenance guy to come let me in. Yet these experiences, while terrible, did not seem to deter me from locking myself out of my apartment. It remained a regular part of my routine until I instituted a new rule: whenever I opened the door to go outside, I was not allowed to close it until I took my key out and held it in my hand. I don’t know if this is how most people prevent themselves from becoming locked out of their apartments, but it worked for me. You will find that similar workarounds can help prevent careless mistakes on math problems. Here are some common careless mistakes and tips for avoiding them:

1. Overlooking a key detail.

This kind of careless mistake occurs when you fail to incorporate an important detail from the given information into your solution. For example, consider the following problem:

If two distinct positive integers add to 38, what is the largest possible value for their product?

A. 37
B. 105
C. 280
D. 360
E. 361

If you picked E. 361, you picked the sucker answer. The correct answer is D. 360. Someone who chooses ‘E’ most likely glossed over the word “distinct”. 19*19=361, but 19 and 19 are not distinct. The best you can do with two distinct positive integers that add to 38 is 20*18=360.

Tip: If you find yourself making this kind of mistake repeatedly, slow down when you read the questions. Stop at each word and ask yourself how its presence will impact the solution.

2. Answering the wrong question.

This can happen if you assume you know what the question is going to be without ever actually verifying it. Here is a simple example:

If 2x-5=13, what is the value of 2x?

A. 2
B. 5
C. 9
D. 18
E. 20

Here the sucker answer is ‘C’. In high school algebra, the goal of practically every single problem is to find the value of x. The GMAT is not so predictable. They like to throw things like this in to make sure that you are paying attention. People who pick ‘C’ are probably guilty of answering the wrong question, “What is x?” instead of the actual question, “What is 2x?” The correct answer, of course, is D. 18.

Tip: Before you submit an answer, always double check that you are answering the exact question that they asked.

3. Answering in the wrong units.

This is  essentially the same as overlooking a key detail, but it comes up frequently enough that it deserves special attention. Here is a simple example of how it could come up:

How many minutes will it take David to drive 120 miles if he drives at an average speed of 40 miles per hour?

A. 2
B. 3
C. 60
D. 120
E. 180

Here, the sucker answer is B. 3. It takes him 3 hours to drive 120 miles, but the question is asking for the number of minutes. So the correct answer is E. 180.

I usually find that people who make a lot of mistakes with units don’t have a proper appreciation for how important units are. So, if you find yourself making these kinds of mistakes, take some time to think about how critical using proper units is to communicating information accurately. For example, let’s say that you and I have the following exchange:

You: How far is it to Lubbock, Texas from here?
Me: Twenty-four!

You would be rightfully angry. What a stupid answer! Twenty-four what? Miles? Kilometers? Feet? Light-years? Maybe I’m even answering in hours. Clearly my answer is useless if I don’t include the units.

Also, try thinking about how changing the units of an answer can change an answer from plausible to ridiculous. 72 inches is a plausible height for a man. 72 feet is probably not. 3 hours is a reasonable amount of time to drive 120 miles. 3 minutes is probably not. Units are everything.

The point of all this is that sometimes there is a disconnect between how we perceive things in the real world and how we perceive them in a GMAT math problem. I don’t expect that any of you would ever, in the real world, make the mistake of saying that a man is 72 feet tall. However, people make mistakes like that on the GMAT without a second thought. GMAT problems tend to come across as artificial, like something that was isolated in a lab, lacking the nuances and messiness of real life. As a result, we let our guard down and don’t apply the same tests of plausibility that we would apply automatically in the real world. You must be vigilant about not letting this happen.

Tip: If you see any kind of unit in a GMAT math problem, keep your guard up. Make a special point to check that the answer you obtained is plausible in terms of the units given and that you handled all unit conversions properly.

4. Making an arithmetic error

Having to do arithmetic mentally or by hand on the GMAT can come as a rude awakening if you have come to rely on calculators. The arithmetic required is not very difficult, but if you’re out of practice it can slow you down significantly. It can also create terrific opportunities for careless mistakes. If you do feel rusty with arithmetic, here are a few things you can try.

First, stop using a calculator for arithmetic in your daily life unless it is absolutely necessary. If arithmetic doesn’t come up much in your daily life, look around for opportunities to practice. See how fast you can add up numbers that you see on license plates or in phone numbers. If something is on sale for 30% off, try to compute the sale price in your head. At the grocery store, try to compute the price per ounce of different products and see how close you can get. If arithmetic comes up that you have to use a calculator for, try to guess a reasonable estimate for the answer first.

Second, find a simple arithmetic game to play online or find some free worksheets that you can print out. The games are easy to find by googling “timed arithmetic challenge”. Once you find one you like, play it repeatedly and see how high you can make your score. If you want to practice on paper, google “free arithmetic worksheets” and find some worksheets to print out.

Third, learn some mental math shortcuts. There are several shortcuts available that you can use to simplify certain calculations. For example, if you want to multiply a number by 15, just add half of that number to itself and multiply the result by 10. So, if you want to multiply 24 by 15, take half of 24(which is 12) and add it to 24 to get 36. Then multiply 36 by 10 to get 360. Thus, 24*15=360. There are plenty of other useful tricks like this that make arithmetic faster and easier. In the future, GMAT Math Pro will have a series of videos detailing the tricks. For now, try googling “mental math tricks” and try to find some that you think are useful.

Finally, when you do arithmetic mentally or on paper, it is important to be able to check if your answer is at least in the right ballpark. To do this, round the numbers you are working with to numbers that are close but more manageable. For example, suppose you are trying to determine what 53% of 612 is. 53% is close to 50% and 612 is close to 600. 50% of 600 is 300. Since the original numbers were slightly more than 50% and 600, we would expect the actual answer to be slightly higher than 300. So if you get something like 31 for your answer, you know you made a serious error. Of course, if you do get an answer slightly higher than 300 it doesn’t necessarily mean you are correct. But you can use this technique to catch big mistakes like putting the decimal point in the wrong spot.

Having a Plan

Note: This is the second entry of a series on preventing careless mistakes.  To start at the beginning, scroll down to the first entry, “Preventing Careless Mistakes”.

Outline a general plan that you will use for every problem.

A good general problem-solving plan for a GMAT math problem looks something like this:

1. Read the problem carefully.
2. Solve the problem, double-checking each step as you go, and mark the answer.
3. Reread the problem and make sure your solution is exactly what the question is asking for. Also look for any details you may  have missed the first time that would impact the solution.
4. Check that you marked the answer you intended to.
5. Submit your answer.

Read. Solve. Reread. Check. Submit.

Some of these steps are obvious; you probably won’t forget to read the problem or submit your answer. But keeping the complete framework in your mind as you solve problems will help you remember to do the important steps. Repeat it in your mind consciously until it becomes second nature.

Read. Solve. Reread. Check. Submit.

Preventing Careless Mistakes

Careless mistakes, by their very nature, are difficult to anticipate and prevent. They can be extremely frustrating because they seem like they could have been so easily avoided. I frequently hear people say things like, “Well, I missed 7 questions, but 5 of them were careless, so…..”

But you shouldn’t dismiss your careless mistakes so cavalierly. Don’t just think, “Well, I’m sure I’ll care on test day!” ‘Careless’ is actually a bit of a misnomer. Careless mistakes don’t really reflect a lack of caring but a lack of attention to detail. And attention to detail is a habit that must be cultivated; it’s not something you can just switch on when you want it. If you consistently make careless mistakes on your practice tests, it is imperative that you take specific steps to establish better problem-solving habits.

There are three main things you should do to prevent careless mistakes:

1. Outline a general plan that you will use for every problem.

2. When you make a careless mistake, determine its nature so that you can be alert for it next time.

3. Get in the habit of writing your work in a clear, legible, organized fashion.

Over the next few days I’ll talk about each of these in more detail.

These three things are important, but how successfully you incorporate them will depend heavily on your state of mind. When you solve math problems, it is crucial that you take the attitude that this is your job and that you are going to do it like a professional. Your boss expects you to be a professional at work. Demand the same of yourself on the GMAT. Do the problems carefully but quickly. Don’t tolerate mistakes. If you discover you have a weakness, do what it takes to correct it. People often give up on new techniques early because they feel awkward and unfamiliar. This is understandable, but if you want to succeed, you cannot be one of these people.