Math Workout for the GMAT
Part II General TestTaking Tips
The GMAT, like all other standardized tests, follows certain predictable patterns. That’s why it’s “standardized.” The test writers must follow these patterns so that everybody who takes a GMAT gets tested on the same criteria. Otherwise, their scores wouldn’t be comparable and schools couldn’t use those scores to evaluate applicants.
By learning the methods and patterns that the test writers follow, you can use some general testtaking strategies that will help you beat the test. These guidelines apply no matter what specific topic the question covers, so they are useful throughout the test. Don’t underestimate how much these tips can help your score.
PACING
Time is your most precious resource on the GMAT. In the Math section, you have only 75 minutes to answer the 37 questions; be sure you use your time wisely. You should keep three pacing goals in mind as you take the test:
1. Answer every question. You’ll receive a significant penalty to your score if you don’t answer all of the questions in each section. Therefore, you should answer every question, even if you have to guess on the last few. However, guessing on more than three or four questions at the end can also have a serious negative impact on you score. So, it’s very important to use your time wisely and pace yourself to at least come close to finishing each section and then guess on the remaining few questions.
2. Start slowly. On a CAT, the computer first gives you a medium question. If you answer it correctly, the computer gives you a slightly harder question. If you answer it incorrectly, the computer gives you a slightly easier question, and so on. The idea is that the computer will zero in on your exact level of ability fairly quickly and make a finely honed assessment of your abilities. Because of this system, the earlier questions have a greater impact than the later questions, in that you are attempting to prove to GMAC that you are a highcaliber student and can breeze through hard questions. Go slower and be more careful on the earlier questions. Don’t spend five minutes on a question; just try to minimize your chances of making a careless mistake. As you progress through the section and get to questions that have less impact on your score, you should gradually pick up some speed. If you make a careless mistake on a question near the end, it will have a very small effect on your final score.
3. Don’t waste time on “impossible” questions. Almost everyone comes across a handful of questions that are too difficult to do. No matter how much time you spend staring at them, you’re not improving your chances of answering them correctly. Cut your losses on these questions. Once you realize that it’s an impossible question, take an educated guess (more on this later) and move on. Spend your time on the questions that will get you points.
Let’s integrate these ideas into an overall strategy. The following chart shows how your pacing and accuracy goals should change as you progress through the Math section.
On questions 1 to 10 of the Math section, accuracy is your primary goal. Your performance on these questions sets the tone for the whole section. Think of it as making a good first impression on someone you just met. You should aim for 90 to 100 percent accuracy, and allow yourself plenty of time to work each question thoroughly so that you avoid careless mistakes. If you do encounter an “impossible” question, spend a reasonable amount of time using good elimination methods to make the best possible guess.
In questions 11 to 20, you need to pick up speed so that you will be on pace to answer every question. You can allow yourself an average of two minutes per question. Therefore, you must become more selective. If a question seems likely to take an excessive amount of time, you should quickly eliminate some answers and take a guess. You cannot afford to spend several minutes working a problem and then end up guessing after all. If you hit your accuracy goal of 75 percent or higher, your score will still be increasing.
With questions 21 to 30, your objective shifts from rapidly raising your score to solidifying and further improving your score. If you hit your accuracy targets in the earlier questions, your score should be pretty high, and the remaining questions will correspondingly be more difficult. If you can keep improving your score, that’s great, but first you must “do no harm.” Time is running short, so you will need to be more selective. You can afford to aggressively eliminate and guess on tougher problems so that you will have time to thoroughly work the others. For example, you could spend two minutes on each of six to seven questions, and make educated guesses on the others.
In the end game, questions 31 to 37, the top priority is answering every question, one way or another. Ideally, you will have time to read each one and either solve the problem or make a quick guess. However, you should make absolutely certain that you answer all 37 questions, even if that means blindly choosing answers in the last minute.
GUESSING
No matter how prepared you are, there’s always the possibility that you could run into a problem that you just don’t know how to solve. If this happens, stay calm, eliminate any answers you can, and make the best guess possible. As a general rule, 3 minutes is as much time as you should spend on any given problem. If you get to 3 minutes and you’re not on the verge of solving the problem, then it’s time to cut your losses, eliminate any answers you can, and take a guess. Remember that the test is adaptive, so the material gets harder the better you’re doing. Therefore, you’re probably going to run into a few problems that you don’t know how to solve. In that case, it’s better to make the best guess you can and move on, rather than to stubbornly waste time on a question. Don’t forget that it could be experimental!
USING NOTEBOARDS
Because the questions are presented on a computer screen, you do not have a test booklet on which you can scribble notes. You can’t write directly on the problem to label diagrams, cross off answers, and so forth. In an effort to be environmentally conscious, the GMAT is no longer administered with scrap paper. Instead of paper, you will be given noteboards, which are reusable dryerase sheets for your scratch work. If you manage to fill up all of the space on your noteboards, you can raise your hand and have the test administrator clean them for you. So, use them! Don’t try to do calculations in your head in order to save time. Chances are the time you’ll spend thinking about the numbers is at least as much time as it would take you to write them down, and the accuracy of mental math is never as good.
When you first sit down at your computer—before the timer starts—take a few minutes to divide your noteboards into eight boxes per board, leaving one column of boxes empty for any formulas you might want to write down or outlines you may want to make for the AWA. In each box, write A, B, C, D, E so that you can physically cross off answers as you eliminate them. This will help you keep track of where you are as you work the problem.
When you practice questions from this book, you should do your work on separate sheets of scratch paper, just as you do with the noteboards on test day. Doing so will help you get used to copying stuff to your noteboards.
READ CAREFULLY
For most people, a substantial number of their wrong answers in the Math section are caused by nothing more than reading errors. You may know exactly how to do the math, but that won’t help you if you answer the wrong question. Be sure to read each question carefully. Take a look at some examples of traps for unwary readers.
1.If 3x + 12 = 21, then x + 4 =
3 

4 

5 

7 

9 
Most people will start solving the equation. Subtract 12 from each side to get 3x = 9. Then divide by 3 to get x = 3. Aha, the answer is (A)! Um…no, actually. Although x = 3, that’s not what the question asked. If x = 3, then x + 4 = 7, so the answer is (D). It is usually a good idea to reread the problem, especially the question stem (at the end), before choosing an answer choice.
Here is another example.
2.In a classroom containing only fifth and sixthgraders, fifth graders are seated in of the desks and sixthgraders are seated in of the remainder. Sixthgraders are seated in what fraction of the desks in use?
A good way to start with such a problem is to choose a number for the desks in the room. (This handy strategy, called Plugging In, will be covered in Part III). Suppose there are 60 desks in the room. That means there are fifthgraders in × 60 = 30 of the desks, with 60 − 30 = 30 desks remaining. So there are × 30 = 20 desks with sixthgraders. Therefore, sixthgraders take up = of the desks, and (E) must be the answer. Right? Again, sloppy reading has led to a wrong answer designed to trap you. The question asked for the fraction of the desks in use containing sixthgraders, so the answer should be = , or answer choice (D).
If you avoid reading mistakes, these questions may seem relatively easy, but that’s exactly the point. You don’t want to choose wrong answers when you know how to do the math. Careless mistakes can wreak havoc on your GMAT score.
There are several ways to decrease the number of reading mistakes you make in the Math section. As suggested, reread the question, particularly the stem, before selecting your answer. Also, when you copy numbers and diagrams to your scratch paper, doublecheck your notes before working the problem. Finally, pay extra attention to key phrases such as “of the…” in word problems.
ANSWERS THAT ARE TOO OBVIOUS
Choose a number and write it down.
Don’t read any further until you’ve chosen your number. You’ll need to refer to this number in a few minutes.
When the test writers write questions for the GMAT, they need to generate questions that cover the range of difficulty from pretty easy to really tough. However, they’re restricted by the topics they’re allowed to test. They can’t write questions about calculus or other difficult topics in order to get hard questions. Instead, they have to use the same topics, yet somehow they have to make the questions hard. One way the test writers make a question hard is to include things that will trick you into choosing the wrong answer.
People tend to think in predictable patterns. For example, look at the number you wrote down earlier. You could have chosen any number, including 0.08, −7.5, , , and similar numbers. However, no one chooses those numbers; almost everyone chooses a whole number such as 1, 2, or 3. In addition, most people choose a whole number from 1 to 10. People choose these numbers because they are “programmed” to think that way.
This type of thinking is called the obvious answer choice response. The obvious answer choice is the choice that is chosen by the completely predictable person who lives inside each of us. When you look at a math problem, watch your initial reaction to the question—your “five seconds or less” solution. That’s the too obvious answer.
The test writers use those predictable thinking patterns against you. They write questions in such a way that being predictable will lead you to the wrong answer. You should be suspicious of a solution that you come up with too quickly. If you don’t use at least 30 seconds to solve the question, you should doublecheck your answer. That doesn’t mean that the easy answers are never correct, just that they are usually traps. Look at this next example.
1.At a certain store silk scarves are sold at an everyday price of 20% off the normal list price. If the scarves are sold at a sales price of an additional 10% off the everyday price, the sales price is what percent of the normal list price?
28% 

30% 

70% 

72% 

80% 
The predictable person inside us says, “Simple. A 20% discount plus a 10% discount is a 30% discount.” The trap answer is (C) or, if you misread the question, (B). So you want to be suspicious of those answers. If you come up with one of those answers, doublecheck your work and see if there’s a different approach you could take. Take a look at another example.
2.If w + 2x = 150, 2w + 3y = 100, and x + 3z = 50, what is the value of w + x + y + z?
12.5 

20 

50 

100 

It cannot be determined from the information provided. 
Your first instinct is probably something like, “Wow. There’s no way I can solve for all those variables. With four variables, I’m going to need four equations!” If so, you’re absolutely right. However, the too obvious answer, (E), is not the correct answer.
EXPLANATIONS FOR THE TOO OBVIOUS EXAMPLES
1.DSuppose a scarf has a retail price of $100. The store sells that scarf at an everyday 20% discount, or $80. During the sale, the store reduces the everyday price another 10%, or $8 (10% of $80). So the sale price is $72, which is 72% of the retail price.
2.DCombine all the equations into one and you get (w + 2x) + (2w + 3y) + (x + 3z) = 150 + 100 + 50. If you simplify each side, you get 3w + 3x + 3y + 3z = 300. Divide everything by 3 and you find that w + x + y + z = 100. Even though you can’t solve for each individual variable, you can answer the question.
DRILL 1
Identify the answer choices that are too obvious for each of the following questions, and then figure out the correct answers. The answers can be found on this page.
1.If Alex drives 80 miles per hour from her house to work and 100 miles per hour from work to her house, and drives along the same route both ways, which of the following is the closest approximation of her average speed, in miles per hour, for the round trip?
80.0 

88.9 

90.0 

91.1 

100.0 
2.For which of the following values of n is (−0.5)^{n} the greatest?
5 

4 

3 

2 

1 
3.( + + )^{2} =
27 

18 

9 

3 

3 
PROCESS OF ELIMINATION
Process of Elimination (POE) is one of the most important tools you can utilize to help answer some of the tougher GMAT questions. By properly utilizing a little bit of question awareness and POE, you can often eliminate answer choices for those tough questions that may have you stumped.
POE is a great tool, but it is no substitute for familiarity with the material. While POE may not get you to the correct answer on a question you don’t know how to do, it can help you eliminate wrong answer choices before you make a guess. When you work through practice problems, make sure to show on your note board which answer choices you have eliminated. Doing so will help you prepare for harder questions. Because the GMAT is an adaptive test and you must answer every question, the ability to guess intelligently on questions you don’t know how to answer is a useful tool. Take a look at this example from Drill 1.
1.If Alex drives 80 miles per hour from her house to work and 100 miles per hour from work to her house, and drives along the same route both ways, which of the following is the closest approximation of her average speed, in miles per hour, for the round trip?
80.0 

88.9 

90.0 

91.1 

100.0 
The correct answer to this problem is (B), as explained in the solutions at the end of this chapter. However, let’s say that you didn’t have time to answer the question. What answers can you eliminate using POE?
The problem asks for Alex’s average speed. If Alex drives 80 miles per hour one way and 100 miles per hour the other way, her average speed must be somewhere in between those two speeds. So, her average speed cannot be 80 or 100 miles per hour. Without doing a single calculation, you can eliminate choices (A) and (E). If you did not know how to proceed any further with this question, you have already narrowed this question down to three possible choices.
Guessing from three answer choices is better than guessing from five, which means that POE has already worked for you. However, you can eliminate one more answer choice. What is the too obvious answer here? Alex travels the same route at two different speeds, 80 and 100 mile per hour. The question then asks for the average of those two speeds for the trip. So, the too obvious answer is choice (C), 90.0. This is the too obvious answer that the test writers want you to pick. But if you are aware that the GMAT test writers use answers that are too obvious, you can eliminate (C). Now, you can guess from only two answer choices. You have a 5050 shot at getting the question correct and you haven’t even done any calculations yet!
The Person Who Wrote the Problem Had to Solve the Problem
When you studied math in school, you probably didn’t see many multiple choice questions. After all, math teachers like to see your work. In fact, they often care more about the work done to solve the problem than the actual answer you got to the problem. Standardized tests such as the GMAT are different. Here, the test writers only care about the answer you get to the problem. So, they give you multiple choice questions.
But when they give you multiplechoice questions, they actually do you a favor. You can often find the answer to the problem more easily by simply using the answers in some way. As we’ve already seen, you may be able to eliminate wrong answers and either get the right answer or get close to the right answer. Another way to use the answers is to Plug In.
Plugging In is a great strategy to use when the problem contains variables or when there are pieces of information that are missing. In these cases, plugging in a number for the variable or missing piece of information is a quick, easy way to take advantage of the fact that the person who wrote the problem also had to solve the problem! You probably learned to check your work by using a number when you studied algebra in school. That’s all Plugging In. Here, the person who wrote the problem came up with the possible expressions.
When applying the Plugging In strategy, it is usually best to try a number that is easy to work with like 2, 3, 5, or 10. When dealing with percentages, the number 100 is the easiest number to begin with. Look at the same problem below to see Plugging In in action.
1.The output of a factory is increased by 10% to keep up with rising demand. To handle the holiday rush, this new output is increased by 20%. By approximately what percent would the holiday output of the factory now have to be decreased in order to restore the original output?
20% 

24% 

30% 

32% 

70% 
The question asks for the output of the factory. It gives information about the increase in production for the factory, and then asks how much that production would need to be decreased to match the original output. However, the problem never gives any information about what the original output was, so this is a good chance to Plug In. Since the problem is dealing with variables, plug in the number 100 for the original output. Now work the problem.
The original output of the factory (100) is increased by 10%. Since 10% of 100 is 10, the new output is 110. Then, the output is increased again by 20%. 20% of 110 is 22, so the new output of the factory is 110 + 22 = 132. The question now asks by what percent the holiday output of the factory (132) would have to be decreased to get back to the original output (100). The difference between 132 and 100 is 32, so to find the percent decrease, determine what percentage 32 is of 132. The equation is 32 = (132). Divide both sides by 132 and multiply by 100 to find that x is approximately 24%. So, the correct answer is (B).
As you can see, Plugging In is a valuable strategy that can take an otherwise difficult problem and turn it into a series of simple steps and calculations. However, just for practice, look at that same problem again and try to eliminate answer choices using POE.
What’s the obvious answer here? Somebody who answers the question too quickly sees the percentages of 10% and 20% and would be tempted to add them together, which results in 30%. Reading a little more of the problem, this person might subtract 30% from 100 to yield 70%. Notice how both of these answer choices are present to trick you. Eliminate them.
Write Things Down
Many GMAT test takers make the mistake of trying to do too much work in their heads. While it may seem that you might save time by not writing your work down, you are more likely to make silly mistakes. In other words, saving a few seconds by not writing your work down isn’t worth getting the wrong answer. The test writers are very good at figuring out the mistakes that people make when doing each problem. So, don’t fall into this trap. Always remember to: write down ABCDE on your note board for every question, redraw any figures in the problem, write down any equations you may need, and label everything. Geometry problems are a good way to illustrate the usefulness of this level of organization. Take a look at the following example:
If, in the figure above, square ABCD has an area of 25 and is inscribed in the circle with center O, what is the area of the circle?
π 

π 

π 

25π 

50π 
Begin working this problem by writing down ABCDE on the note board and redrawing the figure. The problem mentions the area of the square and asks for the area of the circle, so write down those two formulas. The formula for the area of a square is A = s^{2} and the formula for the area of a circle is A = πr^{2}. By writing these down, you now know that to find the area of the circle you have to find a value for the radius of the circle. Notice that by writing things down, you can very quickly discover what information you need to answer a question correctly.
Since the area of the square is 25, then 25 = s^{2} and s = 5. Be sure to go back to the figure and label the side of the square as 5. Now, it’s time to think about the angles that were created when AC was drawn in square ABCD. Each of the four angles in a square is 90°. Since all the sides of the square have the same length, AC bisects angles BAD and ACD. So, each of those is 45°. Again, be sure to write that information on the figure you drew. AC is the hypotenuse of a 45°45°90° triangle that has two sides of length 5. So, the length of AC is 5. (We’ll cover the 45°45°90° triangle relationship when we discuss geometry.) Again, be sure to write that length on the figure. Now, you can see that the radius of the circle is . Go back to the formula for the area of the circle and plug in the length of the radius: A = π = π. So, the answer is (C). But, did you really want to try to do all of that work in your head just to potentially save a few seconds? Accuracy is more important when taking the GMAT. Did you notice how answer (A) is what you would get if you correctly determined the radius of the circle but didn’t square the radius? Or, that answer (E) is what you get if you use the length of AC as the radius? These answers are there to trap test takers who try to do too much of the work in their heads.
DRILL 2
Answer these questions. Try to apply POE and Plugging In where possible. Don’t forget to write things down. The answers can be found on this page.
1.If a $3,000 deposit is made into a savings account that pays 6 percent interest, compounded monthly, and there are no other deposits or withdrawals from the account, how much money, rounded to the nearest dollar, is in the account at the end of one year?
$2,160 

$3,180 

$3,185 

$5,160 

$6,037 
2.If Steve’s original salary is increased by 5 percent and then, 3 months later, his salary is increased again by 20 percent, then Steve’s raises are what percent of his original salary?
25% 

26% 

27% 

30% 

32% 
ANSWERS AND EXPLANATIONS
Drill 1
1.BThe too obvious answer to this question is choice (C). The first instinct is to average 80 and 100 to get 90. Instead, Plug In to solve this question. Pick a number that is easy to work with for the total distance Alex must drive one way. Since Alex drives at 80 miles per hour and 100 miles per hour, pick a number that is divisible by both 80 and 100, such as 400. If Alex drives 80 miles per hour for 400 miles, it takes her 5 hours to reach work. If Alex drives the same route home at 100 miles per hour, the return trip takes her 4 hours. Alex has traveled a total of 800 miles in 9 hours, which makes her average speed equivalent to 800 divided by 9, which is approximately 88.9 miles per hour. The correct answer is choice (B).
2.DChoice (A) is the too obvious answer here. When the question asks for a greatest value, the initial reaction is to look for the greatest number. However, for this problem that is incorrect. To solve this problem, find the value for n that produces the greatest value. Since n is a value that is represented by the answer choices, plug the answer choice values in for n to see which yields the greatest number. Negative fractions that are raised to even powers have a positive result and negatives fractions that are raised to odd powers have negative results, so eliminate choices (A), (C), and (E). Fractions or decimals that are raised to powers have results that are less than the original fraction or decimal. So, choice (D) is the correct answer. These rules will be covered in more detail later in this book.
3.AThe too obvious answers for this question are choices (C) and (D). Many students assume that they can eliminate the square roots and add, to yield 9, which is choice (C). To solve this problem, simplify the like roots. So, ( + + )^{2} = (3)^{2}. Now square both values inside the parentheses to find that (3)^{2} = 9×3 = 27. The correct answer is choice (A). Notice that choice (D) is what you get if you correctly combine the roots but forget to square the result. Don’t worry if you are rusty on exponents and roots; they will be covered in more detail later in the book.
Drill 2
1.CWhen a question asks for a result based on compound interest, the answer is always slightly greater than what it would be if the interest had been paid as simple interest. So if the account had simple interest of 6%, the interest earned would be 6% × $3,000 = $180, for a total of $3,180. However, because compounding the interest monthly earns interest slightly faster because there is interest earned on interest already paid, the answer is slightly greater than $3,180. Use POE. Choice (A) is less money than was in the account at the start, so eliminate (A). Choice (B) is simple annual interest as calculated above, so eliminate (B). Choices (D) and (E) are both too large. The correct answer is choice (C). For more information on compound interest, see Chapter 4.
2.BNotice that choice (A) is the too obvious answer since it is just 5% + 20%. Choices (D) and (E) are also too large and can be eliminated. To answer the question, Plug In. The problem is about increases in Steve’s original salary, so plug in for his salary. Since the problem involves percents, make Steve’s salary $100. If he receives a 5% raise, then his salary is 1.05 × $100 = $105. He then receives another raise of 20%, so 1.20 × $105 = $126. Now solve the problem, which asks for the percent of Steve’s original salary that his raises represent. Since he raises total $26 and his original salary is $100, the raises represent 26% of his salary. The correct answer is choice (B).