## Cracking the SAT

# Part III

# How to Crack the Math Section

9 Joe Bloggs and the Math Section

10 The Calculator

11 Fun with Fundamentals

12 Algebra: Cracking the System

13 Advanced Arithmetic

14 Geometry

15 Grid-Ins: Cracking the System

16 Putting It All Together

## A FEW WORDS ABOUT SAT MATH

As we’ve mentioned before, the SAT isn’t your normal school test. The same is true of the Math portion of the SAT. There are two types of questions that you’ll run into: multiple-choice and student-produced response questions. We’ve talked before about multiple-choice, so let’s talk about these strange questions known as student-produced response questions. These questions are the only non-multiple-choice questions on the SAT, other than the essay; instead of selecting ETS’s answer from among several choices, you will have to find the answer on your own and mark it in a grid, which is why we call them Grid-Ins. The Grid-In questions on your test will be drawn from arithmetic, algebra, and geometry, just like regular SAT math questions. However, the format has special characteristics, so we will treat them a bit differently. You’ll learn more about them later in this book.

### What Does the Math SAT Measure?

ETS says that the Math SAT measures “mathematical reasoning abilities” or “higher-order reasoning abilities.” Unfortunately, or fortunately for you, this is not true. The Math section is merely a brief test of arithmetic, algebra, and a bit of geometry—when we say a “bit,” we mean it. The principles you’ll need to know are few and simple. We’ll show you which ones are important. Most of them are listed for you at the beginning of each section.

### Order of Difficulty

As was true of the Sentence Completion questions, questions on the Math section are arranged in order of difficulty. The first question in each section will be the easiest in that section, and the last will be the hardest; in this case, harder doesn’t mean tougher—it means trickier. In addition, the questions within the grid-in question groups will also be arranged in order of difficulty. The difficulty of a problem will help you determine how to attack it.

### You Don’t Have to Finish

We’ve all been taught in school that when you take a test, you have to finish it. If you answered only two-thirds of the questions on a high school math test, you probably wouldn’t get a very good grade. But as we’ve already seen, the SAT is not at all like the tests you take in school. Most students don’t know about the difference, so they make the mistake of doing all the problems on each Math section of the SAT.

Because they have only a limited amount of time to answer all the questions, most students are always in a rush to get to the end of the section. At first, this seems reasonable, but think about the order of difficulty for a minute. All the easy questions are at the beginning of a Math section, and the hard questions are at the end. So when students rush through a Math section, they’re actually spending less time on the easier questions (which they have a good chance of getting right), just so they can spend more time on the harder questions (which they have very little chance of getting right). Does this make sense? Of course not.

Here’s the secret: On the Math section, you don’t have to answer every question in each section. In fact, unless you’re trying to score 600 or more, you shouldn’t even look at the difficult last third of the Math questions. Most students can raise their math scores by concentrating on getting all the easy and medium questions correct. In other words…

#### Slow Down!

Most students do considerably better on the Math section when they slow down and spend less time worrying about the hard questions (and more time working carefully on the easier ones). Haste causes careless errors, and careless errors can ruin your score. In most cases, you can actually*raise* your score by answering *fewer* questions. That doesn’t sound like a bad idea, does it? If you’re shooting for an 800, you’ll have to answer every question correctly. But if your target is 550, you should ignore the hardest questions in each section and use your limited time wisely.

To make sure you’re working at the right pace in each Math section, refer to the Math Pacing Chart on this page. The chart will tell you how many questions you need to answer in each section to achieve your next score goal.

### Calculators

Students are permitted (but not required) to use calculators on the SAT. You should definitely take a calculator to the test. It will be extremely helpful to you, as long as you know how and when to use it and don’t get carried away. We’ll tell you more about calculators as we go along.

### The Princeton Review Approach

We’re going to give you the tools you need to handle the easier questions on the Math section, along with several great techniques to help you crack some of the more difficult ones. But you must concentrate first on getting the easier questions correct. Don’t worry about the difficult third of the Math section until you’ve learned to work carefully and accurately on the easier questions.

When it does come time to look at some of the harder questions, the Joe Bloggs principle will help you once again—this time to zero in on ETS’s answer. You’ll learn what kinds of answers appeal to Joe in math, and how to avoid those answers. Just as you did in the Critical Reading section, you’ll learn to use POE to find ETS’s answer by getting rid of obviously incorrect answers.

Generally speaking, each chapter in the Math section of this book begins with the basics and then gradually moves into more advanced principles and techniques. If you find yourself getting lost toward the end of the chapter, don’t worry. Concentrate your efforts on principles you can understand but still need to master.

# Chapter 9

# Joe Bloggs and the Math Section

In Chapter 4 you learned how Joe Bloggs, our model average student, can help you avoid trap answers on the Critical Reading sections of the SAT. Well, that’s not all he can do. In this chapter you will see that we can learn from Joe on the Math sections of the test too. By keeping in mind what Joe is likely to do on difficult math questions, you can avoid those same mistakes and maximize your math score.

## HEY, JOE!

Joe Bloggs has already been a big help to you on the Critical Reading section of the SAT. By learning to anticipate which answer choices would attract Joe on difficult questions, you know how to avoid careless mistakes and eliminate obvious incorrect answers.

You can do the same thing on the SAT Math section. In fact, Joe Bloggs answers are even easier to spot on math questions. ETS is quite predictable in the way it writes incorrect answer choices, and this predictability will make it possible for you to zero in on its answers to questions that might have seemed impossible to you before.

### How Joe Thinks

As was true on the SAT Critical Reading section, Joe Bloggs gets the easy questions right and the hard questions wrong. In Chapter 3, we introduced Joe by showing you how he approached a particular math problem. That problem, you may remember, involved the calculation of total miles in a trip. Here it is again:

**20.** Graham walked to school at an average speed of 3 miles an hour and jogged back along the same route at 5 miles an hour. If his total traveling time was 1 hour, what was the total number of miles in the round trip?

(A) 3

(B)

(C)

(D) 4

(E) 5

When we showed this problem the first time, you were just learning about Joe Bloggs. Now that you’ve made him your invisible partner on the SAT, you ought to know a great deal about how he thinks. Your next step is to put Joe to work for you on the Math sections.

Here’s How to Crack It

This problem was the last in a 20-question Math section. Therefore, it was the hardest problem in that section. Naturally, Joe got it wrong.

The answer choice most attractive to Joe on this problem is D. The question obviously involves an average of some kind, and 4 is the average of 3 and 5, so Joe picked it. Choice D just seemed like the right answer to Joe. (Of course, it wasn’t the right answer; Joe gets the hard ones wrong.)

Because this is true, we know which answers we should avoid on hard questions: answers that seem obvious or that can be arrived at simply and quickly. If the answer really were obvious and if finding it really were simple, the question would be easy, not hard.

Joe Bloggs is also attracted to answer choices that simply repeat numbers from the problem. This means, of course, that you should avoid such choices. In the problem about Graham’s going to school, you can also eliminate choices A and E, because 3 and 5 are numbers repeated directly from the problem. Therefore, they are extremely unlikely to be ETS’s answer.

We’ve now eliminated three of the five answer choices. Even if you couldn’t figure out anything else about this question, you’d have a fifty-fifty chance of guessing correctly. Those are excellent odds, considering that we really didn’t do any math. By eliminating answer choices that we knew were wrong, we were able to beat ETS at its own game. (ETS’s answer to this question is C, by the way.)

### Putting Joe to Work on the Math SAT

Generally speaking, the Joe Bloggs principle teaches you to

· trust your hunches on easy questions.

· double-check your hunches on medium questions.

· eliminate Joe Bloggs answers on difficult questions.

The rest of this chapter is devoted to using Joe Bloggs to zero in on ETS’s answers to difficult questions. Of course, your main concern is still to answer all easy and medium questions correctly. But if you have some time left at the end of a Math section, the Joe Bloggs principle can help you eliminate answers on a few difficult questions, so that you can venture some good guesses. And as we’ve already seen, smart guessing means more points. (In Chapter 15, you’ll learn how he can help you with grid-ins.)

## DON’T GET TRAPPED

Let’s take a look at some common situations you will come across on the SAT and find out how best to handle them.

### Hard Questions = Hard Answers

As we’ve just explained, hard questions on the SAT simply don’t have correct answers that are obvious to the average person. Avoiding the “obvious” choices will take some discipline on your part, but you’ll lose points if you don’t. Even if you’re a math whiz, the Joe Bloggs principle will keep you from making careless mistakes.

Here’s an example:

**18.** A 50-foot wire runs from the roof of a building to the top of a 10-foot pole 14 feet across the street. How much taller would the pole have to be if the street were 16 feet wider and the wire remained the same length?

(A) 2 feet

(B) 8 feet

(C) 14 feet

(D) 16 feet

(E) 18 feet

Here’s How to Crack It

Which answer seems simple and obvious? Well, if the wire stays the same length, and the street is 16 feet wider, then it seems obvious that the pole would have to be 16 feet higher.

What does that mean? It means that we can eliminate choice D. If 16 feet were the correct answer, then Joe Bloggs would get this problem right and it would be an easy question, not one of the hardest in the section.

Choice C repeats a number from the problem, which means we can be certain that it’s wrong too.

If you don’t know how to do this problem, working on it further probably won’t get you anywhere. You’ve eliminated two choices; guess if you need the question to reach your pacing goal and then move on. (ETS’s answer is B. Use the Pythagorean theorem—see Chapter 14.)

### Simple Operations = Wrong Answers on Hard Questions

Joe Bloggs doesn’t usually think of difficult mathematical operations, so he is attracted to solutions that use very simple arithmetic. Therefore, any answer choice that is the result of simple arithmetic should be eliminated on hard SAT math questions.

Here’s an example:

**17.** A dress is selling for $100 after a 20 percent discount. What was the original selling price?

(A) $200

(B) $125

(C) $120

(D) $80

(E) $75

Here’s How to Crack It

When Joe Bloggs looks at this problem, he sees “20 percent less than $100” and is attracted to choice D. Therefore, you must eliminate it. If finding the answer were that easy, Joe Bloggs would have gotten it right. Joe is also attracted to choice C, which is 20 percent more than $100. Again, eliminate.

With two Joe Bloggs answers out of the way, you ought to be able to solve this problem quickly. The dress is on sale, which means that its original price must have been more than its current price. That means that ETS’s answer has to be greater than $100. Two of the remaining choices, A and B, fulfill this requirement. Now you can ask yourself these questions:

(A) Is $100 20 percent less than $200? No. Eliminate.

(B) Is $100 20 percent less than $125? Could be. This must be ETS’s answer. (It is.)

Here’s another example:

**16.** The length of each side of a cube is doubled to create a second cube. The volume of the second cube will be how many times the area of the original cube?

(A) 2

(B) 4

(C) 8

(D) 16

(E) 64

Here’s How to Crack It

The problems asks for how much larger the volume of the new cube is than the original. What’s the Joe Bloggs answer here? Since each side doubled, Joe Bloggs would guess that the volume doubles as well. That’s too easy for a difficult question such as this one. Eliminate A.

You may have also noticed that you can eliminate E, 64, because it is too large.

The problem never gives you an actual cube, right? In that case, you can always make one up. This is a Hidden Plug In question, which we’ll talk more about later. Say that each side of the cube is 3. That makes the current volume 3 × 3 × 3 = 27. The second cube has sides which are twice as long, so each side is 3 × 2 = 6. The volume of the second cube is therefore 6 × 6 × 6 = 216. Now, go back and reread the question. Since the questions asks how many times larger the second cube is than the first, divide 216 by 27, to find that the second cube is 8 times larger than the first one, and the answer is C.

### “It Cannot Be Determined”

Occasionally on the Math section, the fifth answer choice on a problem will be

(E) It cannot be determined from the information given.

The Joe Bloggs principle makes these questions easy to crack. Why? Joe Bloggs can never determine the correct answer on difficult SAT problems. Therefore, when Joe sees this answer choice on a difficult problem, he is greatly attracted to it.

### What Does This Mean?

It means that if “it cannot be determined” is offered as an answer choice on a difficult problem, it is usually wrong.

Here’s an example:

**19.** If the average of *x, y,* and 80 is 6 more than the average of *y, z,* and 80, what is the value of *x* – *z* ?

(A) 2

(B) 3

(C) 6

(D) 18

(E) It cannot be determined from the information given.

Here’s How to Crack It

This problem is the next-to-last question in a section. It looks absolutely impossible to Joe. Therefore, he assumes that the problem must be impossible to solve. Of course, he’s wrong. Eliminate choice E. If E were ETS’s answer, Joe would be correct and this would be an easy problem.

Choice C simply repeats a number from the problem, so you can eliminate that choice also. Because you already eliminated two answer choices, the odds are in your favor if you need to guess.

ETS’s answer is D. Don’t worry about how to solve this problem right now. It’s only important that you understand how to eliminate Joe Bloggs answers to get closer to ETS’s answer. If you have to guess, that’s okay. Besides, that was a hard question; you should be concentrating on answering all the easy and medium questions correctly.

## Summary

· Joe Bloggs gets the easy math questions right and the hard ones wrong.

· On difficult problems, Joe Bloggs is attracted to easy solutions arrived at with methods he understands. Therefore, you should eliminate obvious, simple answers on difficult questions.

· On difficult problems, Joe Bloggs is also attracted to answer choices that simply repeat numbers from the problem. Therefore, you should eliminate any such choices.

· On difficult problems, you can almost always eliminate any answer choice that says, “It cannot be determined from the information given.”

· The point of Joe Bloggs is not to get to ONE answer choice; it’s to improve your odds when you must guess, and eliminate answer choices that could distract you or seem right if you made a careless error.