﻿ Be Strategic When Doing Algebra - Ten Steps to Scoring Higher on the GRE - Crash Course for the New GRE ﻿

## Crash Course for the New GRE, 4th Edition (2011)

### Step 6. Be Strategic When Doing Algebra

A normal person might say, “I have ten dollars. I’m going to buy three pieces of candy, and each one costs 50 cents. How much change do I get?” The normal answer would be \$8.50. ETS is not normal. They will ask you the same question, but when they ask it, they say, “I have x dollars. I’m going to buy y pieces of candy and each piece costs z cents. Now how much change do I get (z − xy would be a tempting answer choice for this question, but wrong)? This is called algebra. In algebra, normal numbers are replaced with abstract symbols (who has ever heard of z dollars?). We know the logic of numbers so well, that finding an answer requires nothing more than common sense, but with symbols it’s different. You never really know you have the right answer with symbols, and then, because they don’t follow common sense, you have to learn all sorts of rules to tell you what you’re allowed to do with them. Anytime the GRE gives us algebra, all we’re going to do is take the symbols out, and put the numbers back in. When you do this, the only math functions you ever have to perform are addition, subtraction, multiplication, and division. Here’s an example:

Kyle has four fewer toys than Scott, but seven more toys than Jody. If Kyle has k toys, then how many toys do Scott and Jody have together? 2k + 11 2k + 7 2k + 3 2k − 3 2k − 11

Step 1—Recognize the opportunity. The minute you see variables in the question and variables in the answers, you know you can Plug In.

Step 2—Engage the hand. As soon as you recognize the problem as a Plug In, write A, B, C, D, and E in a column in the upper left corner of your scratch paper.

Step 3—Plug In. Who’s ever heard of k toys? Give k a number. Let’s say k = 10. Write this down on your scratch paper. Now work the problem. Kyle has four fewer toys than Scott. If Kyle has 10, then Scott has 14. Write down s= 14 on your scratch paper. Kyle has seven more toys than Jodi, so write j = 3 on your scratch paper.

Step 4—Identify and circle your target number. The target number is the number the question asks you to find. In this case, the question asks us, “How many toys do Scott and Jody have together?” Well, Scott has 14 and Jody has 3, so our target number is 17. Write it down and circle it.

Step 5—Use POE. Now check all answer choices. Anywhere you see a k, Plug In 10. You’re shooting for a 17.

(A) 2(10) + 11—Nope. Cross it off.

(B) 2(10) + 7—Nope. Cross it off.

(C) 2(10) + 3—Nope. Cross it off.

(D) 2(10) − 3 = 17. This looks good, but always check all answer choices.

(E) 2(10) − 11—Nope.

The answer is (D). Here is what your scratch paper should look like: When Plugging In for variables, there are a few things to keep in mind:

a. Do all work on your scratch paper. When you’re done you should always see all terms labeled, a target number circled, and all answer choices checked.

b. Plug In nice, happy numbers that make your life easier. If you Plug In a number and the math gets creepy, don’t sweat it; just change your Plug In. Avoid Plugging In 0, 1, the same number for multiple variables, or numbers that you find in the question. They’re not necessarily wrong; they’re just likely to yield more than one answer choice.

c. Check to see if more than one answer choice works. If so, Plug In a new number, find a new target number, and check only the answer choices that remain.

Plugging In on Quant Comp

So you’re sitting in your cubical at the test center, and this question pops up:

Question 11 of 20

 y ≠ 0 Quantity A Quantity B −10y −y Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.

Step 1—Recognize the opportunity. The first thing you see is that it’s a quant comp problem. The second thing you see is variables. This is a Plug In. This all takes about 4 seconds to process.

Step 2—Engage the hand. On the fifth second, reach for your scratch paper. When you see variables on quant comp, make the set-up. It looks like this:

Step 3—Plug In something happy and use POE. Throw a nice happy number in there. Let’s make y = 2. Now, when y = 2, then Quantity A is −20, and Quantity B is −2. Write these down in columns as you go. Which one is bigger? Quantity B. Remember what the answer choices mean on quant comp questions. Choice (A) means that the value in Quantity A is always bigger, no matter what. Here we have a case where we’ve followed all of the rules and A is not bigger, so choice (A) can’t be the answer. Cross off choice (A). Since they’re not the same, you can cross off (C) as well. Step 4—Keep Plugging In according to ZONE F. On a regular Plug In we always want to Plug In nice, happy numbers. On a quant comp Plug In, we have to Plug In the weird stuff. ZONE F is a list of the weird stuff. It stands for Zero, One, Negatives, Extra big/small numbers, and Fractions. Be strategic. When you look at this problem, how can you mess with it? The problem asks about negative numbers, so try Plugging In a negative. Make y = −2. Now, Quantity A is 20, and Quantity B is 2. A is bigger so cross of choice (B). Since only (D) is left, that must be the answer.

Here’s what your scratch paper should look like: Always keep Plugging In until you’ve eliminated choices (A), (B), and (C), or you’ve tried everything on ZONE F. If you’ve tried everything on ZONE F and either (A), (B), or (C) is still standing, then that’s your answer.

Let’s try one more:

Question 17 of 20

 0 < x < 100 < y < 1 Quantity A Quantity B x − y 9 Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.

Step 1—Recognize the opportunity. Quant comp? Yup. Variables? Yup.

Step 3—Plug In. Here we have ranges for x and y, so let’s Plug In our biggest x and our smallest y. Make x = 9 and y = .1. In Quantity A we have 8.9, and in Quantity B we have 9. Quantity B is bigger, so cross off choices (A) and (C). Now let’s try our smallest x with our biggest y. Make x = .1 and y = 9. In Quantity A we have .1. In Quantity B we still have 9. Quantity B is still bigger. Now is a good time to use the ZONE F checklist. We can’t Plug In zero. We tried 1. We can’t Plug In a negative. We used large and small numbers. What about fractions? We Plugged In fractions for y, but what about x? There is no law that says that we can Plug In only integers. Try x = 9.9 and y = .1. In Quantity A we get 9.8, and in Quantity B we still have 9. Quantity A is bigger, so cross off choice (B) and pick choice (C). Getting the hang of it?

Must Be

Take a look at this question:

The positive difference between the squares of any two consecutive integers, x and x + 1, must be the square of an integer a multiple of 5 an even integer an odd number a prime number

The phrase “must be” is a lot like the word “always” we just saw in the quant comps. This is like asking, which of the following must always be true, no matter what? We can treat these the same way.

Step 1—Recognize the opportunity. The phrase “must be” in a question is a trigger that provokes a specific action. Sensitize yourself to it.

Step 2—Engage the hand. The minute you see the phrase “must be,” make your set-up. Variables go across the top, and answer choices go across the bottom. It looks like this: Step 3—Plug In and use POE. Start with something nice and happy. Try x = 2. If x = 2, then x + 1 = 3. The squares are 4 and 9, and the positive difference is 5. Remember to always park your set-ups on the left-hand side of your scratch paper so that the right-hand side is free for any scratch work you might have to do. (A) does not work. Put an x next to it and cross off (A). (B) works. Give it a check. (C) does not work. Put and x next to it and cross off (C). (D) works. Give it a check. (E) works. Give it a check.

Step 4—Keep Plugging In. “Must be” is really code for “Plug In more than once.” Keep going until there’s only one left. Try x = 10. Our squares are 100 and 121. The difference is 21. You need to check only answer choices that are still standing. (B) does not work. Put an x next to it and cross it off. (D) works. Give it a check. (E) does not work. Give it an x and cross it off. The only one left is (D). That’s our answer. Your scratch paper should look like this: PITA

There is one more kind of Plugging In. It’s called PITA which stands for Plugging In The Answers. It is one of the most powerful types of Plugging In because it can take some of the hardest problems on the GRE and turn them into simple arithmetic. The hardest thing about this technique, however, is remembering to use it. Let’s look at a question:

Two positive integers, x and y, have a difference of 15. If the smaller integer, y, is of x, then what is the value of y? 40 25 20 15 10

Step 1—Recognize the opportunity. There are three signs that you can Plug In the answers:

a. The question contains the phrase “how much…,” “how many.…,” or “what is the value of.…”

b. You have specific numbers in the answer choices in ascending or descending order.

c. You are tempted to design or write your own algebraic formula to solve the problem.

Step 2—Engage the hand. The minute you recognize the opportunity, list the answer choices in the upper left corner of your scratch paper. In this case, list 40, 25, 20, 15, and 10 in a vertical column in the upper left hand side of your scratch paper.

Step 3—Label the First Column. The question asks, “What is the value of y?” The answer choices, therefore, represent possible values for y. Label that first column “y.”

Step 4—Assume (C) to be correct. If y = 20, what else do we know?

Step 5—Work the problem in bite-sized pieces. Make a new column for every step. The problem tells us that y is of x. So if y = 20, then x = 32. Make a column labeled “x” and write 32 in the column next to the 20.

Step 6—Use POE. How do we know if (C) is the correct answer? There must be a difference of 15 between x and y. With choice (C) there is only a difference of 12. Cross off choice (C). For a bigger difference we will need bigger numbers, so cross off choices (D) and (E) as well. Now, not only are we down to a 50/50 shot, but we also have a little spreadsheet designed specifically to calculate the answer to this problem. Just fill in the cells. If x = 25, then y = 40. The difference is 15, and we have an answer. When you’re Plugging In The Answers, only one can work. So when you get one that works, you’re done. Here’s what your scratch paper should look like after this problem: Let’s try one more.

Question 1 of 3

Vicken, Roger, and Adam went to buy a \$90 radio. If Roger agrees to pay twice as much as Adam and Vicken agrees to pay three times as much as Roger, how much must Roger pay? \$10 \$20 \$30 \$45 \$65

Step 1—Recognize the opportunity. The question asks “How much must Roger pay?”

Step 2—Engage the hand. List the answer choices, \$10, \$20, \$30, \$45, and \$65 on your scratch paper.

Step 3—Label the first column. What do those numbers represent? The amount Roger pays.

Step 4—Assume (C) to be correct. Reread the problem, but assume that it says that Roger pays \$30.

Step 5—Work the problem in bite-size pieces. Make a new column for every step. If Roger pays \$30, then Adam pays \$15. Make a column for Adam. Vicken pays \$90, so make a column for Vicken.

Step 6—Use POE. Now we have a problem because the whole radio costs \$90. Cross off choice (C). We also know that (C) is too big, so cross off (D) and (E). Now fill in the columns for answer choice (B). If Roger pays \$20, then Adam pays \$10, and Vicken pays \$60. All three together pay \$90, which is exactly what they should be paying, so we are done and the answer is (B).

Your scratch paper should look something like this: There are four kinds of Plugging In we’ve reviewed here:

1. Plugging In for variables. When you see variables in the question and the answer choices, you can Plug In. Make sure you have your terms labeled, your target number circled, and all answer choices checked.

2. Quant comp Plug In. When you see quant comp and variables, make your set-up. Plug In according to ZONE F, eliminating as you go.

3. Must be. “Must be” is code for “Plug In more than once.” Make your set-up with variables across the top and answer choices down the side. Plug In according to ZONE F, eliminating as you go.

4. PITA. When the question asks, How much? How many? or What is the value of? you know you can Plug In The Answers. Label the first column, start with choice (C), and make a new column for every step. When you find one that works, you’re done.

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