Part II. Ten Steps to Scoring Higher on the GRE

Step 8. Ballpark the Equations

MATH SENTENCES?

Many GRE math problems involve words and letters, or variables, such as n, x, or y, in equations. It’s time to learn how to deal with those.

Solving for One Variable

Any equation with one variable can be solved by manipulating the equation. You get the variables on one side of the equation and the numbers on the other side. To do this, you can add, subtract, multiply, or divide both sides of the equation by the same number. Just remember that anything you do to one side of an equation, you have to do to the other side. Be sure to write down every step. Look at a simple example:

4x − 3 = 9

You can get rid of negatives by adding something to both sides of the equation, just as you can get rid of positives by subtracting something from both sides of the equation.

4x − 3 = 9

+3  +3

4x     =12

You may already see that x = 3. But don’t forget to write down that last step. Divide both sides of the equation by 4.

x = 3

MORE MATH VOCAB

1. F.O.I.L.—F.O.I.L. stands for First, Outer, Inner, Last—the four steps of multiplication when you see two sets of parentheses. Here’s an example:

= (x × x) + (x × 3) + (4 × x) + (4 × 3)

= x² + 3x + 4x + 12

= x² + 7x + 12

This also works in the opposite direction.

2. Factoring—If you rewrite the expression xy + xz as x(y + z), you are said to be factoring the original expression. That is, you take the factor common to both terms of the original expression (x) and “pull it out.” This gives you a new, “factored” version of the expression you began with. If you rewrite the expression x(y + z) as xy + xz, you are unfactoring the original expression.

3. Functions—No, not real mathematical functions. On the GRE, a function is a funny-looking symbol that stands for an operation. For example, say you’re told that m @ n is equal to . What’s the value of 4 @ 6? Just follow directions: , or , or 2. Don’t worry that “@” isn’t a real mathematical operation; it could have been a “#” or an “&,” or any other symbol. The point is, just do what you are told to do.

4. Inequalities—Here are the symbols you need to know: ≠ means not equal to; > means greater than; < means less than; ≥ means greater than or equal to; ≤ means less than or equal to. You can manipulate any inequality in the same way you can an equation, with one important difference. For example,

10 − 5x > 0

You can solve this by subtracting 10 from both sides of the equation, and ending up with −5x > − 10. Now you have to divide both sides by −5.

With inequalities, any time you multiply or divide by a negative number, you have to flip the sign.

x < 2

5. Percent—Percent means “per 100” or “out of 100” or “divided by 100.” If your friend finds a dollar and gives you 50 cents, your friend has given you 50 cents out of 100, or  of a dollar, or 50 percent of a dollar. When you have to find exact percentages it’s much easier if you know how to translate word problems, which lets you express them as equations. Here’s a translation “dictionary.”

Word	Translates to
percent	/100 (example: 40 percent translates to )
is	=
of	x
what	any variable (x, k, b)
what percent

What is 30 percent of 200?

First, translate it, using the “dictionary” above.

x =  × 200

Now reduce that 100 and 200, and solve for the variable, like this

x = 30 × 2

x = 60

So, 30 percent of 200 is 60.

6. Percent change—To find a percentage increase or decrease, first, find the difference between the original number and the new number. Then, divide that by the original number, and then multiply the result by 100. In other words:

Percent Change =  × 100

For example, if you had to find the percent decrease from 4 to 3, first, figure out what the difference is. The difference, or decrease, from 4 to 3 is 1. The original number is 4. So,

Percent Change =  × 100

Percent Change =  × 100

Percent Change = 25

So, the percent decrease from 4 to 3 is 25 percent.

7. Quadratic equations—Three equations that sometimes show up on the GRE. Here they are, in their factored and unfactored forms.

Factored form		Unfactored form
x2 − y2	=	(x + y)(x − y)
(x + y)2	=	x2 + 2xy + y2
(x − y)2	=	x2 − 2xy + y2

8. Simultaneous equations—Two algebraic equations that include the same variables. For example, what if you were told that 5x + 4y = 6 and 4x + 3y = 5, and asked what x + y equals? To solve a set of simultaneous equations, you can usually either add them together or subtract one from the other (just remember when you subtract that everything you’re subtracting needs to be made negative). Here’s what we get when we add them:

5x + 4y = 6

+ 4x + 3 y = 5

9x + 7 y = 11

A dead end. So, try subtraction.

5x + 4y = 6

– 4x − 3y = − 5

x + y = 1

Eureka. The value of the expression (x + y) is exactly what we’re looking for.

BALLPARKING

Say you were asked to find 30 percent of 50. Don’t do any math yet. Now let’s say that you glance at the answer choices and you see these:

          5

          15

          30

          80

          150

Think about it. Whatever 30 percent of 50 is, it must be less than 50, right? So any answer choice greater than 50 can’t be right. That means you should eliminate both 80 and 150 right off the bat, without doing any math. You can also eliminate 30, if you think about it. Half, or 50 percent, of 50 is 25, so 30 percent must be less than 25. Congratulations, you’ve just eliminated three out of five answer choices without doing any math.

What we’ve just done is known as Ballparking. Ballparking will help you eliminate answer choices and increase your odds of zeroing in on ETS’s answer. Remember to eliminate any answer choice that is “out of the ballpark” by crossing them off on your scratch paper (remember, you’ll be writing down A, B, C, D, E for each question).

Charts

Ballparking will also help you on the few chart questions that every GRE math section will have. You should Ballpark whenever you see the word “approximately” in a question, whenever the answer choices are far apart in value, and whenever you start to answer a question and you justifiably say to yourself, “This is going to take a lot of calculation!”

To help you ballpark, here are a few percents and their fractional equivalents:

If, on a chart question, you were asked to find 9.6 percent of 21.4, you could ballpark by using 10 percent as a “friendlier” percentage and 20 as a “friendlier” number. Ten percent of 20 is 2. That’s all you need to do to answer most chart questions.

Try out Ballparking on a real chart. Keep in mind that while friends give you charts to display the information they want you to see and to make that information easier to understand, ETS constructs charts to hide the information you need to know and to make that information hard to understand. So read all titles and small print, to make sure you understand what the charts are conveying.

Looking over these charts, notice that they are for 1975 and 1985, and that all you know are percentages. There are no total numbers for the survey, and because the percentages are pretty “ugly,” you can anticipate doing a lot of Ballparking to answer the questions. Try one:

To the nearest one percent, what percentage decrease in popularity occurred for chocolate from 1975 to 1985 ?

          9%

          10%

          11%

          89%

          90%

First, we need to find the difference between 28.77 (the 1975 figure) and 25.63 (the 1985 figure). The difference is 3.14. Second, notice that ETS has asked for an approximate answer (“to the nearest one percent”) which is screaming “Ballpark!” Could 3.14 really be 89 or 90 percent of 28.77? No way; it’s closer to the neighborhood of 10 percent. Eliminate choices (D) and (E). Is it exactly 10 percent? No; that means choice B is out. Is it more or less than 10 percent? It’s more—exactly 10 percent would be 2.877, and 3.14 is more than 2.877. That means the answer is (C).

Try another one:

In 1985, if 20 percent of the “other” category is lemon flavor, and 4,212 people surveyed preferred lemon, then how many people were surveyed?

          1,000

          10,000

          42,120

          100,000

          1,000,000

The first piece of information you have is a percentage of a percentage. The percentage of people who preferred lemon in 1985 is equal to 20 percent of 21.06 percent. Make sure you see that before you go on. Now, notice that the numbers in the answer choices are very widely separated—they aren’t consecutive integers. If you can just get in the ballpark, the answer will be obvious.

Rather than try to use 21.06 percent, we’ll call it 20 percent. And rather than use 4,212, we’ll use 4,000. The question is now: “20 percent of 20 percent of what is 4,000?” So, using translation, your equation looks like this:

Do a little reducing.

 ×  × x= 4,000

 × x= 4,000

x = 100,000

That’s (D).