Cracking the GRE Premium (2015)
Part III How to Crack the Math Section
Chapter 11 Real World Math
Real world math is our title for the grab bag of math topics that will be heavily tested on the GRE. This chapter details a number of important math concepts, many of which you’ve probably used at one point or another in your daily adventures, even if you didn’t recognize them. After completing this chapter, you’ll have brushed up on important topics such as fractions, percents, ratios, proportions, and average. You’ll also learn some important Princeton Review methods for organizing your work and efficiently and accurately answering questions on these topics.
EVERYDAY MATH
A few years ago when ETS reconfigured the GRE, they wanted the Math section test more of what they call “real life” scenarios that a typical graduate student might see. You can therefore expect the math questions on the GRE to heavily test topics such as fractions, percents, proportions, averages, and ratios—mathematical concepts that are theoretically part of your everyday life. Regardless of whether that’s true of your daily life or not, you’ll have to master these concepts in order to do well on the GRE Math section.
The math on the GRE
is supposed to reflect
the math you use in your
day-to-day activities.
FRACTIONS, DECIMALS, AND PERCENTAGES
In the previous chapter we spent most of our time working with integers. Now we’ll expand our discussion to include concepts like fractions, decimals, and percentages—all of which will appear frequently on the GRE.
Fractions
A fraction expresses the number of parts out of a whole. In the fraction , for instance, the top part, or numerator, tells us that we have 2 parts, while the bottom part of the fraction, the denominator, indicates that the whole, or total, consists of 3 parts. We use fractions whenever we’re dealing with a quantity that’s less than one.
Notice that the fraction bar is simply another way of expressing division. Thus, the fraction is just expressing the idea of “2 divided by 3.”
Fractions are important on
the GRE. Make sure you’re
comfortable with them.
Reducing and Expanding Fractions
Fractions express a relationship between numbers, not actual amounts. For example, saying that you did of your homework expresses the same idea whether you had 10 pages of homework to do and you’ve done 5, or you had 50 pages to do and you’ve done 25 pages. This concept is important because on the GRE you’ll frequently have to reduce or expand fractions.
To reduce a fraction, simply express the numerator and denominator as the products of their factors. Then cross out, or “cancel,” factors that are common to both the numerator and denominator. Here’s an example:
You can achieve the same result by dividing the numerator and denominator by the factors that are common to both. In the example you just saw, you might realize that 4 is a factor of both the numerator and the denominator. That is, both the numerator and the denominator can be divided evenly (without a remainder) by 4. Doing this yields the much more manageable fraction .
When you confront GRE math problems that involve big fractions, always reduce them before doing anything else.
Remember: You can only reduce across a multiplication sign.
Look at each of the following fractions:
What do you notice about each of these fractions? They all express the same information! Each of these fractions expresses the relationship of “1 part out of 4 total parts.”
Why Bother?
You may be wondering why, if the GRE allows the use of a calculator, you should bother learning how to add or subtract fractions or to reduce them or even know any of the topics covered in the next few pages. While it’s true that you can use a calculator for these tasks, for many problems it’s actually slower to do the math with the calculator than without. Scoring well on the GRE Math section requires a fairly strong grasp of the basic relationships among numbers, fractions, percents, and so on, so it’s in your best interest to really understand these concepts rather than to rely on your calculator to get you through the day. In fact, if you put in the work now, you’ll be surprised at how easy some of the problems become, especially when you don’t have to refer constantly to the calculator to perform basic operations.
Adding and Subtracting Fractions
Adding and subtracting fractions that have a common denominator is easy—you just add the numerators and put the sum over the common denominator. Here’s an example:
In order to add or subtract fractions that have different denominators, you need to start by finding a common denominator. You may remember your teachers from grade school imploring you to find the “lowest common denominator.” Actually, any common denominator will do, so find whichever one you find most comfortable working with.
Here, we expanded the fraction into the equivalent fraction by multiplying both the numerator and denominator by 3. Similarly, we converted to by multiplying both denominator and numerator by 2. This left us with two fractions that had the same denominator, which meant that we could simply subtract their numerators.
When adding and subtracting fractions, you can also use a technique we call the Bowtie. The Bowtie method accomplishes exactly what we just did in one fell swoop. To use the Bowtie, first multiply the denominators of each fraction. This gives you a common denominator. Then multiply the denominator of each fraction by the numerator of the other fraction. Take these numbers and add or subtract them—depending on what the question asks you to do—to get the numerator of the answer. Then reduce if necessary.
and
The Bowtie method is a
convenient shortcut to use
when you’re adding and
subtracting fractions.
Multiplying Fractions
There’s nothing tricky about multiplying fractions: All you do is multiply straight across—multiply the first numerator by the second numerator and the first denominator by the second denominator. Here’s an example:
When multiplying fractions, you can make your life easier by reducing before you multiply. Do this once again by dividing out common factors.
Multiplying fractions
is a snap: Just multiply
straight across, numerator
times numerator and
denominator times
denominator.
Also remember that when you’re multiplying fractions, you can even reduce diagonally; as long as you’re working with a numerator and a denominator of opposite fractions, they don’t have to be in the same fraction. So you end up with
Of course, you get the same answer no matter what method you use, so attack fractions in whatever fashion you find easiest.
Dividing Fractions
Dividing fractions is just like multiplying fractions, with one crucial difference: Before you multiply, you have to turn the second fraction upside down (that is, put its denominator over its numerator, or to use fancy math lingo, find its reciprocal). In some cases, you can also reduce before you multiply. Here’s an example:
ETS sometimes gives you problems that involve fractions which have numerators or denominators that are themselves fractions. These problems might look intimidating, but if you’re careful, you won’t have any trouble with them. All you have to do is remember what we said about a fraction being shorthand for division. Always rewrite the expression horizontally. Here’s an example:
Comparing Fractions
The GRE might also present you with math problems that require that you to compare two fractions and decide which is larger, especially on quant comp questions. There are a couple of ways to accomplish this. One is to find equivalent fractions that have a common denominator. This works with simpler fractions, but on some problems the common denominator might be hard to find or hard to work with.
As an alternative, you can use a variant of the Bowtie technique. In this variant, you don’t have to multiply the denominators, just the denominators and the numerators. The fraction with the larger product in its numerator is the greater fraction. Let’s say we had to compare the following fractions:
Multiplying the first denominator by the second numerator gives you 49. This means the numerator of the second fraction is 49. Multiplying the second denominator by the first numerator gives you 36, which means the first fraction has a numerator of 36. Since 49 is greater than 36, is greater than . Remember that when you use this method, it’s the numerators that matter.
You can also use the
calculator feature to
change the fractions
into decimals.
Comparing More Than Two Fractions
You may also be asked to compare more than two fractions. On these types of problems, don’t waste time trying to find a common denominator for all of them. Simply use the Bowtie to compare two of the fractions at a time.
Here’s an example:
Which of the following statements is true?
Here’s How to Crack It
As you can see, it would be a nightmare to try to find common denominators for all these fractions, so instead we’ll use the Bowtie method. Simply multiply the denominators and numerators of a pair of fractions and note the results. For example, to check answer choice (A), we first multiply 8 and 2, which gives us a numerator of 16 for the fraction . But multiplying 9 and 3 gives us a numerator of 27 for the first fraction. This means that is greater than , and we can eliminate choice (A), because the first part of it is wrong. Here’s how the rest of the choices shape up:
The answer is choice (C). Make sure you are doing all of this work in an organized fashion on your scratch paper.
Converting Mixed Numbers into Fractions
A mixed number is a number that is represented as an integer and a fraction, such as . In most cases on the GRE, you should get rid of mixed fractions by converting them to improper fractions. How do you do this? By multiplying the denominator of the fraction by the integer, then adding that result to the numerator, and then putting the whole thing over the denominator. In other words, for the fraction above we would get or .
Improper fractions have a
numerator that is greater
than the denominator.
When you convert mixed
numbers, you’ll get
an improper fraction as
the result.
The result, , is equivalent to . The only difference is that is easier to work with in math problems. Also, answer choices are usually not given in the form of mixed numbers.
Decimals
Decimals are just fractions in disguise. Basically, decimals and fractions are two different ways of expressing the same thing. Every decimal can be written as a fraction, and every fraction can be written as a decimal. For example, the decimal 0.35 can be written as the fraction : These two numbers, 0.35 and , have the same value.
To turn a fraction into its decimal equivalent, all you have to do is divide the numerator by the denominator. Here, for example, is how you would find the decimal equivalent of :
Try this problem:
7 < x < 8
y = 9
Quantity A |
Quantity B |
0.85 |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Here’s How to Crack It
So, you’re sitting at your cubicle at the Prometric testing center and this problem pops up. What do you see? Before we even talk about fractions, the first thing you should note is that this is a quant comp with variables. Set up your scratch paper. It should look like this:
Now they’ve told us that x is going to be seven point something. Try Plugging In the smallest value you can think of for x. Write down x = 7.1 and y = 9. The value in Quantity A is 0.79. The value in Quantity B is 0.85. Quantity B is greater, so eliminate choices (A) and (C). Now try makingx as big as you can make it. Write down x = 7.9 and y = 9. The value in column A is 0.88 and the value in Quantity B is 0.85. Quantity A is greater so eliminate choice B, and you’re done. The answer is (D).
Your scratch paper should look like this:
Comparing Decimals
Which is greater: 0.00099 or 0.001? ETS loves this sort of problem. You’ll never go wrong, though, if you follow these easy steps.
· Line up the numbers by their decimal points.
· Fill in the missing zeros.
Here’s how to answer the question we just asked. First, line up the two numbers by their decimal points.
0.00099
0.001
Now fill in the missing zeros.
0.00099
0.00100
Can you tell which number is greater? Of course you can. 0.00100 is greater than 0.00099, because 100 is greater than 99.
Digits and Decimals
Remember our discussion about digits, earlier? Well, sometimes the GRE will ask you questions about digits that fall after the decimal point as well. Suppose you have the number 0.584.
· 0 is the units digit.
· 5 is the tenths digit.
· 8 is the hundredths digit.
· 4 is the thousandths digit.
Percentages
The final member of our numbers family is percentages. A percentage is just a special type of fraction, one that always has 100 as the denominator. Percent literally means “per 100” or “out of 100” or “divided by 100.” If your best friend finds a dollar and gives you 50¢, your friend has given you 50¢ out of 100, or of a dollar, or 50 percent of the dollar. To convert fractions to percentages, just expand the fraction so it has a denominator of 100:
Another way to
convert a fraction into
a percentage is to
divide the numerator
by the denominator
and multiply the result
by 100. So, = 3 ÷ 5 =
0.6 × 100 = 60%.
For the GRE, you should memorize the following percentage-decimal-fraction equivalents. Use these friendly fractions and percentages to eliminate answer choices that are way out of the ballpark.
Converting Decimals to Percentages
In order to convert decimals to percentages, just move the decimal point two places to the right. For example, 0.8 turns into 80 percent, 0.25 into 25 percent, 0.5 into 50 percent, and 1 into 100 percent.
Translation
One of the best ways to handle percentages in word problems is to know how to translate them into an equation that you can manipulate. Use the following table to help you translate percentage word problems into equations you can work with.
These translations apply to
any word problem, not just
percent problems.
Here’s an example:
56 is what percent of 80 ?
66%
70%
75%
80%
142%
Here’s How to Crack It
To solve this problem, let’s translate the question and then solve for the variable. So, “56 is what percent of 80,” in math speak, is equal to
Don’t forget to reduce the fraction: 56 = x.
Now multiply both sides of the equation by the reciprocal, .
Don’t forget to reduce again before you calculate:
That’s answer choice (B). Did you notice choice (E)? Because 56 is less than 80, the answer would have to be less than 100 percent, so 142 percent is way too big, and you could have eliminated it from the get-go by Ballparking.
Let’s try a quant comp example.
5 is r percent of 25
s is 25 percent of 60
Quantity A |
Quantity B |
r |
s |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Here’s How to Crack It
First translate the first statement.
That takes care of Quantity A. Now translate the second statement.
That takes care of Quantity B. The answer is (A).
Percentage Increase/Decrease
To find the percentage by which something has increased or decreased, use the following formula.
On percent increase
problems, the original is
always the smaller number.
On percent decrease
problems, the original is
the larger number.
Percent Change = × 100
The “difference” is simply what you get when you subtract the smaller number from the larger number. The “original” is whichever number you started with. If the question asks you to find a percent increase, then the original number is the smaller number. If the question asks you to find a percent decrease, then the original number is the larger number.
On the GRE, a percent change will not be stated as a negative number. Instead, the problem will ask for a percent decrease. So, if something declined by 50%, the problem will ask for a percent decrease and the answer will be stated as 50%. Note that when you use the percent change formula, you always subtract the smaller number from the larger number to find the difference. Doing so ensures that you get a positive result.
Here’s an example.
What number goes on the
bottom of the fraction?
Vandelay Industries reported a $6,000 profit over the three-month period from March to May of the current year. Over the previous three month period, if Vandelay Industries realized a $3,500 profit, by approximately what percent did its profit increase?
25%
32%
42%
55%
70%
Here’s How to Crack It
Let’s use the percent change formula we just learned. The first step is to find the difference between the two numbers. The initial profit was $3,500 and the final profit is $6,000. The difference between these two numbers is 6,000 – 3,500 = 2,500. Next, we need to divide this number by the original, or starting, value.
One way to help you figure out what value to use as the original value is to check to see whether you’re dealing with a percent increase or a percent decrease question. Remember that on a percent increase question, you should always use the smaller of the two numbers as the denominator and that on percent decrease you need to use the larger of the two numbers as the denominator. Because here we want to find the percent increase, the number we want to use for our denominator is 3,500. So our percent increase fraction looks like this: . We can reduce this to by dividing by 100, and reduce even further by dividing by 5. This leaves us with , which is approximately 70% (remember that the fraction bar means divide, so if you divide 5 by 7, you’ll get 0.71). Thus, choice (E) is the answer.
Here’s another question.
The table above shows the original price and the sale price for six different models of cars. For which car models is the discount between the original price and the sale price at least 25%?
Indicate all such models.
A
B
C
D
E
F
Here’s How to Crack It
First list A, B, C, D, E, and F in a column on your scratch paper. You are asked to identify a 25% change or greater between the two prices. You know the formula for this. It is percent change = × 100. Using the calculator, subtract 9,500 from 12,000 to get 2,500. This is the difference. Divide it by the original, 12,000, to get 0.2, which when multiplied by 100 is 20%. Since 20% is less than 25%, cross it off on your scratch paper. Try the next one. 16,000 – 13,000 = 3,000. Divide by 16,000. Too small. Cross it off. Repeat this process for each of the answer choices. Choices (C), (D), and (F) all work.
PLUGGING IN ON FRACTION AND PERCENT PROBLEMS
Now that you’ve become familiar with fractions and percents, we’ll show you a great method for solving many of these problems. When you come to regular multiple-choice questions, or multiple choice, multiple answers, that involve fractions or percents, you can simply Plug In a number and work through the problem using that number. This approach works even when the problem doesn’t have variables in it. Why? Because, as you know, fractions and percents express only a relationship between numbers—the actual numbers don’t matter. For example, look at the following problem:
Plugging In on fraction
and percent problems is
a great way to make your
life easier.
A recent survey of registered voters in City X found that of the respondents support the mayor’s property tax plan. Of those who did not support the mayor’s plan, indicated they would not vote to reelect the mayor if the plan were implemented. Of all the respondents, what fraction indicated that they would not vote for the mayor if the plan were enacted?
What important information
is missing from the
problem?
Here’s How to Crack It
Even though there are no variables in this problem, we can still Plug In. On fraction and percent problems, ETS will often leave out one key piece of information: the total. Plugging In for that missing value will make your life much easier. What crucial information did ETS leave out of this problem? The total number of respondents. So let’s Plug In a value for it. Let’s say that there were 24 respondents to the survey. 24 is a good number to use because we’ll have to work with and , so we want a number that’s divisible by both those fractions. Working through the problem with our number, we see that of the respondents support the plan. of 24 is 8, so that means 16 people do not support the plan. Next, the problem says that of those who do not support the plan will not vote for the mayor. of 16 is 2, so 2 people won’t vote for the mayor. Now we just have to answer the question: Of all respondents, how many will not vote for the mayor? Well, there were 24 total respondents and we figured out that 2 aren’t voting. So that’s , or . Answer choice (B) is the one we want.
RATIOS AND PROPORTIONS
If you’re comfortable working with fractions and percentages, you’ll be comfortable working with ratios and proportions, because ratios and proportions are simply special types of fractions. Don’t let them make you nervous. Let’s look at ratios first and then we’ll deal with proportions.
What Is a Ratio?
Recall that a fraction expresses the relationship of a part to the whole. A ratio expresses a different relationship: part to part. Imagine yourself at a party with 8 women and 10 men in attendance. Remembering that a fraction expresses a part-to-whole relationship, what fraction of the partygoers are female? , or 8 women out of a total of 18 people at the party. But what’s the ratio, which expresses a part to part relationship, of women to men? , or as ratios are more commonly expressed, 8 : 10. You can reduce this ratio to 4 : 5, just like you would a fraction.
A ratio is just another
type of fraction.
On the GRE, you may see ratios expressed in several different ways:
x : y
the ratio of x to y
x is to y
In each case, the ratio is telling us the relationship between parts of a whole.
Every Fraction Can Be a Ratio, and Vice Versa
Every ratio can be expressed as a fraction. A ratio of 1 : 2 means that the total of all the parts is either 3 or a multiple of 3. So, the ratio 1 : 2 can be expressed as the fraction . Likewise, the fraction means that we are looking at one part out of a total of three so the other part must be 2. That means that the ratio is 1 : 2.
Treat a Ratio Like a Fraction
Anything you can do to a fraction you can also do to a ratio. You can cross-multiply, find common denominators, reduce, and so on.
Find the Total
The key to dealing with ratio questions is to find the whole, or the total. Remember: A ratio tells us only about the parts, not the total. In order to find the total, add the numbers in the ratio. A ratio of 2 : 1 means that there are three total parts. A ratio of 2 : 5 means that we’re talking about a total of 7 parts. And a ratio of 2 : 5 : 7 means there are 14 total parts. Once you have a total you can start to do some fun things with ratios.
For example, let’s say you have a handful of pennies and nickels. If you have 30 total coins and the pennies and nickels are in a 2 : 1 ratio, how many pennies do you have? The total for our ratio is 3, meaning that out of every 3 coins, there are 2 pennies and 1 nickel. So if there are 30 total coins, there must be 20 pennies and 10 nickels. Notice that is the same as , is the same as 2 : 1!
Like a fraction, a ratio
expresses a relationship
between numbers.
When you are working with ratios, there’s an easy way not only to keep track of the numbers in the problem but also to quickly figure out the values in the problem. It’s called the Ratio Box. Let’s try the same question, but with some different numbers; if you have 24 coins in your pocket and the ratio of pennies to nickels is 2 : 1, how many pennies and nickels are there? The Ratio Box for this question is below, with all of the information we’re given already filled in.
The minute you see the
word “ratio,” draw a ratio
box on your scratch paper.
Remember that ratios are relationships between numbers, not actual numbers, so the real total is 24; that is, you have 24 actual coins in your pocket. The ratio total (the number you get when you add the number of parts in the ratio) is 3.
The middle row of the table is for the multiplier. How do you get from 3 to 24? You multiply by 8. Remember when we talked about finding equivalent fractions? All we did was multiply the numerator and denominator by the same value. That’s exactly what we’re going to do with ratios. This is what the ratio box looks like now:
The multiplier is the key
concept in working
with ratios. Just
remember that whatever
you multiply one
part by, you must
multiply every part by.
Now let’s finish filling in the box by multiplying everything else.
Therefore, of the 24 coins 16 are pennies and 8 are nickels.
Let’s try a GRE example.
Flour, eggs, yeast, and salt are mixed by weight in the ratio of 11 : 9 : 3 : 2, respectively. How many pounds of yeast are there in 20 pounds of the mixture?
2
Need More Math
Review?
Check out Math Workout
for the GRE. If you’re in
a hurry, pick up Crash
Course for the GRE.
Here’s How to Crack It
The minute you see the word ratio, draw a ratio box on your scratch paper and fill in what you know.
First, add all of the numbers in the ratio to get the ratio total.
Now, what do we multiply 25 by to get 20?
So is our “multiply by” number. Let’s fill it in.
The question asks for the amount of yeast, so we don’t have to worry about the other ingredients. Just look at the yeast column. All we have to do is multiply 3 by and we have our answer: , which is answer choice (D).
What Is a Proportion?
So you know that a fraction is a relationship between part and whole, and that a ratio is a relationship between part and part. A proportion is an equivalent relationship between two fractions or ratios. Thus, and are proportionate because they are equivalent fractions. But and are not in proportion because they are not equal ratios.
The GRE often contains problems in which you are given two proportional, or equal, ratios from which one piece of information is missing. These questions take a relationship or ratio, and project it onto a larger or smaller scale. Proportion problems are recognizable because they always give you three values and ask for a fourth value. Here’s an example:
The key to proportions is
setting them up correctly.
If the cost of a one-hour telephone call is $7.20, what would be the cost in dollars of a 10-minute telephone call at the same rate?
dollars
Click on the answer box and type in a number.
Backspace to erase.
Here’s How to Crack It
It’s very important to set up proportion problems correctly. That means placing your information on your scratch paper. Be especially careful to label everything. It takes only an extra two or three seconds, but doing this will help you catch lots of errors.
Relationship Review
You may have noticed a trend in the preceding pages. Each of the major topics covered—fractions, percents, ratios, and proportions—described a particular relationship between numbers. Let’s review:
· A fraction expresses the relationship between a part and the whole.
· A percent is a special type of fraction, one that expresses the relationship of part to whole as a fraction with the number 100 in the denominator.
· A ratio expresses the relationship between part and part. Adding the parts of a ratio gives you the whole.
· A proportion expresses the relationship between equal fractions, percents, or ratios.
· Each of these relationships shares all the characteristics of a fraction. You can reduce them, expand them, multiply them, and divide them using the exact same rules you used for working with fractions.
For this question, let’s express the ratios as dollars over minutes, because we’re being asked to find the cost of a 10-minute call. That means that we have to convert 1 hour to 60 minutes (otherwise it wouldn’t be a proportion).
Now cross-multiply.
Now we can enter 1.20 into the box.
AVERAGES
The average (arithmetic mean) of a list of numbers is the sum, or total value, of all the numbers in the list divided by the number of numbers in the list. The average of the list 1, 2, 3, 4, 5 is equal to the total of the numbers (1 + 2 + 3 + 4 + 5, or 15) divided by the number of numbers in the list (which is 5). Dividing 15 by 5 gives us 3, so 3 is the average of the list.
GRE average problems
always give you two of
the three numbers needed.
ETS always refers to an average as an “average (arithmetic mean).” This confusing parenthetical remark is meant to keep you from being confused by other more obscure kinds of averages, such as geometric and harmonic means. You’ll be less confused if you simply ignore the parenthetical remark and know that average means total of the elements divided by the number of elements.
Think Total
Don’t try to solve average problems all at once. Do them piece by piece. The key formula to keep in mind when doing problems that involve averages is
The minute you see the
word average, draw
an average pie on your
scratch paper.
Drawing an Average Pie will help you organize your information.
Here’s how the Average Pie works. The total is the sum of the numbers being averaged. The number of things is the number of different elements that you are averaging. And the average is, naturally, the average.
Say you wanted to find the average of 4, 7, and 13. You would add those numbers to get the total and divide that total by three.
Mathematically, the Average Pie works like this:
Which two pieces of the
pie do you have?
The horizontal bar is a division bar. If you divide the total by the number of things, you get the average. If you divide the total by the average, you get the number of things. If you have the number of things and the average, you can simply multiply them together to find the total. This is one of the most important things you need to be able to do to solve GRE average problems.
Using the Average Pie has several benefits. First, it’s an easy way to organize information. Furthermore, the Average Pie makes it clear that if you have two of the three pieces, you can always find the third. This makes it easier to figure out how to approach the problem. If you fill in the number of things, for example, and the question wants to know the average, the Average Pie shows you that the key to unlocking that problem is finding the total.
Try this one.
The average (arithmetic mean) of seven numbers is 9 and the average of three of these numbers is 5. What is the average of the other four numbers?
4
5
7
10
12
Draw a new average pie
each time you encounter
the word average in a
question.
Here’s How to Crack It
Let’s take the first sentence. You have the word average, so draw an average pie and fill in what you know. We have seven numbers with an average of 9, so plug those values into the Average Pie and multiply to find the total.
Now we also know that three of the numbers have an average of 5, so draw another Average Pie, plug those values into their places, and multiply to find the total of those three numbers.
The question is asking for the average of the four remaining numbers. Draw one more Average Pie and Plug In 4 for the number of things.
In order to solve for the average, we need to know the total of those four numbers. How do we find this? From our first Average Pie we know that the total of all seven numbers is 63. The second Average Pie tells us that the total of three of those numbers was 15. Thus, the total of the remaining four has to be 63 – 15, which is 48. Plug 48 into the last Average Pie, and divide by 4 to get the average of the four numbers.
The average is 12, which is answer choice (E).
Let’s try one more.
The average (arithmetic mean) of a set of 6 numbers is 28. If a certain number, y, is removed from the set, the average of the remaining numbers in the set is 24.
Quantity A |
Quantity B |
y |
48 |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Here’s How to Crack It
All right, let’s attack this one. The problem says that the average of a set of six numbers is 28, so let’s immediately draw an average pie and calculate the total.
If a certain number, y, is removed from the set, there are now five numbers left. We already know that the new average is 24, so draw another Average Pie.
The difference between the totals must be equal to y. 168 – 120 = 48. Thus, the two quantities are equal, and the answer is (C).
Up and Down
Averages are very predictable. You should make sure you automatically know what happens to them in certain situations. For example, suppose you take three tests and earn an average score of 90. Now you take a fourth test. What do you know?
If your average goes up as a result of the fourth score, then you know that your fourth score was higher than 90. If your average stays the same as a result of the fourth score, then you know that your fourth score was exactly 90. If your average goes down as a result of the fourth score, then you know that your fourth score was less than 90.
MEDIAN, MODE, AND RANGE
The median is the middle value in a list of numbers; above and below the median lie an equal number of values. For example, in the list of numbers (1, 2, 3, 4, 5, 6, 7) the median is 4, because it’s the middle number (and there are an odd number of numbers in the list). If the list contained an even number of integers such as (1, 2, 3, 4, 5, 6) the median is the average of 3 and 4, or 3.5. When looking for the median, sometimes you have to put the numbers in order yourself. What is the median of the list of numbers (13, 5, 6, 3, 19, 14, 8)? First, put the numbers in order from least to greatest, (3, 5, 6, 8, 13, 14, 19). Then take the middle number. The median is 8. Just think median = middle and always make sure the numbers are in order.
Don’t confuse
median and mode!
The mode is the number in a list of numbers that occurs most frequently. For example, in the list (2, 3, 4, 5, 3, 8, 6, 9, 3, 9, 3) the mode is 3, because 3 shows up the most. Just think mode = most.
The minute you see the
word median in a question,
find a bunch of numbers
and put them in order.
The range is the difference between the greatest and the least numbers in a list of numbers. So, in the list of numbers (2, 6, 13, 3, 15, 4, 9), the range is 15 (the greatest number in the list) – 2 (the least number in the list), or 13.
Here’s an example:
Set F = {4, 2, 7, 11, 8, 9}
Quantity A |
Quantity B |
The range of Set F |
The median of Set F |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
What do we need to do to
the numbers in this set?
Here’s How to Crack It
Let’s put the numbers in order first, so it’ll be easier to see what we have: {2, 4, 7, 8, 9, 11}. First let’s look at Quantity A: The range is the greatest number, or 11, minus the least number, or 2. That’s 9. Now let’s look at Quantity B: The minute you see the word median, be sure to put the numbers in order. The median is the middle number of the set, but because there are two middle numbers, 7 and 8, we have to find the average. Or do we? Isn’t the average of 7 and 8 clearly going to be smaller than the number in Quantity A, which is 9? Yes, in quant comp questions, we compare, not calculate. The answer is (A).
STANDARD DEVIATION
Standard deviation is one of those phrases that math people like to throw around to scare non-math people, but it’s really not that scary. The GRE might ask you questions about standard deviation, but you’ll never have to actually calculate it; instead, you’ll just need a basic understanding of what standard deviation is. In order to understand standard deviation, we must first look at something all standardized testers should be familiar with, the bell curve.
You’ll never have to calculate
the standard deviation
on the GRE.
Your Friend the Bell Curve
The first thing to know about a bell curve is that the number in the middle is the mean.
The minute you see the phrase “standard deviation” or “normal distribution,” draw a bell curve and fill in the percentages.
Imagine that 100 students take a test and the results follow a normal distribution. The minute you see the phrase “normal distribution,” draw a bell curve. Let’s say that the average score on this test is an 80. Put 80 in the middle of the curve. You know, however, that a few of those students were extremely well prepared and got a really high score, let’s say that 2% of them got a 96 or higher. Put a 96 above the right 2% line on the curve.
Standard deviation measures how much a score differs from the norm (the average) in even increments. The curve tells us that a score earned by only 2% of the students is two standard deviations from the norm. If the norm is 80 and 96 is two standard deviations away, then one standard deviation on this test is 8 points. Why? Remember that standard deviations are even increments. If the average is 80 and the score 2 standard deviations from the norm is 96, then the difference is 16. So, one standard deviation is half of that difference or 8. The score at 1 standard deviation greater than the norm is, therefore, 88. Two standard deviations above the norm is 96, while two standard deviations below the norm is 64. One standard deviation above the norm is 88, and one standard deviation below the norm is 72. Fill these in on your bell curve.
Now you know quite a bit about the distribution of scores on this test. 68 percent of the students received a score between 72 and 88. 98 percent scored above a 64. That’s all there is to know about standard deviations. The percentages don’t change, so memorize those. When you see the phrase, just draw a bell curve and fill in what you know. Here’s what the curve looks like for this test:
When it comes to standard deviation, the percentages don’t change, so memorize those: 2, 14, and 34.
Here’s an example of how ETS might test standard deviation:
Quantity A |
Quantity B |
The standard deviation of a list of data consisting of 10 integers ranging from –20 to –5 |
The standard deviation of a list of data consisting of 10 integers ranging from 5 to 20 |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Here’s How to Crack It
ETS is hoping you’ll make a couple of wrong turns on this problem. The first trap they set is that one list of numbers contains negative integers while the other doesn’t—but this doesn’t mean that one list has a negative standard deviation. Standard deviation is defined as the distance a point is from the mean, so it can never be negative. The second trap is that ETS hopes you’ll waste a lot of time trying to calculate standard deviation based on the information given. But you know better than to try to do that. Remember that ETS won’t ask you to calculate standard deviation; it’s a complex calculation. Plus, as you know, you need to know the mean in order to calculate the standard deviation and there’s no way we can find it based on the information here. Thus, we have no way of comparing these two quantities, and our answer is (D).
Now let’s try a question that will make use of the bell curve.
The fourth grade at School X is made up of 300 students who have a total weight of 21,600 pounds. If the weight of these fourth graders has a normal distribution and the standard deviation equals 12 pounds, approximately what percentage of the fourth graders weighs more than 84 pounds?
12%
16%
36%
48%
60%
Here’s How to Crack It
This one’s a little tougher than the earlier standard deviation questions. The first step is to determine the average weight of the students, which is = 72 pounds. If the standard deviation is 12 pounds, then 84 pounds places us exactly one standard deviation above the mean, or at the 84th percentile (remember the bell curve?). Because 16 percent of all students weigh more than 84 pounds, the answer is (B).
RATE
Rate problems are similar to average problems. A rate problem might ask for an average speed, distance, or the length of a trip, or how long a trip (or a job) takes. To solve rate problems, use the Rate Pie.
A rate problem is really
just an average problem.
The Rate Pie works exactly the same way as the Average Pie. If you divide the distance or amount by the rate, you get the time. If you divide the distance or amount by the time, you get the rate. If you multiply the rate by the time, you get the distance or amount.
Let’s take a look.
It takes Carla three hours to drive to her brother’s house at an average speed of 50 miles per hour. If she takes the same route home, but her average speed is 60 miles per hour, what is the time, in hours, that it takes her to drive home?
2 hours
2 hours and 14 minutes
2 hours and 30 minutes
2 hours and 45 minutes
3 hours
Here’s How to Crack It
The trip to her brother’s house takes three hours, and the rate is 50 miles per hour. Plug those numbers into a Rate Pie and multiply to find the distance.
So the distance is 150 miles. On her trip home, Carla travels at a rate of 60 miles per hour. Draw another Rate Pie and Plug In 150 and 60. Then all you have to do is divide 150 by 60 to find the time.
So it takes Carla two and a half hours to get home. That’s answer choice (C).
Try another one.
A machine can stamp 20 envelopes in 4 minutes. How many of these machines, working simultaneously, are needed to stamp 60 envelopes per minute?
5
10
12
20
24
Here’s How to Crack It
First we have to find the rate per minute of one machine. Plug 20 and 4 into a Rate Pie and divide to find the rate.
The rate is 5. If one machine can stamp 5 envelopes per minute, how many machines do you need to stamp 60 per minute? 60 ÷ 5 = 12, or answer choice (C).
CHARTS
Every GRE Math section has a few questions that are based on a chart or graph (or on a group of charts or graphs). But don’t worry; the most important thing that chart questions test is your ability to remember the difference between real-life charts and ETS charts.
Chart questions frequently
test percents, percent
change, ratios, proportions,
and averages.
In real life, charts are often provided in order to display information in a way that’s easier to understand. Conversely, ETS constructs charts to hide information you need to know and to make that information harder to understand.
Chart Questions
There are usually two or three questions per chart or per set of charts. Like the Reading Comprehension questions, chart questions appear on split screens. Be sure to click on the scroll bar and scroll down as far as you can; there may be additional charts underneath the top one, and you want to make sure you’ve seen all of them.
Chart problems just recycle the basic arithmetic concepts we’ve already covered: fractions, percentages, and so on. This means you can use the techniques we’ve discussed for each type of question, but there are two additional techniques that are especially important to use when doing chart questions.
On charts, look for the
information ETS is trying
to hide.
Don’t Start with the Questions: Start with the Charts
Take a minute to note the following key bits of information from any chart you see.
· Information in titles: Make sure you know what each chart is telling you.
· Asterisks, footnotes, parentheses, and small print: Often there will be crucial information hidden away at the bottom of the chart. Don’t miss it!
· Funny units: Pay special attention when a title says “in thousands” or “in millions.” You can usually ignore the units as you do the calculations, but you have to use them to get the right answer.
Approximate, Estimate, and Ballpark
Like some of our other techniques, you have to train yourself to estimate when working with charts and graphs questions. You should estimate, not calculate exactly, in the following situations:
· Whenever you see the word approximately in a question
· Whenever the answer choices are far apart in value
· Whenever you start to answer a question and you justifiably say to yourself, “This is going to take a lot of calculation!”
Don’t try to work
with huge values.
Ballpark instead!
Review those “friendly” percentages and their fractions from earlier in the chapter. Try estimating this question:
What is approximately 9.6 percent of 21.4?
Here’s How to Crack It
Use 10 percent as a friendlier percentage and 20 as a friendlier number. One-tenth of 20 is 2 (it says “approximately”—who are you to argue?). That’s all you need to do to answer most chart questions.
Chart Problems
Make sure you’ve read everything on the chart carefully before you try the first question.
Approximately how many tons of aluminum and copper combined were purchased in 1995 ?
125
255
325
375
515
How much did Company X spend on aluminum in 1990 ?
$675,000
$385,000
$333,000
$165,000
$139,000
Approximately what was the percent increase in the price of aluminum from 1985 to 1995 ?
8%
16%
23%
30%
42%
Here’s How to Crack the First Question
As you can see from the graph on the previous page, in 1995, the black bar (which indicates aluminum) is at 250, and the dark grey bar (which indicates copper) is at approximately 125. Add those figures and you get the number of tons of aluminum and copper combined that were purchased in 1995: 250 + 125 = 375. That’s choice (D). Notice that the question says “approximately.” Also notice that the numbers in the answer choices are pretty far apart.
Here’s How to Crack the Second Question
We need to use the chart and the graph to answer this question, because we need to find the number of tons of aluminum purchased in 1990 and multiply it by the price per ton of aluminum in 1990 in order to figure out how much was spent on aluminum in 1990. The bar graph tells us that 175 tons of aluminum was purchased in 1990, and the little chart tells us that aluminum was $2,200 per ton in 1990. 175 × $2,200 = $385,000. That’s choice (B).
Here’s How to Crack the Third Question
Remember that percent increase formula from earlier in this chapter?
Percent change = × 100
We’ll need to use the little chart for this one. In 1985, the price of aluminum was $1,900 per ton. In 1995, the price of aluminum was $2,700 per ton. Now let’s use the formula. 2,700 – 1,900 = 800, so that’s the difference. This is a percent increase problem, so the original number is the smaller one. Thus, the original is 1,900, and our formula looks like this: Percent change = × 100. By canceling the 0’s in the fraction you get × 100, and multiplying gives you . At this point you could divide 800 by 19 to get the exact answer, but because they’re looking for an approximation, let’s round 19 to 20. What’s 800 ÷ 20? That’s 40, and answer choice (E) is the only one that’s close.
Real World Math Drill
Now it’s time to try out what you have learned on some practice questions. Try the following problems and then check your answers in Part V.
1 of 18
If 3 (r + s) = 7, then, in terms of r, s =
– r
+ r
7 – 3r
2 of 18
Sadie sells half the paintings in her collection, gives one-third of her paintings to friends, and keeps the remaining paintings for herself. What fraction of her collection does Sadie keep?
Click on each box and type in a number.
Backspace to erase.
3 of 18
During a sale, a store decreases prices on all its scarves by 25 to 50 percent. If all of the scarves in the store originally cost $20, which of the following could be the sale price of a scarf?
Indicate all such prices.
$8
$10
$12
$14
$16
4 of 18
Quantity A |
Quantity B |
θ8 |
θ4 |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
5 of 18
5x – 2y = 2y – 3x
Quantity A |
Quantity B |
x |
y |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Questions 6 through 9 refer to the following graph.
6 of 18
If the six New England states are ranked by population in Year X and Year Y, how many states would have a different ranking from Year X to Year Y?
None
One
Two
Three
Four
7 of 18
In Year X, the population of Massachusetts was approximately what percent of the population of Vermont?
50%
120%
300%
800%
1,200%
8 of 18
By approximately how much did the population of Rhode Island increase from Year X to Year Y ?
750,000
1,250,000
1,500,000
2,250,000
3,375,000
9 of 18
A water jug with a capacity of 20 gallons is 20 percent full. If an amount of water equal to 50 percent of the amount of water currently in the jug is added to the jug every 3 days, how many days does it take for the jug to be at least 85% full?
4
6
12
15
20
10 of 18
Towns A, B, C, and D are all in the same voting district. Towns A and B have 3,000 people each who support referendum R and the referendum has an average (arithmetic mean) of 3,500 supporters in towns B and D and an average of 5,000 supporters in Towns A and C.
Quantity A |
Quantity B |
The average number of supporters of Referendum R in Towns C and D |
The average number of supporters of Referendum R in Towns B and C |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
11 of 18
A company paid $500,000 in merit raises to employees whose performances were rated A, B, or C. Each employee rated A received twice the amount of the raise that was paid to each employee rated C; and each employee rated B received one-and-a-half times the amount of the raise that was paid to each employee rated C. If 50 workers were rated A, 100 were rated B, and 150 were rated C, how much was the raise paid to each employee rated A ?
$370
$625
$740
$1,250
$2,500
12 of 18
The original price of an item at a store is 40 percent more than the price the retailer paid for it. To encourage sales, the retailer reduces the price of the item by 15 percent from the original selling price. If the retailer sells the item at the reduced cost, his profit is what percent of his cost?
percent
Click on the answer box and type in a number.
Backspace to erase.
Questions 13 through 15 refer to the following graphs.
13 of 18
In 2013, the median reading test score for ninth grade students was in which score range?
Below 65 points
65–69 points
70–79 points
80–89 points
90–100 points
14 of 18
If the number of students in grades 9 through 12 comprised 35 percent of the number of students in School District X in 1995, then approximately how many students were in School District X in 1995 ?
9,700
8,700
3,400
3,000
1,200
15 of 18
Assume that all students in School District X took the reading test each year. In 2013, approximately how many more ninth grade students had reading test scores in the 70–79 point range than in the 80–89 point range?
470
300
240
170
130
16 of 18
Quantity A |
Quantity B |
x |
21 |
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
17 of 18
One ounce of Solution X contains only ingredients a and b in a ratio of 2 : 3. One ounce of Solution Y contains only ingredients a and b in a ratio of 1 : 2. If Solution Z is created by mixing solutions X and Y in a ratio of 3 : 11, then 630 ounces of Solution Z contains how many ounces of a ?
68
73
89
219
236
18 of 18
On Sunday, Belmond Public Library has 160 books, none of which have been checked out. On Monday, 40 of the books are checked out. On Tuesday, of the borrowed books are returned. Wednesday, of the books still checked out are returned and then 20 more are checked out. On Thursday, a wealthy patron donates 80 books, and of the books still checked out are returned. On Friday 30 more books are borrowed, and on Saturday 35 are checked out. What is the percent change from the books in the library at the end of the day on Monday to the books in the library at end of the day the following Saturday?
percent
Click on the answer box and type in a number.
Backspace to erase.
Summary
· Fractions, decimals, and percents are all ways of expressing parts of integers.
· Translation is a useful tool for converting fraction and percent problems into mathematical equations.
· Percent change is expressed as the difference between two numbers divided by the original number × 100.
· Plug In on questions that ask about percents or fractions of an unknown amount.
· A ratio expresses a part to part relationship. The key to ratio problems is finding the total. Use the ratio box to organize ratio questions.
· A proportion expresses the relationship between equal fractions, percents, or ratios. A proportion problem always provides you with three pieces of information and asks you for a fourth.
· Use the Average Pie to organize and crack average problems.
· The median is the middle number in a set of values. The mode is the value that appears most frequently in a set. The range of a set is the difference between the largest and smallest values in the set.
· You will never have to calculate standard deviation on the GRE.
· Standard deviation problems are really average and percent problems. Make sure you know the percentages associated with the bell curve: 34%–14%–2%.
· Use the Rate Pie for rate questions.
· On chart questions, make sure you take a moment to understand what information the chart is providing. Estimate answers to chart questions whenever possible.