Math Workout for the GRE, 3rd Edition (2013)

Chapter 3. The Nuts and Bolts

Whenever you decide to learn a new language, what do they start with on the very first day? Vocabulary. Well, math has as much of its own lexicon as any country’s mother tongue, so now is as good a time as any to familiarize yourself with the terminology. These vocabulary words are rather simple to learn—or relearn—but they’re also very important. Any of the terms you’ll read about in this chapter could show up in a GRE math question, so you should know what the test is talking about. (For a complete list, you can consult the glossary in Chapter 13.)

We’ll start our review with the backbone of all Arabic numerals: the digit.

DIGITS

You might think there are an infinite number of digits in the world, but in fact there are only ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the mathematical “alphabet” that serves as the building block from which all numbers are constructed.

Modern math uses digits in a decile system, meaning that every digit in a number represents a multiple of ten. For example, 1,423.795 = (1 × 1,000) + (4 × 100)+ (2 × 10) + (3 × 1) + (7 × 0.1) + (9 × 0.01) + (5 × 0.001).

You can refer to each place as follows:

• 1 occupies the thousands place.

• 4 occupies the hundreds place.

• 2 occupies the tens place.

• 3 occupies the ones, or units, place.

• 7 occupies the tenths place, so it’s equivalent to “seven tenths,” or .

• 9 occupies the hundredths place, so it’s equivalent to “nine hundredths,” or .

• 5 occupies the thousandths place, so it’s equivalent to “five thousandths,” or .

When all the digits are situated to the left of the decimal place, you’ve got yourself an integer.

INTEGERS

When we first learn about addition and subtraction, we start with integers, which are the numbers you see on a number line.

Integers and digits are not the same thing; for example, 39 is an integer that contains two digits, 3 and 9. Also, integers are not the same as whole numbers, because whole numbers are only positive. Conversely, integers include negatives and zero, and any integer is considered greater than all of the integers to its left on the number line. So just as 5 is greater than 3 (which can be written as 5 > 3), 0 is greater than −4, and −4 is greater than −10. (For more about greater than, less than, and solving for inequalities, see the next chapter.)

Consecutive Integers and Sequences

Integers can be listed consecutively (such as 3, 4, 5, 6…) or in patterned sequences such as odds (1, 3, 5, 7…), evens (2, 4, 6, 8…), and multiples of 6 (6, 12, 18, 24…). The numbers in these progressions will always get larger, except when explicitly noted otherwise. Note also that, because zero is an integer, a list of consecutive integers that progresses from negative to positive numbers must include it (−2, −1, 0, 1…).

ZERO

Zero is a special little number that deserves your attention. It isn’t positive or negative, but it is even. (So a list of consecutive even integers might look like −4, −2, 0, 2, 4….) Zero might also seem insignificant because it’s what’s called the additive identity, which basically means that adding zero to any other number doesn’t change anything. (This will be an important consideration when you start plugging numbers into problems in Chapter 4.)

POSITIVES AND NEGATIVES

On either side of zero, you’ll find positive and negative numbers. For the GRE, the best thing to know about positives and negatives is what happens when you multiply them together.

• A positive times a positive yields a positive (3 × 5 = 15).

• A positive times a negative yields a negative (3 × −5 = −15).

• A negative times a negative yields a positive (−3 × −5 = 15).

EVEN AND ODD

As you might have guessed from our talk of integers above, even numbers (which include zero) are multiples of 2, and odd numbers are not multiples of 2. If you were to experiment with the properties of these numbers, you would find that

• any number times an even number yields an even number

• the product of two or more odd numbers is always odd

• the sum of two or more even numbers is always even

• the sum of two odd numbers is always even

• the sum of an even number and an odd number is always odd

Obviously, there’s no need to memorize stuff like this. If you’re ever in a bind, try working with real numbers. After all, if you want to know what you get when you multiply two odd numbers, you can just pick two odd numbers—like 3 and 7, for example—and multiply them. You’ll see that the product is 21, which is also odd.

Digits Quick Quiz

Question 1 of 3

If x, y, and z are consecutive even integers and x < 0 and z > 0, then xyz =

Question 2 of 3

The hundreds digit and ones digit of a three-digit number are interchanged so that the new number is 396 less than the old number. Which of the following could be the number?

293

327

548

713

801

Question 3 of 3

a, b, and c are consecutive digits, and a > b > c

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanations to Digits Quick Quiz

1. If x, y, and z are consecutive even integers and x < 0 and z > 0, then x must be −2, y must be 0, and z must be 2. Therefore, their product is 0, and you would enter this number into the box.

2. Take the answer choices and switch the hundreds digit and ones digit. When the result is 396 less than the old number, you have a winner. Answer choices (A), (B), and (C) are out, because their ones digits are greater than their hundreds digits; therefore, the result will be greater (for example, 293 becomes 392). If you rearrange 713, the result is 317, which is 396 less than 713. The answer is (D).

3. Pick three consecutive digits for a, b, and c, such as 2, 3, and 4. Quantity A becomes 2 × 3 × 4, or 24, and Quantity B becomes 2 + 3 + 4, or 9. Quantity A is greater so eliminate (B) and (C). But if a, b, and c are −1, 0, and 1, respectively, then both quantities become 0 so eliminate (A). Therefore, the answer is (D).

PRIME NUMBERS

Prime numbers are special numbers that are only divisible by two distinct factors: themselves and 1. Since neither 0 nor 1 is prime, the least prime number is 2. The rest, as you might guess, are odd, because all even numbers are divisible by two. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Note that not all odd numbers are prime; 15, for example, is not prime because it is divisible by 3 and 5. Said another way, 3 and 5 are factors of 15, because 3 and 5 divide evenly into 15. Let’s talk more about factors.

FACTORS AND MULTIPLES

As we said, a prime number has only two distinct factors: itself and 1. But a number that isn’t prime—like 120, for example—has several factors. If you’re ever asked to list all the factors of a number, the best idea is to pair them up and work through the factors systematically, starting with 1 and itself. So, for 120, the factors are

• 1 and 120

• 2 and 60

• 3 and 40

• 4 and 30

• 5 and 24

• 6 and 20

• 8 and 15

• 10 and 12

Notice how the two numbers start out far apart (1 and 120) and gradually get closer together? When the factors can’t get any closer, you know you’re finished. The number 120 has 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Of these factors, three are prime (2, 3, and 5).

That’s also an important point: Every number has a finite number of factors.

Prime Factorization

Sometimes the best way to analyze a number is to break it down to its most fundamental parts—its prime factors. To do this, we’ll break down a number into factors, and then continue breaking down those factors until we’re stuck with a prime number. For instance, to find the prime factors of 120, we could start with the most obvious factors of 120: 12 and 10. (Although 1 and 120 are also factors of 120, because 1 isn’t prime, and no two prime numbers can be multiplied to make 1, we’ll ignore it when we find prime factors.) Now that we have 12 and 10, we can break down each of those. What two numbers can we multiply to make 12? 3 and 4 work, and since 3 is prime, we can break down 4 to 2 and 2. 10 can be broken into 2 and 5, both of which are prime. Notice how we kept breaking down each factor into smaller and smaller pieces until we were stuck with prime numbers? It doesn’t matter which factors we used, because we’ll always end up with the same prime factors: 12 = 6 × 2 = 3 × 2 × 2, or 12 = 3 × 4 = 3 × 2 × 2. So the prime factor tree for 120 could look something like this:

The prime factorization of 120 is 2 × 2 × 2 × 3 × 5, or 2³ × 3 × 5. Note that these prime factors (2, 3, and 5) are the same ones we listed earlier.

MULTIPLES

Since 12 is a factor of 120, it’s also true that 120 is a multiple of 12. It’s impossible to list all the multiples of a number, because multiples trail off into infinity. For example, the multiples of 12 are 12 (12 × 1), 24 (12 × 2), 36 (12 × 3), 48 (12 × 4), 60 (12 × 5), 72 (12 × 6), 84 (12 × 7), and so forth.

If you ever have trouble distinguishing factors from multiples, remember this:

Factors are Few; Multiples are Many.

Divisibility

If a is a multiple of b, then a is divisible by b. This means that when you divide a by b, you get an integer. For example, 65 is divisible by 13 because 65 ÷ 13 = 5.

Divisibility Rules

The most reliable way to test for divisibility is to use the calculator that they give you. If a problem requires a lot of work with divisibility, however, there are several cool rules you can learn that can make the problem a lot easier to deal with. As you’ll see later in this chapter, these rules will also make the job of reducing fractions much easier.

1. All numbers are divisible by one. (Remember that if a number is prime, it is only divisible by itself and 1.)

2. A number is divisible by 2 if the last digit is even.

3. A number is divisible by 3 if the sum of the digits is a multiple of 3. For example: 13,248 is divisible by 3 because 1 + 3 + 2 + 4 + 8 = 18, and 18 is divisible by 3.

4. A number is divisible by 4 if the two digits at the end form a number that is divisible by 4. For example, 13,248 is divisible by 4 because 48 is divisible by 4.

5. A number is divisible by 5 if it ends in 5 or 0.

6. A number is divisible by 6 if it is divisible by both 2 and 3. Because 13,248 is even and divisible by 3, it must therefore be divisible by 6.

7. There is no easy rule for divisibility by 7. It’s easier to just try dividing by 7!

8. A number is divisible by 8 if the three digits at the end form a number that is divisible by 8. For example, 13,248 is divisible by 8 because 248 is divisible by 8.

9. A number is divisible by 9 if the sum of the digits is a multiple of 9. For example: 13,248 is divisible by 9 because 1 + 3 + 2 + 4 + 8 = 18, and 18 is divisible by 9.

10.A number is divisible by 10 if it ends in 0.

Remainders

If an integer is not evenly divisible by another integer, whatever integer is left over after division is called the remainder. You can find the remainder by finding the greatest multiple of the number you are dividing by that is still less than the number you are dividing into. The difference between that multiple and the number you are dividing into is the remainder. For example, when 19 is divided by 5, 15 is the greatest multiple of 5 that is still less than 19. The difference between 19 and 15 is 4, so the remainder when 19 is divided by 5 is 4.

WORKING WITH NUMBERS

A lot of your math calculation on the GRE will require you to know the rules for manipulating numbers using the usual hit parade of mathematical operations: addition, subtraction, multiplication, and division.

PEMDAS (Order of Operations)

When simplifying an expression, you need to perform mathematical operations in a specific order. This order is easily identified by the mnemonic device that most of us come in contact with sooner or later at school—PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. (You might have remembered this as a kid by saying “Please Excuse My Dear Aunt Sally,” which is a perfect mnemonic because it’s just weird enough not to forget. What the heck did Aunt Sally do, anyway?)

In order to simplify a mathematical term using several operations, perform the following steps:

1. Perform all operations that are in parentheses.

2. Simplify all terms that use exponents.

3. Perform all multiplication and division from left to right. Do not assume that all multiplication comes before all division, as the acronym suggests, because you could get a wrong answer.
WRONG: 24 ÷ 4 × 6 = 24 ÷ (4 × 6) = 24 ÷ 24 = 1.
RIGHT: 24 ÷ 4 × 6 = (24 ÷ 4) × 6 = 6 × 6 = 36.

4. Fourth, perform all addition and subtraction, also from left to right.

It’s important to remember this order, because if you don’t follow it your results will very likely turn out wrong.

Try it out in a GRE example.

Question 3 of 20

(2 + 1)³ + 7 × 2 + 7 − 3 × 4² =

−16

0

288

576

1,152

Here’s How to Crack It

Simplify (2 + 1)³ + 7 × 2 + 7 − 3 × 4² like this:

The answer is (B).

Working with Numbers Quick Quiz

Question 1 of 7

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 2 of 7

Which of the following number has the same distinct prime factors as 42 ?

63

98

210

252

296

Question 3 of 7

p and r are factors of 100

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 4 of 7

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 5 of 7

6 (3 − 1)³ + 12 ÷ 2 + 3² =

24

39

63

69

105

Question 6 of 7

If r, s, t, and u are distinct, consecutive prime integers less than 31, then which of the following could be the average (arithmetic mean) of r, s, t, and u ?

Indicate all such numbers.

4

4.25

6

9

14

22

24

Question 7 of 7

The greatest prime number that is less than 36 is represented by x. If y represents the least even number greater than 19 that is divisible by 3, and x is divided by y, what is the result when 2 is added to that quotient?

Answers to Working with Numbers Quick Quiz

1. The only even multiples of 11 between 1 and 100 are 22, 44, 66, and 88, so Quantity A equals 4. Quantity B is tricky, because if 22 is even, all multiples of 22 are also even. There are no odd multiples of 22, so Quantity B equals 0. The answer is (A).

2. The prime factorization of 42 is 2 × 3 × 7 so those are the distinct prime factors. Answer choice (A) can be eliminated, because 63 is odd. The prime factorization of 252 is 2 × 2 × 3 × 3 × 7, so its distinct prime factors are the same. The answer is (D).

3. If p and r are factors of 100, then each must be one of these numbers: 1, 2, 4, 5, 10, 20, 25, 50, or 100. If you Plug In p = 1 and r = 2, for example, then pr = 2 and Quantity B is greater so eliminate (A) and (C). If p = 50 and r = 100, however, then pr = 5,000, which is much greater than 100 so eliminate (B). Therefore, the answer is (D).

4. 12 goes into 33 two times. The remainder is 9. Since 9 is greater than 7, there is no need to calculate the remainder for column B because it can’t possibly be greater than 6. Remember that the remainder is always less than the number you are dividing by. The answer is (A).

5. Follow PEMDAS and calculate the parentheses and exponents first: 6 × (3 − 1)³ + 12 ÷ 2 + 3² = 6 × 8 + 12 ÷ 2 + 9. Second, perform all multiplication and division: 6 × 8 + 12 ÷ 2 + 9 = 48 + 6 + 9. Now, it’s just a matter of addition: 48 + 6 + 9 = 63. The answer is (C).

6. This one can be tricky because of the math vocabulary; the question is really asking for the average of four prime numbers in a row. Start by making a list of all the consecutive prime numbers less than 31. Remember that 1 is not prime, and that 2 is the least prime number. Your list is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Starting with 2, 3, 5, 7, use the on-screen calculator to work out the different possible averages for four consecutive primes. Choices (B), (D), and (F) are the answers.

7. Work this question one piece at a time. The greatest prime number less than 36 is 31, so x = 31. The least even number greater than 19 that is divisible by 3 is 24, so y = 24. Thus you have + 2 and the correct answer is or .

PARTS OF THE WHOLE (FRACTIONS)

It’s still necessary to be knowledgeable when it comes to fractions, decimals, and percents. Each of these types of numbers has an equivalent in the form of the other two, and fluency among the three of them can save you precious time on test day.

For example: Say you had to figure out 25% of 280. You could take a moment to realize that the fractional equivalent of 25% is . At this point, you might see that of 280 is 70, and your work would be done.

If nothing else, memorizing the following table will increase your math IQ and give you a head start on your calculations.

The Conversion Table

Fractions

Each fraction is made up of a numerator (the number on top) divided by a denominator (the number down below). In other words, the numerator is the part, and the denominator is the whole. By most accounts, the part is less than the whole, and that’s the way a fraction is “properly” written.

Improper Fractions

For a fraction, when the part is greater than the whole, the fraction is considered improper. The GRE won’t quiz you on the terminology, but it usually writes its multiple-choice answer choices in proper form. A proper fraction takes this form:

Integer

Converting from Improper to Proper

To convert the improper fraction into proper form, find the remainder when 16 is divided by 3. Because 3 goes into 16 five times with 1 left over, rewrite the fraction by setting aside the 5 as an integer and putting the remainder over the number you divided by (in this case, 3). Therefore, is equivalent to 5, because 5 is the integer, 1 is the remainder, and 3 is the divisor.

The latter format is also referred to as a mixed number, because the number contains both an integer and a fraction.

Converting from Proper to Improper

Sometimes you’ll want to convert a mixed number into its improper format. Converting to an improper fraction from a mixed number is a little easier, because all you do is multiply the divisor by the integer, then add the remainder.

4 = = =

Improper formats are much easier to work with when you have to add, subtract, multiply, divide, or compare fractions. The important thing to stress here is flexibility; you should be able to work with any fraction the GRE gives you, regardless of what form it’s in.

Adding and Subtracting Fractions

There’s one basic rule for adding or subtracting fractions: You can’t do anything until all of the fractions have the same denominator. If that’s already the case, all you have to do is add or subtract the numerators, like this:

+ = =

− = =

When the fractions have different denominators, you must convert one or both of them first in order to find their common denominator.

When you were a kid, you may have been trained to follow a bunch of complicated steps in order to find the “lowest common denominator.” It might be a convenient thing to learn in order to impress your math teacher, but on the GRE it’s way too much work.

The Bowtie

The Bowtie method has been a staple of The Princeton Review’s materials since the company began in a living room in New York City in 1981. It’s been around so long because it works so simply.

To add and , for example, follow these three steps:

Step One: Multiply the denominators together to form the new denominator.

+ = =

Step Two: Multiply the first denominator by the second numerator (5 × 4 = 20) and the second denominator by the first numerator (7 × 3 = 21) and place these numbers above the fractions, as shown below.

See? A bowtie!

Step Three: Add the products to form the new numerator.

+ = =

Subtraction works the same way.

Note that with subtraction, the order of the numerators is important. The new numerator is 21 − 20, or 1. If you somehow get your numbers reversed and use 20 − 21, your answer will be −, which is incorrect. One way to keep your subtraction straight is to always multiply up from denominator to numerator when you use the Bowtie so the product will end up in the right place.

Question 13 of 25

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, eliminate choice (D) because Quantity A and Quantity B both contain numbers. Next, the Bowtie can also be used to compare fractions. Multiply the denominator of the fraction in Quantity B by the numerator of the fraction in Quantity A and write the product (40) over the fraction in Quantity A. Next, multiply the denominator of the fraction in Quantity A by the numerator of the fraction in Quantity B and write the product (39) over the fraction in Quantity B. Since 40 is greater than 39, the answer is (A).

Multiplying Fractions

Multiplying fractions isn’t nearly as complicated as adding or subtracting, because any two fractions can be multiplied by each other exactly as they are. All you have to do is multiply all the numerators to find the new numerator, and multiply all the denominators to find the new denominator, like this:

× = =

The great thing is that it doesn’t matter how many fractions you have; all you have to do is multiply across.

× × × = =

Dividing Fractions

Dividing fractions is almost exactly like multiplying them, except that you need to perform one extra step:

When dividing fractions, don’t ask why; just flip the second fraction and multiply.

Dividing takes two forms. When you’re given a division sign, flip the second fraction and multiply them, like this:

÷ = × = =

Sometimes, you’ll be given a compound fraction, in which one fraction sits on top of another, like this:

=

The fraction bar might look a little intimidating, but remember that a fraction bar is just another way of saying “divide.” In this case, flip the bottom and multiply.

= ÷ = × =

Reciprocals of Fractions

When a fraction is multiplied by its reciprocal, the result is always 1. You can think of the reciprocal as being the value you get when the numerator and denominator of the fraction are “flipped.” The reciprocal of , for example, is .

× = = = 1

Knowing this will help you devise a nice shortcut for working with problems such as the following.

Question 8 of 20

Which of the following is the reciprocal of ?

Here’s How to Crack It

To solve this problem, multiply each answer choice by the original expression, . If the product is 1, you know you have a match. In this case, the only expression that works is , so the answer is (C).

Reducing Fractions

Are you scared of reducing, or canceling, fractions because you’re not sure what the rules are? If so, there’s only one rule to remember:

You can do anything to a fraction as long as you do exactly the same thing to both the numerator and the denominator.

When you reduce a fraction, you divide both the top and bottom by the same number. If you have the fraction , for example, you can divide both the numerator and denominator by a common factor, 3, like this:

Be Careful

If you are worried about when you can cancel terms in a fraction, here’s an important rule to remember. If you have more than one term in the numerator of a fraction but only a single term in the denominator, you can’t divide into one of the terms and not the other.

WRONG: = 17

The only way you can cancel something out is if you can factor out the same number from both terms in the numerator, then divide.

Decimals

Decimals are just fractions with a hidden denominator: Each place to the right of the decimal point represents a fraction.

0.146 =

Comparing Decimals

To compare decimals, you have to look at the decimals place by place, from left to right. As soon as the digit in a specific place of one number is greater than its counterpart in the other number, you know which is bigger.

For example, 15.345 and 15.3045 are very close in value because they have the same digits in their tens, units, and tenths places. But the hundredths digit of 15.345 is 4 and the hundredths digit of 15.304 is 0, so 15.345 is greater.

Rounding Decimals

In order to round a decimal, you have to know how many decimal places the final answer is supposed to have (which the GRE will usually specify) and then base your work on the decimal place immediately to the right. If that digit is 5 or higher, round up; if it’s 4 or lower, round down.

For example, if you had to round 56.729 to the tenths place, you’d look at the 2 in the hundredths place, see that it was less than 5, and round down to 56.7. If you rounded to the hundredths place, however, you’d consider the 9 in the thousandths place and round up to 56.73.

Fractions and Decimals Quick Quiz

Question 1 of 8

The fraction is equivalent to what percent?

30%

33

35%

37.5%

62.5%

Question 2 of 8

What is the sum of and ?

Question 3 of 8

Indicate all such values.

59,049

3¹⁰

Question 4 of 8

If x is the 32^nd digit to the right of the decimal point when is expressed as a decimal, and y is the 19^th digit to the right of the decimal point when is expressed as a decimal, what is the value of xy ?

Question 5 of 8

If two-thirds of 42 equals four-fifths of x, what is the value of x ?

28

35

42

63

84

Question 6 of 8

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 7 of 8

How many digits are there between the decimal point and the first even digit after the decimal point in the decimal equivalent of ?

Question 8 of 8

A set of digits under a bar indicates that those digits repeat infinitely. What is the value of ()(10⁶ − 10⁴) ?

0.023

22.77

23

Explanations for Fractions and Decimals Quick Quiz

1. Reduce to and remember the common conversion table to convert the fraction to 37.5%. The answer is (D).

2. Use the Bowtie method. First, multiply the denominators: 5 × 12 = 60. When you multiply diagonally, as in the diagram given in the text, the numerator becomes 35 + 24, or 59. The new fraction is , and the answer is (D).

3. Simplify the negative exponents by taking the reciprocal of the corresponding positive exponent, which gives you . Now you have three reciprocals, so flip them over and calculate: = 59,049. You can also express each number as a power of 3, which gives you (3³)(3⁴)(3³) = 3¹⁰, which makes choice (B) correct. 3¹⁰ can also be expressed as . Thus, the correct answer is choices (A), (B), and (C).

4. Divide to convert the fractions into decimals. First, . This is really a pattern question: The odd numbered terms are 2 and the even numbered terms are 7. The 32^nd digit to the right of the decimal is an even term, so x= 7. Next, . This time, the odd numbered terms are 6 and the even numbered terms are 3. The 19th digit on the right side of the decimal place is an odd term, so y = 6. Lastly, xy = 7 × 6 = 42.

5. One-third of 42 is 14 so two-thirds is 28 translates to 28 = x. Multiply both sides by the reciprocal, to eliminate the fraction and isolate the x. The fractions on the right cancel out and on the left you have (28) = 35, so 35 = x. The answer is (B).

6. To compare two fractions, just cross multiply and compare the products. 7 × 8 = 56 and 11 × 5 = 55, so Quantity A is greater.

7. The answer is 0. First, divide to convert into a decimal. The on-screen calculator doesn’t do exponents, so you may want to factor the denominator: = 0.00003125. Since 0 is even, there are no digits between the decimal point and the first even digit after the decimal point, and the correct answer is 0. In fact, if you noticed that any decimal starting with a 0 would have the same answer, you only needed to make sure the denominator was larger than 10.

8. Before you break out the calculator, distribute the (). This gives you ()(10⁶) − ()(10⁴). Now move the decimal point 6 places to the right for the first term and 4 places to the right for the second term, which gives you = 23. The correct answer is choice (D).

Percents

As you may have noted from the conversion chart, decimals and percents look an awful lot alike. In fact, all you have to do to convert a decimal to a percent is to move the decimal point two places to the right and add the percent sign: 0.25 becomes 25%, 0.01 becomes 1%, and so forth. This is because they’re both based on multiples of 10. Percents also represent division with a denominator that is always 100.

Calculating Percents

Now, let’s review four different ways to calculate percents. Some people find that one way makes more sense than others. There is no best way to find percents. Try out all four, and figure out which one seems most natural to you. You will probably find that some methods work best for some problems, and other methods work best for others.

Translating

Translating is one of the more straightforward and versatile methods of calculating percents. Each word in a percent problem is directly translated into a mathematical term, according to the following chart:

This table can be a great help. For example, take a look at this Quant Comp problem.

Question 7 of 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

Since it’s not immediately obvious which quantity is larger, we’re going to have to do some actual calculation. Let’s start with Quantity A. 24% of 15% of 400 can be translated, piece by piece, into math. Remember that % means to divide by 100 and of means to multiply. As always, write down everything on your scratch paper. After translation, we get . Cancel out one of the 100s in the denominator with the 400 in the numerator of the last fraction to get . Now feel free to use the calculator. Multiply the top of the fractions first, then the bottom, giving you = 14.4. Now let’s do the same with Quantity B. 52% of 5% of 600 becomes . Cancel out the first 100 with the 600 and you’ll have = 15.6. Quantity B is therefore bigger, and the answer is (B).

Word Problems and Percents

Word problems are also far less onerous when you apply the math translation table.

Question 16 of 20

Thirty percent of the graduate students at Hardcastle State University are from outside the state, and 75% of out-of-state students receive some sort of financial aid. If there are 3,120 graduate students at Hardcastle State, how many out-of-state students do NOT receive financial aid?

234

468

702

936

1,638

Here’s How to Crack It

Note the range in answer choices. This is another problem that is ripe for ballparking. Assume there are 3,000 graduate students. 10% of 3,000 is 300, so 30% is 900. 25% do not receive financial aid, which is what you’re looking for, so of 900 is 225. Since there are few more than 3,000 graduate students the answer must be a few more than 225. Only answer (A) is even close. Note that answer choice (C) represents the 75% who do receive financial aid. Make sure you read slowly and carefully and take all problems in bite sized pieces.

Conversion

The second way to deal with percentage questions is to use the chart on this page. This will allow you to quickly change each percentage into a fraction or a decimal. This method works well in tandem with Translating, as reduced fractions are often easier to work with when you are setting up a calculation.

Proportions

You can also set up a percent question as a proportion by matching up the part and whole. Let’s look at a couple quick examples:

1) 60 is what percent of 200?
Because 60 is some part of 200, we can set up the following proportion:

Notice that each fraction is simply the part divided by the whole. On the left side, 60 is part of 200. We want to know what that part is in terms of a percent, so on the right side we set up x (the percent we want to find), divided by 100 (the total percent).

2) What is 30% of 200?
Now we don’t know how much of 200 we’re dealing with, but we know the percentage. So we’ll put our unknown, x, over the whole, 200, and our percentage on the right:

3) 60 is 30% of what number?
We know our percentage, and we know our part, but we don’t know the whole, so that’s our unknown:

Notice that the setup for each problem (they’re actually just variations of the same problem: 30% of 200 is 60) is essentially the same. We had one unknown, either the part, the whole, or the percentage, and we wrote down everything we knew.

Every proportion will therefore look like:

Once you’ve set up the proportion, cross multiply to solve.

Question 17 of 20

Ben purchased a computer and paid 45% of the price immediately. If he paid $810 immediately, what is the total price of the computer?

Here’s How to Crack It

Let’s find the parts of the proportion that we know. We know he paid 45%, so we know the percentage, and we know the part he paid: $810. We’re missing the whole price. On your scratch paper, set up the proportion . Cross multiply, and you’ll get 45x = 81,000. Divide both sides by 45 (feel free to use the calculator here), and you’ll get x = $1,800.

Tip Calculation

The last method for calculating percentages is a variation on a method many people use to calculate the tip for a meal.

To find 10% of any number, simply move the decimal one place to the left.

10% of 100 = 10.0

10% of 30 = 3.0

10% of 75 = 7.5

10% of 128 = 12.8

10% of 87.9 = 8.79

To find 1% of any number, move the decimal two places to the left.

1% of 100 = 1.00

1% of 70 = 0.70

1% of 5 = 0.05

1% of 2,145 = 21.45

You can then find the value of any percentage by breaking the percentage into 1%, 10%, and 100% pieces. Remember that 5% is half of 10%, and 50% is half of 100%.

5% of 60 = half of 10% = half of 6 = 3

20% of 35 = 10% + 10% = 3.5 + 3.5 = 7

52% of 210 = 50% + 1% + 1% = 105 + 2.1 + 2.1 = 109.2

40% of 70 = 10% + 10% + 10% + 10% = 7 + 7 + 7 + 7 = 28

Generally, we’ll use this most often with Ballparking, especially in Charts and Graphs questions.

Question 18 of 20

Last month Dave spent at least 20% but no more than 25% of his monthly income on groceries. If his monthly income was $2,080.67, which of the following could be the amount Dave spent on groceries?

Indicate all such values.

$391.92

$432.88

$456.02

$497.13

$530.17

$545.60

$592.43

Here’s How to Crack It

Ugh. Those are some ugly numbers. Let’s Ballpark a little bit, and use some quick tip calculations to simplify. First, write down A B C D E F G vertically on your scratch paper. We’re looking for a number greater than 20% of $2,080.67. Ignore the 67 cents for now. If any answers are only a couple pennies away from our answer, then we can go back and use more exact numbers, but that’s fairly unlikely. 10% of $2,080 is $208, which means that 20% is $208 + $208 = $416. Dave must have spent at least $416 on groceries, so cross off answer (A). We know he spent no more than 25% of his income. 5% of $2,080 is half of 10%, which means that 5% of $2,080 is $104, and 25% of $2,080 is 10% + 10% + 5% = $208 + $208 + $104 = $520. He couldn’t have spent more than $520 on groceries, so cross off answers (E), (F), and (G). The answers are everything between $416 and $520: (B), (C), and (D).

Percent Change

As we explored in the introduction, percent change is based on two quantities: the change and the original amount.

% change = × 100

Question 19 of 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

At first glance, you might assume that the answer is (C), because the numbers in each quantity look so similar. However, even though both quantities changed by a value of 1, the original amounts are different. When you follow the formula, you find that Quantity A equals × 100, or 10%, while Quantity B equals × 100, or 9%. The answer is (A).

Trigger: A question asks
for “percent change,”
“percent increase,” or
“percent decrease.”

Response: Write the
percentage change
formula.

Percentage change also factors into a lot of “real-world math,” so we’ll talk about it more in Chapter 5. From here, we head to a powerful little device that helps us convey very big and very small numbers with very little effort: exponents. Before we get into exponents, though, let’s practice those percents.

Percents Quick Quiz

Question 1 of 6

What is 26% of 3,750 ?

753

975

1,005

2,775

9,750

Question 2 of 6

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 3 of 6

A think tank projects that people over the age of 65 will comprise 25% of the U.S. population by 2010. If the current population of 300 million is expected to grow by 8% by 2010, how many people over age 65, in millions, will there be in 2010?

Question 4 of 6

Marat sold his condo at a price that was 18% more than the price he paid for it. If he bought his condo at a price that was 42% less than the buyer’s $175,000 asking price, which of the following must be true?

Indicate all such values.

The person who sold Marat the condo lost money.

Marat bought the condo for $101,500.

36 percent of the price at which Marat sold the condo is $43,117.20.

Question 5 of 6

If 25% of p equals 65% of 80 and if q is 50% of p, which of the following must be true?

Indicate all such values.

65 is 62.5% of q

q is 130% of 80

p is 200% of q

= 50% of 1

Question 6 of 6

When John withdraws x% of his $13,900 balance from his checking account, his new balance is less than $10,000.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanations for Percents Quick Quiz

1. Convert. 26% is close to 25%, so what is of 3,750? Only answer choice (B) is close.

2. In Quantity A, 20% or of 300 is 60. 100% of 60 is 60, so 200% must be 120. In Quantity B, 25% or of 400 is 100. 20% of 100 is 20, so 120% is 120. The two quantities are equal, and the answer is (C).

3. 1% of 300 million is 3 million, so 8% is 24 million. If the population grew by 8% that means that it added another 24 million people, so the new total is 324 million. 25% or of 324 million is 81. So there will be approximately 81 million people over 65 in the U.S. in 2010 and 81 would be entered in the box.

4. Take bite-sized pieces. Marat bought his condo for 58 percent of $175,000 = $101,500. (Alternatively, you could take 42 percent of $175,000 and subtract that from $175,000 to get $101,500.)
Choice (B) is correct. Marat later sold his condo for 18 percent more than he paid for it. 1.18 × $101,500 = $119,770. 36 percent of that selling price is 0.36 × $119,700 = $43,117.20. Choice (C) is correct. Because you only know the asking price of the person who sold Marat the condo, and not how much he or she paid for it, choice (A) may or may not be true. Eliminate it. The correct answers are choices (B) and (C).

5. Start by translating: 0.25p = 0.65(80), so 0.25p = 52 and p = 208. That means q = 0.50(208) = 104. Now replace p and q in the answer choices with the appropriate values. For choice (A), you have 65 = × 104, which is true. Choice (A) is correct. For choice (B), 104 = × 80. This equation is also true, so keep choice (B). Choice (C) is also correct because 208 = × 104. Finally, choice (D) is correct because = , which is 50 percent of 1. The correct answers are choices (A), (B), (C), and (D).

6. Whenever you see a variable in one column and an actual number in the other column, try plugging the number in for the variable. In this case, if John withdrew 25% of his savings that would be $3,475 and his balance would still be $10,425. Therefore he must withdraw more than 25% in order for his balance to dip below $10,000. The answer is (A).

EXPONENTS

The superscripted number to the upper right corner of an integer or other math term is called an exponent, and it tells you how many times that number or variable is multiplied by itself. For example, 5⁴ = 5 × 5 × 5 × 5. The exponent is 4, and the base is 5.

You can add or subtract two exponential terms as long as both the base and the exponents are the same.

5x⁵ + x⁵ = 6x⁵

6b³ − 4b³ = 2b³

15ab²c³ − 9ab²c³ + 2ab²c³ = 8ab²c³

Multiplying and Dividing Exponential Terms

Let’s say you’re multiplying a² × a³. If we expand out those terms, then we get (a × a)(a × a × a) = a⁵ = a⁽²⁺³⁾. So when multiplying terms that have the same base, you add the exponents.

Note that the terms have to have the same base. If we’re presented with something like a² × b³, then we can’t simplify it any further than that.

Now let’s deal with division. Let’s start by expanding out an exponent problem using division, to see what we can eliminate.

Notice that the three a terms on the bottom canceled out with three a terms on top? We ended up with a², which is the same as a⁽⁵⁻³⁾. So when dividing terms that have the same base, subtract the exponents.

Parentheses with exponents work exactly as they do with normal multiplication. (ab)³ = (ab)(ab)(ab) = a³b³. When a term inside parentheses is raised to a power, the exponent is applied to each individual term within the parentheses.

What if a term inside the parentheses already has an exponent? For example, what if we have (a²)³ ? Well, that’s (a²)(a²)(a²) = (a × a)(a × a)(a × a) = a⁶ = a^2×3. So when you raise a term with an exponent to another power, multiply the two exponents.

To review, the big rules with exponents are:

· When Multiplying terms with the same base, Add the exponents.

· When Dividing terms with the same base, Subtract the exponents.

· When raising a term with an exponent to another Power, Multiply the exponents.

Most exponent questions will use one or more of these rules. You can memorize them as MADSPM: Multiply, Add, Divide, Subtract, Power, Multiply.

When in Doubt, Expand It Out

If you ever have trouble remembering any of these rules, you can always fall back on a very valuable guideline: “When in doubt, expand it out.” Here’s an example of how this works:

Question 8 of 20

Which of the following expressions is equivalent to (2x)³(5x²)(x⁴)⁵ ?

10x¹⁵

20x¹⁵

20x²⁵

40x¹⁵

40x²⁵

Here’s How to Crack It

When you expand everything out, the factors look like this:

(2x)³(5x²)(x⁴)⁵ = [(2x) · (2x) · (2x)] · [5 · x · x] · [(x⁴) · (x⁴) · (x⁴) · (x⁴) · (x⁴)]

= 2 · 2 · 2 · 5 · x¹ · x¹ · x¹ · x¹ · x¹ · x⁴ · x⁴ · x⁴ · x⁴ · x⁴

= 40 · x^{(1+1+1+1+1+4+4+4+4+4)} = 40x²⁵

The answer is (E).

Rules, Quirks, Anomalies, and Other Weirdness

There are several other general peculiarities about exponential terms that you should at least appreciate.

· Any number raised to the first power equals itself: 5¹ = 5.

· Any nonzero number raised to the zero power equals one: 5⁰ = 1.

· Raising a negative number to an even power results in a positive number: (−2)⁴ = 16.

· Raising a negative number to an odd power results in a negative number: (−2)⁵ = −32.

· Raising a fraction between 0 and 1 to a power greater than one results in a smaller number: .

· Finding a root of a fraction between 0 and 1 results in a greater number: .

Comparing Exponential Terms

Comparing two exponential terms is easy if all of the numbers are positive and the two terms have the same base. Just by looking, you can determine that 3⁵ is greater than 3⁴, because 5 is greater than 4. So what happens when the bases are different?

Question 9 of 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

Holy smokes. Those two numbers sure look huge, don’t they? And they are, but lucky for you their exact value isn’t important. All you have to do is compare them. So resist the temptation to plug 64 × 64 × 64 × 64 × 64 into your calculator. That’s exactly what ETS wants you to do, because it’s clumsy and time-consuming. The way to improve your math score is to recognize patterns that save time.

So which number is greater? Is it Quantity A, which has the greater base? Or is it Quantity B, which has the greater exponent? Let’s find out by first looking for common bases.

Both 16 and 64 are multiples of 4. In fact, 16 = 4 × 4, or 4², and 64 = 4 × 4 × 4, or 4³. Therefore, you can rewrite each of the numbers above using 4 as the common base, and if you apply your newfound knowledge of exponential rules, the comparison becomes much more apparent:

64⁵ = (4³)⁵ = 4³ × ⁵ = 4¹⁵

16⁸ = (4²)⁸ = 4² × ⁸ = 4¹⁶

Do we care how much either of those two quantities is? Absolutely not. Because all the numbers are positive, it must be true that 4 raised to the greater power is the greater number. Therefore, the answer is (B).

Trigger: Exponent
problem with large
numbers.

Response: Factor the
base to compare
exponents.

Negative Exponents

Any number raised to a negative exponent can be rewritten in reciprocal form with a positive exponent: x^-3 = , so 8^-3 = . You can analyze a Quant Comp question that involves negative exponents in much the same way.

Question 4 of 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

Because 27 and 9 are both multiples of 3, you can rewrite each quantity using 3 as the common base.

27⁻⁴ = (3³)⁻⁴ = 3^{3 × −4} = 3⁻¹²

9⁻⁸ = (3²)⁻⁸ = 3^{2 × −8} = 3⁻¹⁶

Again, we don’t care about the actual values of these numbers, which are very, very small. All we need to do is compare the exponents: Because −12 is greater than −16, the answer is (A).

Scientific Notation

In real life, the chief purpose of exponents is to express humongous or teeny-tiny numbers conveniently. The sciences are chock full of these sorts of numbers—such as Avogadro’s number (6.022 × 10²³), which helps chemistry students determine the molecular weight of each element in the periodic table. On the GRE, you probably won’t see Avogadro’s number, but you may well see a number expressed in scientific notation. Scientific notation is basically a number multiplied by 10 raised to a positive or negative power.

Scientific notation merely serves to make unwieldy numbers a little more manageable. If you see one on the GRE, just remember these rules:

· If the exponent is positive, then move the decimal point that many spaces to the right (6.022 × 10²³ = 602,200,000,000,000,000,000,000); and

· If the exponent is negative, then move the decimal point that many spaces to the left (4.5 × 10⁻¹⁵ = 0.0000000000000045).

Fractional Exponents

Any number raised to a fractional exponent can be rewritten as a root: (or (or “the cube root of x”), so = 2.

Question 11 of 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

Quantity A can be rewritten as , which equals 4. Quantity B is a little trickier, but you can figure it out if you follow a few of the rules we’ve discussed in this chapter.

Both quantities are equal to 4, so the answer is (C).

What’s that? Never heard of roots? Well, roots are covered in the next section, right after the quick quiz.

Exponents Quick Quiz

Question 1 of 4

What is the product of 2a³ and 5a⁷ ?

7a¹⁰

7a²¹

10a¹⁰

10a²¹

14a¹⁵

Question 2 of 4

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 3 of 4

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 4 of 4

Which of the following is equivalent to ?

2³ × 3²

Explanations for Exponents Quick Quiz

1. Multiply the coefficients first: 2 × 5 = 10. When you multiply the exponential terms, add 3 and 7 to get 10. The combined term is 10a¹⁰, and the answer is (C).

2. This is a tricky one, because it looks like Quantity A will always be greater due to the greater exponent and greater coefficient. If x is negative, however, it’s a different story. Because −1 to any odd power is negative and −1 to any even power is positive, Quantity B is greater. The answer is (D).

3. Quantity A translates to , or . You can use the Bowtie to add these and get , which happens to equal . The answer is (C).

4. An exponent of is equivalent to a square root, so you can rewrite the term as ÷ 4, which equals 12 ÷ 4, or 3. An exponent of is equivalent to a cube root, and because 3 is the cube root of 27, the answer is (A).

ROOTS

A square root denoted by the funny little check mark with an adjoining roof: . The number inside the house is called a radicand.

Square roots can cause a lot of confusion and despair because it can be hard to remember when you can combine them and when you can’t. As a result, you see people adding square roots like this:

WRONG: + =

This is absolutely wrong, and it’s easy to prove it: = 2 and = 3, so the left side of the equation can be rewritten as 2 + 3, or 5. 5 is equal to , not .

If you ever find yourself in a jam when it comes to remembering square-root rules, try fiddling around with some numbers as we did in that last example. You’ll start to see which manipulations of square roots work and which don’t, and that will help you understand the rules more easily.

Perfect Squares

If the square root of a number is an integer, then that number is known as a perfect square. Perfect squares will come up a lot in the rest of the chapter, and it pays to be able to recognize them on sight. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 (and it couldn’t hurt if you added the next five—121, 144, 169, 196, and 225—to your list).

Knowing your perfect squares is a great tool to use when estimating. Because 70 is between 64 and 81, for example, it must be true that is between 8 and 9, because = 8 and = 9.

The on-screen calculator can find any square root you’re unsure of, but get in the habit of being able to estimate square roots for numbers up to 200. It’ll save you time, since you won’t have to keep bringing up the calculator, and it’ll help you realize when you’ve made a mistake entering something in your calculator. Feel free to go back to the calculator if you’re not sure, but learn those perfect squares.

Question 19 of 20

How many even integers are there between and ?

3

5

10

50

100

Here’s How to Crack It

7² is 49. 8² is 64. The , therefore, is a number just greater than 7 but much less than 8. 12² is 144, and 13 squared is 169. The , therefore, is a number just greater than 12 but a lot less than 13. Count the even integers between 7 and 12, including the 12: 8, 10, 12. The answer is (A).

Multiplying Roots

Multiplying two square roots is a rather straightforward process; you just multiply the numbers and put the result under a new square root sign.

× = =

If there are numbers both outside and inside the square-root sign, you multiply them separately and then put the pieces together.

6 × 2 = (6 × 2) = = 12

Dividing Roots

Dividing roots involves much the same process, but in reverse. When you divide roots, it’s often helpful to set the division up as a fraction, like this:

Again, if you have numbers both outside and inside the square root sign, divide them separately, like this:

Simplifying Roots

If the radicand has a factor that is a perfect square, the term can be simplified. The GRE doesn’t ask you to do this very often, but it pays to have the ability, just in case your answer is not in its simplest form but the answer choices are.

When the two expressions were multiplied in the above example, the answer included the term . Because 45 = 9 × 5, and 9 is a perfect square, you can simplify the expression using factoring and the rules of multiplication that you just learned. In fact, you just follow the directions in reverse order.

= = × = 3

If there’s already a number sitting outside the square-root sign, you will need to combine it with the simplified term like this:

12 = 12× = 12 × × = 12 × 3 × = 36

Adding and Subtracting Roots

You can add and subtract square roots just like variables as long as they have the same radicands.

5 + 3 = 8

9 − 2 = 7

If the radicands are different, however, you can’t do a thing with them; + , for example, cannot be combined or simplified. The only way you can hope to combine two square roots that have different radicands is if you factor them to find a common radicand. The key to factoring is to determine if the radicands have factors that are perfect squares.

For example, look at the expression 2 + :

2 = 2 × = 2 × × = 2 × 2 × = 4

= = × = 5 × = 5

Therefore, 2 + can be rewritten as 4 + 5, which equals 9.

Rationalizing Roots

There’s a rule in lots of moldy old math textbooks that says you have to rationalize a square root in the denominator of a fraction. You should be able to recognize equivalent values of fractions with square roots in them, especially since answer choices on the GRE are most commonly in rationalized form. (This is especially important for geometry questions, as we’ll discuss in Chapter 7.) In order to rationalize a fraction with a square root in its denominator, you need to multiply both the numerator and the denominator by the square root.

To rationalize , for example, watch what happens when you multiply the top and bottom by .

See? The denominator is now an integer, and everything’s all nice and legal.

Roots Quick Quiz

Question 1 of 4

5 × 2 =

20

30

40

50

60

Question 2 of 4

12 ÷ 2

6

12

18

60

240

Question 3 of 4

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 4 of 4

3 + 2 =

10

12

15

19

180

Explanations for Roots Quick Quiz

1. Multiply the numbers outside the square-root sign first: 5 × 2 = 10. Next, multiply the radicands: 18 × 10 = 180. The combined expression is 10, but you’re not done. The greatest perfect square that is a factor of 180 is 36, so factor it out: = = × = 6. The new expression is 10 × 6, which combines to 60. The answer is (E).

2. Divide the coefficients first: 12 ÷ 2 = 6. Now you can divide the radicands to get a new one: ÷ = = , which, when simplified, becomes 2. The new expression is 6 × 2, which combines to 12. The answer is (B).

3. The is 10. Use a calculator to find that is a number between 31 and 32. Now count the multiples of 3: 12, 15, 18, 21, 24, 27, and 30. There are seven. The answer is (B).

4. You can’t do a thing until you convert the terms so that they have the same radicand. The first term, 3, can be simplified to 3 × 3, or 9. The second term, 2, can be rewritten as 2 × 5, or 10. Now, you can add the terms, which combine to 19. The answer is (D).

This was a long chapter, chock full of mathematical goodness. To see how well you retained it, try these questions and review the answers that immediately follow. We’ll see you over in the next chapter, which tells you how much you should know about algebra—and how little of it you should use.

Nuts and Bolts Drill

Let’s review and test your new skills on the nuts and bolts of GRE math in the following drill. Remember to work carefully!

Question 1 of 19

What is the sum of the distinct prime factors of 36 ?

2

5

6

8

10

Question 2 of 19

If a certain fraction with a numerator of 12 has a value of 0.25, then the denominator is

3

12

25

48

300

Question 3 of 19

If 5 less than is −1, then x =

−10

 −4

 4

 8

 10

Question 4 of 19

The numbers that correspond to points A, B, and C on the number line are −, −, and, respectively. Which of the following values fall between the average (arithmetic mean) of A and B and the average of B and C ?

Indicate all such values.

−1

−

 0

 1

Question 5 of 19

If > 1, then each of the following could be the value of x EXCEPT

0.0011

0.0013

0.0015

0.0017

0.0091

Question 6 of 19

27⁰ + 36⁰ =

0

2

9

15

63

Question 7 of 19

How many positive integers that are multiples of 3 are also divisors of 42 ?

One

Two

Three

Four

Five

Question 8 of 19

In a certain hardware store, 3 percent of the lawnmowers need new labels. If the price per label is $4 and the total cost for new lawnmower labels is $96, how many lawnmowers are in the hardware store?

1,600

800

240

120

24

Question 9 of 19

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 10 of 19

What is the greatest possible value of integer n if 6ⁿ < 36¹⁰ ?

6

9

12

15

19

Question 11 of 19

5 × 10³ is what percent of × 10² ?

2,500%

4,900%

20,000%

24,900%

25,000%

Question 12 of 19

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 13 of 19

A is a positive odd number less than 5. Which of the following could be the value of + 3 ?

9

9.5

10

10.5

11

Question 14 of 19

In the number 4,A34, A represents a digit. For which of the following values of A is 4,A34 a multiple of 3 ?

2

3

4

6

9

Question 15 of 19

At a certain animal shelter, the previous annual budget for cat food was $3,900 and the current cat food budget is 125 percent greater than the previous budget. What is the amount, in dollars, of the current annual cat food budget?

Question 16 of 19

Which of the following are true statements?

I. 5³ × = 1

II. 5 ×

III.  = 1

I only

II only

III only

I and II only

II and III only

Question 17 of 19

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Question 18 of 19

If 15% of j is greater than 55% of k, and 325 < k < 375, then which of the following are possible values for j ?

Indicate all such values.

57

332

782

1,087

1,192

1,314

Question 19 of 19

How many more integers, between 552 and 652, exclusive, are divisible by 3 than are divisible by 4 ?

1

6

8

9

12

EXPLANATIONS FOR NUTS AND BOLTS DRILL

1. B

Find the prime factors of 36: 2, 2, 3, and 3. Distinct means different, so the distinct prime factors are 2 and 3, and the sum is 5.

2. D

0.25 is equivalent to . If you multiply the top and bottom of the fraction by 12, you get . The denominator is 48.

3. E

Translate “5 less than is −1” into an equation: − 5 = −1, or = 4. Solve for x; the answer is choice (E) because = 4.

4. A and B

The average of A and B is . The average of B and C, using the same type of calculation, equals 0. Only choices (A) and (B) fall within this range.

5. E

Write out each answer choice as the denominator of a fraction with 0.0017 as the numerator. Move the decimal points to the right the same number of places in both the numerator and the denominator until it’s clear whether the fraction is greater than or equal to 1. Either way, be careful about choice (D), which equals 1, and so isn’t correct. Only choice (E) is not greater than or equal to 1: , which is less than 1.

6. B

By definition, any number raised to the power of 0 is equal to 1, and 1 + 1 = 2.

7. D

You’re looking for numbers that are both multiples of 3 and divisors of 42. There is an infinite number of multiples of 3 but a limited number of divisors of 42, so start there. The divisors (factors) of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Which of these are also multiples of 3? There are four numbers that fit the bill—3, 6, 21, and 42, so the answer is choice (D).

8. B

The question states that the total cost for the labels is $96 so divide that by the $4 cost per label to get the total number of lawnmowers in need of one. That gives 24 which is 3% of the total number of lawnmowers. 0.03 × total = 24 so the total is 800 and the answer is choice (B).

9. C

The order of operations is the key here: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Quantity A simplifies to 15 − 18 ÷ 9 × 8 + 2. The division and multiplication go in left to right order, so 15 − (18 ÷ 9) × 8 + 2 = 15 − (2 × 8) + 2 = 15 − 16 + 2 = 1. The two quantities are therefore always equal and choice (C) is correct.

10. E

Start by finding a common base. 36 is also 6². So 36¹⁰ = (6²)¹⁰ or 6²⁰. 19 is the greatest possible value for n for which 6ⁿ is still less than 6²⁰. The answer is choice (E).

11. E

Translate into an equation: 5 × 10³ = × × 10² so = × , and 50 = . Therefore, x = 25,000.

12. C

If a problem asks you to add or subtract large exponents, it is an opportunity to factor and look for common bases. You can factor out 9¹⁵ from the numerator of the fraction in Quantity A, giving you 9¹⁵ (9² − 1) which equals 9¹⁵ (80). The next step is to see if you can factor the 12² in the denominator. 12 is a multiple of both 3 and 4, so 12² contains 3² and 4². Since 3² = 9, you can subtract one 9 from the numerator; 4² = 16, which does not contain any threes or nines to cancel. But 80 ÷ 16 = 5, so after simplifying the fraction you get 9¹⁴ × 5. Since this matches the value in Quantity B, the correct answer is choice (C).

13. B

There are only two positive odd numbers less than 5: 1 and 3. Since the calculation is pretty straightfoward, start by plugging in 1 for A. The result is 8.5 which is not one of the answer choices. Next Plug In 3 for A and the result is 9.5. Only answer choice (B) works.

14. C

Test the answer choices using the divisibility-of-three rule: The sum of the digits should be divisible by three. For choice (A), the number is 4,234. The sum of the digits is 4 + 2 + 3 + 4 = 13. Because 13 is not divisible by 3, 4,234 is not divisible by three. For choice (B), the sum is 4 + 3 + 3 + 4 = 14, which is not divisible by three. For the choice (C), the sum is 4 + 4 + 3 + 4 = 15, which is divisible by three. For choice (D), the sum is 4 + 6 + 3 + 4 = 17, which is not divisible by three. For choice (E), the sum is 4 + 9 + 3 + 4 = 20, which is not divisible by three. The answer, therefore, is (C).

15. 8,775

Since the question says 125 percent greater than, use the percent change formula: Percent change is × 100. This means that 125% = and you can solve this to get $4,875. This is the difference, not the total for the current year. To get the current total, add $4,875 to the original $3,900 to get the correct amount: 8,775.

16. E

Evaluate each equation. The first one equals , which does not equal 1, so eliminate choices (A) and (D). The next equation equals 5 × , or 5 × ; this equation does equal 1, so eliminate choice (C). The third equation equals , which also equals 1, so this equation works as well; choice (E) is correct.

17. B

Simplify the value in Quantity A to make it easier to approximate: . Since = 12, the value in Quantity A is less than 12, and the value in Quantity B is greater. Alternately, you could turn the value in Quantity B into a root: 15 = , and is greater than .

18. E and F

Use percent translation to establish the least possible value of j: × j > × 325. Solve the inequality for j to get j > 1191 . Since no limits are set on the greatest possible value of j, any value greater than 1191 will work so the answer is choices (E) and (F).

19. D

The most reliable way to approach this question is to count out the multiples of 3 and of 4, then subtract. It appears you have a range of 100, which when divided by 3 equals approximately 33, and when divided by 4 equals 25. 33 − 25 = 8, which is choice (C), but this is incorrect. There are 33 multiples of 3, but since 552 and 652 are excluded, there are only 24 multiples of 4. 33 − 24 = 9, which is choice (D).