GMAT Foundations of Math

CHAPTER 1

Arithmetic

In This Chapter

•Quick-Start Definitions

•Basic Numbers

•Greater Than and Less Than

•Adding and Subtracting Positives and Negatives

•Multiplying and Dividing

•Distributing and Factoring

•Multiplying Positives and Negatives

•Fractions and Decimals

•Divisibility and Even and Odd Integers

•Exponents and Roots (and π)

•Variable Expressions and Equations

•PEMDAS

•Combining Like Terms

•Distribution

•Pulling Out a Common Factor

•Long Addition and Subtraction

•Long Multiplication

•Long Division

In this chapter, you will learn to recognize math vocabulary so that you can set problems up correctly. You’ll also learn how to apply PEMDAS, simplify math, and solve in ways that avoid tedious calculations.

CHAPTER 1 Arithmetic

This book has three goals for you. First, you will review fundamental math skills. Second, you will learn how to approach the math with a business focus: how to get to the correct answer quickly and effectively while avoiding drawn-out math calculations that aren’t necessary on the GMAT. Third, you will practice applying these skills.

To practice applying these skills, there are a number of Check Your Skills questions throughout each chapter. After learning a topic, try these problems, checking your answers at the end of the chapter as you go.

If you find these skill checks challenging, reread the section you just finished. Then, try the questions again. Whenever needed, use the solution to help you work through the math step-by-step. When you get stuck, don’t read the entire solution immediately. Read as much as you need to get yourself unstuck, then continue to try to do the work on your own. Return to the solution each time you get stuck and use only the portion of the solution you need to help unstick yourself.

Quick-Start Definitions

Whether you work with numbers every day or only do math when you absolutely have to, read this first section carefully. It provides definitions for certain core concepts in order to give your studies a quick start. You’ll come back to many of these concepts during your studies. You can also find a glossary of common math terms at the back of this guide.

Basic Numbers

All the numbers that come into play on the GMAT can be shown as a point somewhere on the number line:

Another word for number is value.

Positive integers are the counting numbers 1, 2, 3, and so on. These are the first numbers that you ever learned—the numbers that you use to count individual items. They are whole numbers; they do not include any fractions or decimals:

Digits are 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) used to represent numbers. If the GMAT asks you specifically for a digit, it wants one of these 10 symbols.

Counting numbers above 9 are represented by two or more digits. The number four hundred twelve is represented by three digits in this order: 412.

Place value tells you how much a digit in a specific position is worth. The 4 in 412 is worth 4 hundreds (400), so 4 is the hundreds digit of 412. Meanwhile, 1 is the tens digit and is worth 1 ten (10). Finally, 2 is the units digit and is worth 2 units, or just plain old 2:

The GMAT always separates the thousands digit from the hundreds digit by a comma. For readability, big numbers are broken up by commas placed three digits apart:

1,298,023 equals one million two hundred ninety-eight thousand twenty-three.

Addition (+, or plus) is the most basic operation in arithmetic. If you add one positive number to another, you get a third positive number farther to the right on the number line. For example:

Therefore, 12 is the sum of 7 and 5. By the way, the statement 7 + 5 = 12 is called an equation.

When you add two numbers in either order, you get the same result:

Subtraction (−, or minus) is the opposite of addition. Subtraction undoes addition. For example:

Order matters in subtraction: 6 − 2 = 4, but 2 − 6 = something else (more on this in a minute). By the way, since 6 − 2 = 4, the difference between 6 and 2 is 4.

Any number minus itself equals zero (0):

Any number plus zero equals that starting number. The same is true if you subtract zero. For example:

In either case, you’re moving zero units away from the original number on the number line, or staying right where you started.

Numbers can be positive, negative, or neither. Positive numbers lie to the right of zero on a number line. Negative numbers lie to the left of zero. Zero itself is neither positive nor negative—it’s the only number that is neither:

The sign of a number indicates whether the number is positive or negative.

Negative integers are −1, −2, −3, and so on:

When you subtract a greater number from a smaller number, you will end up with a negative number. For example:

Negative numbers can be used to represent deficits. If you have $2 but you owe $6, your net worth is −$4.

Most people tend to make more mistakes when dealing with negative numbers than with positive ones. If you’re asked to find small minus big, you may instead want to figure out big minus small, then make the result negative. For example:

What is 35 − 57 ?

Since big minus small is equal to 22, the reverse—small minus big—is equal to −22.

The work above represents what is called textbook math. There are other, often better, ways to think your way through the math that needs to be done on the GMAT—and these methods are more useful in the real world as well.

For example, when you want to find 57 minus 35, you are really looking for the distance between the two numbers. Visualize 35 and 57 on a number line, and then count the distance between them using whatever increments seem reasonable to you:

As the diagram above shows, first count by 10 up to 45, then by another 10 up to 55, and finally by 2 to get to 57. The distance between the two numbers is 10 + 10 + 2 = 22.

If the problem asks for 57 − 35, then the answer is 22. If the problem asks for 35 − 57, then add a negative sign: The answer is −22.

Check Your Skills

Perform addition and subtraction.

1.What is 37 + 141 ?

2.What is 23 − 136 ?

Answers can be found on page 37.

Greater Than and Less Than

Greater than (>) means to the right of on a number line:

The statement 7 > 3 is an example of an inequality.

In the real world, people also say bigger than or larger than, but there is one drawback to using this terminology. Take a look at this example:

A lot of people will intuitively call −7 the bigger number because, when the numbers are positive, 7 is greater than 3. But that thinking doesn’t work the same way for negative numbers! The value of −7 is less than the value of −3.

So stick to greater than and less than. Greater numbers are to the right of smaller numbers on the number line.

Any positive number is greater than any negative number. For example:

Likewise, zero is greater than every negative number:

Less than (<) means to the left of on a number line. You can always reexpress a greater than relationship as a less than relationship—just flip it around. For example:

If 7 is greater than 3, then 3 is less than 7.

If you find yourself making mistakes with negatives, look for opportunities to think about negatives in terms of the number line. Test out the following true statements, or inequalities, on a number line:

Inequalities are very similar to equations, but equations always use equals signs (=), while inequalities use greater than (>) or less than (<) signs. Later in this guide, you’ll learn more about how to work with both equations and inequalities.

Check Your Skills

3.What is the sum of the greatest negative integer and the smallest positive integer?

For questions 4 and 5, plug in > and < symbols and say the resulting statement aloud.

4.  5 16

5.−5 −16

Answers can be found on page 37.

Adding and Subtracting Positives and Negatives

Positive plus positive gives you a positive result. For example:

When adding two positives, you move farther to the right of zero, so the result is always greater than either starting number.

Positive minus positive, on the other hand, could give you either a positive or a negative:

Either way, the result is less than where you started because you move left when you subtract a positive.

Adding a negative is the same as subtracting a positive—you move left. For example:

In fact, you can write 8 + (−3) as 8 − 3. They both mean the exact same thing.

Negative plus negative always gives you a negative because you move even farther to the left of zero:

Subtracting a negative is the same as adding a positive—you move right. Add in parentheses to keep the two minus signs straight:

The equation above is the same as the equation below:

In both cases, you end up with the following:

In general, any subtraction can be rewritten as addition. If you’re subtracting a positive, you can instead add a negative. If you’re subtracting a negative, you can instead add a positive.

Check Your Skills

6.Which is greater, a positive minus a negative or a negative minus a positive?

Answer can be found on page 37.

Multiplying and Dividing

Multiplication (×, or times) is the same as adding the same number multiple times. Imagine that there are three tennis balls per can and you have four cans. How many tennis balls do you have?

Therefore, 12 is the product of 4 and 3. Also, 4 and 3 are called factors of 12.

Parentheses can be used to indicate multiplication. Parentheses are usually written with ( ), but brackets [ ] can also be used, especially if you have parentheses within parentheses. If a set of parentheses bumps up right against something else, multiply that something by whatever is in the parentheses:

4(3) = (4)3 = (4)(3) = 4 × 3 = 12

When writing multiplication, if you use a × symbol, make sure to write it in such a way that you won’t confuse it for the variable x. Also, you may have learned to use a dot to mean multiplication:

4 • 3 = 4 × 3 = 12

The potential danger in using a dot is that you might mistake the dot for a decimal point. If you do want to write a dot, make that dot big and high, as shown above, so it doesn’t look like a decimal point. In general, it’s a good idea to get in the habit of using parentheses rather than • or × so that you minimize your chances of making a careless mistake on the real test.

You can always multiply in either order; you will get the same result, as shown below. (Pop quiz: What other operation that you learned earlier in this section can also be done in either order?)

Pop quiz answer: Addition can also be performed in either order.

Since you are allowed to multiply in any order, don’t necessarily multiply in the given order. Rather, invest a few seconds to think about the easiest order in which to multiply. Try this:

2 × 7 × 5

Most people work left to right, so the second step would be 14 × 5. That’s not terrible, but it’s also not the easiest calculation. Go back to the first step and play around with the numbers. Can you find a way that’s easier?

In general, it’s easy to multiply by multiples of 10, so whenever you see 2’s and 5’s in the mix, rearrange to group them together:

Try this one:

6 × 5 × 7 =

This one has a 5 but no 2. Now what? Actually, there is a 2, lurking inside the 6. Split up the 6 into 2 × 3, and rearrange to pair the 2 with the 5:

In general, think before you multiply. Do the math in whatever order looks best to you—and keep an eye out for 2’s and 5’s.

Division (÷, or divided by) is the opposite of multiplication. Division undoes multiplication:

As with subtraction, order matters in division. 12 ÷ 3 = 4, but 3 ÷ 12 = something else (more on this soon).

Multiplying any number by 1 leaves the number the same. One times anything is equal to whatever you started with:

Multiplying any number by zero (0) gives you zero. Anything times zero is zero:

Since order doesn’t matter in multiplication, this means that zero times anything is zero, too:

Multiplying a number by zero destroys it permanently, in a sense. So you’re not allowed to undo that destruction by dividing by zero.

Never divide by zero: 13 ÷ 0 = undefined. Stop right there—don’t do this! The GMAT avoids getting into undefined-number territory, thankfully.

You are allowed to divide zero by any nonzero number. The answer is—surprise!—zero:

Check Your Skills

Complete the operations.

7.What is 7 × 6 ?

8.What is 52 ÷ 13 ?

Answers can be found on page 37.

Distributing and Factoring

What is 4 × (3 + 2) ? Here’s one way to solve it. Turn (3 + 2) into 5, then multiply 4 by that 5:

If you’re working with real numbers, the process above will usually be the fastest way to solve. But you can solve another way—and that other way is usually necessary when working with algebra or other more advanced math. The other way to solve this problem is to distribute the 4 to both the 3 and the 2:

Notice that you multiply the 4 by both the 3 and the 2. Distributing is extra work in this case, but the technique will come in handy later when you’re doing algebra.

Another way to see how distributing works is to put the sum in front:

In a sense, you’re splitting up the sum 3 + 2. Just be sure to multiply both the 3 and the 2 by 4.

You distribute when the terms inside the parentheses are connected by addition or subtraction signs. Do not distribute if the terms inside the parentheses are connected by multiplication or division; in that case, drop the parentheses and multiply or divide straight across. For example:

Distributing works similarly for subtraction. Just keep track of the minus sign:

Again, for that problem, distribution is more work than it’s worth. You can do the subtraction first, 5 − 2 = 3, and then multiply 6 × 3 to get 18. Just learn how distribution works because you’ll need to do it later with algebra.

You can also go in reverse. You can factor the sum of two products if the products contain the same factor. Pull out the common factor of 4 from each of the products 4 × 3 and 4 × 2. Next, put the sum of 3 and 2 into parentheses:

Most of the time, you’ll see the result written in this form: 4(3 + 2). By the way, common here doesn’t mean frequent or typical. Rather, it means belonging to both products. A common factor is a factor in common (like a friend in common).

You can also write the common factor at the back of each product, if you like:

This is the equivalent of writing (3 + 2)4. Since the order of multiplication doesn’t matter, 4(3 + 2) = (3 + 2)4. Reminder: You can distribute when the math in the parentheses is added or subtracted. Don’t distribute if the math in the parentheses is multiplied or divided.

You will use both distributing and factoring in more advanced ways later in this guide. Keep in mind that using simple numbers to understand the concepts now will help you to apply these same concepts to more complex problems in the future. As you move further in your studies, you will find yourself confused about a problem at times. Try some real numbers to help understand what’s going on, and you may find that you can solve that problem after all!

Check Your Skills

9.Use distribution to solve. What is 5 × (3 + 4) ?

10.Factor a 6 out of the following expression: 36 − 12.

Answers can be found on page 38.

Multiplying Positives and Negatives

Positive × positive is always positive:

Positive × negative is always negative:

Since order doesn’t matter in multiplication, the same outcome happens when you have negative times positive. You again get a negative:

What is negative × negative? Positive. This fact may seem weird, but it’s consistent with the rules developed so far. In the same way that something minus a negative turns into something plus a positive (7 − (−3) = 7 + 3), a negative times a negative also turns positive. In either case, two negatives make a positive.

All the same rules hold true for division:

Notice any pattern there? (Really—examine the information in that table before you keep reading!)

Regardless of whether you’re doing multiplication or division, if the two starting numbers have the same sign (both positive or both negative), then you end up with a positive.

But if the two starting numbers have opposite signs (one positive and one negative), then you end up with a negative.

Check Your Skills

11.What is (2)(−5) ?

12.Use distribution to solve. What is −6 × (−3 + (−5)) ?

Answers can be found on page 38.

Fractions and Decimals

Adding, subtracting, or multiplying integers always results in an integer, whether positive or negative:

Integer + Integer = Integer

Integer − Integer = Integer

Integer × Integer = Integer

However, dividing an integer by another integer does not always give you an integer:

Integer ÷ Integer = Sometimes an integer, sometimes not!

When you don’t get an integer, you get a fraction or a decimal—a number that falls between the integers on the number line:

A horizontal fraction line, or bar, expresses the division of the numerator (above the fraction line) by the denominator (below the fraction line):

In fact, the division symbol ÷ is just a miniature fraction. People often say things such as "seven over two rather than seven halves" to express a fraction.

You can express division in three ways: with a fraction line, with the division symbol (÷), or with a slash (/), as shown here:

A decimal point is used to extend place value to the right for decimals. Each place to the right of the decimal point is worth a tenth , a hundredth , and so on. For example:

A decimal such as 3.5 has an integer part (3) and a fractional part or decimal part (0.5). In fact, an integer is just a number with no fractional or decimal part.

Every fraction can be written as a decimal, although you might need an unending string of digits in the decimal to properly express the fraction:

Fractions and decimals obey all the rules you’ve seen so far about how to add, subtract, multiply, and divide. Everything you’ve learned for integers applies to fractions and decimals as well: how positives and negatives work, how to distribute, etc.

Check Your Skills

13.Which arithmetic operation involving integers does NOT always result in an integer?

14.Write 2 ÷ 7 in fraction form, then multiply that fraction by 0. What is the result?

Answers can be found on page 38.

Divisibility and Even and Odd Integers

Sometimes you do get an integer out of integer division:

In this case, 15 and 3 have a special relationship. You can express this relationship in any of the following ways:

15 is divisible by 3.

15 divided by 3 equals an integer: 15 ÷ 3 = int

3 is a divisor of 15.

15 divided by 3 equals an integer: 15 ÷ 3 = int

15 is a multiple of 3.

15 equals 3 times an integer: 15 = 3 × int

3 is a factor of 15.

3 times an integer equals 15: 3 × int = 15

People also say the following, though you won’t see this language on the test:

3 goes into 15 evenly.

3 divides 15 evenly.

Even integers are divisible by 2:

14 is even because 14 ÷ 2 = 7 = an integer.

All even integers have 0, 2, 4, 6, or 8 as their units digit.

Zero is even because it is divisible by 2:

0 ÷ 2 = 0, which is an integer.

Odd integers are not divisible by 2:

15 is odd because 15 ÷ 2 = 7.5 = not an integer.

All odd integers have 1, 3, 5, 7, or 9 as their units digit.

Even and odd integers alternate on the number line:

Only integers can be said to be even or odd. Fractions or decimals are not considered even or odd.

Check Your Skills

15.Fill in the blank. If 7 is a factor of 21, then 21 is a of 7.

16.Is 2,284,623 divisible by 2 ?

Answers can be found on page 38.

Exponents and Roots (and π)

Earlier, you learned that multiplication is really just repeated addition. Exponents are one level up in the food chain: They represent repeated multiplication.

In 5², the exponent is 2 and the base is 5. The exponent tells you how many of the bases to multiply together to get your answer. When the exponent is 2, you usually say squared:

When the exponent is 3, you usually say cubed:

For other exponents, you say to the _____ power or raised to the _____ power. You could also say to the power of _____. For example:

Be careful when writing exponents on your own paper. Make them much tinier than regular numbers and put them clearly up to the right. You don’t want to mistake 5² for 52, or vice versa.

By the way, a number raised to the first power equals the original number:

A perfect square is the square of an integer:

25 is a perfect square because 25 = 5² = int².

A perfect cube is the cube of an integer:

64 is a perfect cube because 64 = 4³ = int³.

Roots undo exponents. The simplest and most common root is the square root, which undoes squaring. The square root is written with the radical sign ; if a problem refers to the radical, it’s talking about that symbol. Here’s how exponents and roots connect:

As a shortcut, the square root of twenty-five can just be called root twenty-five.

Asking for the square root of 49 is the same as asking what number, times itself, gives you 49:

The square root of a perfect square is an integer, because a perfect square is an integer squared:

The square root of any non-perfect square is an unending decimal that never even repeats, as it turns out:

The square root of 2 can’t be expressed as a simple fraction, either. So you can leave it as is , or you can approximate it .

It’s useful on this test to know that root 2 is approximately 1.4 and root 3 is approximately 1.7. Here’s an easy way to remember that:

Speaking of dates, have you ever heard of Pi Day? (Feel free to google it.) You’ll encounter one other common number with an ugly decimal in geometry: pi (π).

Pi is the ratio of a circle’s circumference to its diameter. It’s about 3.14159265 . . . without ever repeating.

Since pi can’t be expressed as a simple fraction, it is typically represented by the Greek letter π, or you can approximate it (π ≈ 3.14, or a little more than 3). And—you guessed it—Pi Day is 3/14!

Cube roots undo cubing. The cube root has a little 3 tucked into its notch :

Other roots occasionally show up. For example, the fourth root undoes the process of raising a base to the fourth power:

Check Your Skills

17.2⁶ =

18.

Answers can be found on page 38.

Variable Expressions and Equations

Up to now, every number you’ve dealt with has been an actual, known number. Algebra is the art of dealing with unknown numbers.

A variable is an unknown number, or an unknown for short. You represent a variable with a single letter, such as x or y.

When you see y, imagine that it represents a number that you don’t happen to know. At the start of a problem, the value of y is hidden from you. It could be anywhere on the number line, in theory, unless you’re told something about y.

The letter x is the stereotypical letter used for an unknown. Since x looks so much like the multiplication symbol ×, you can prevent mistakes by not using the × symbol when writing out algebra. To represent multiplication, do other things.

To multiply variables, put them next to each other:

To multiply a known number by a variable, write the known number in front of the variable:

Here, 3 is called the coefficient of x. If you want to multiply x by 3, write 3x, not x3, which could look too much like x³ (x cubed).

All the operations besides multiplication look the same for variables as they do for known numbers:

By the way, be careful when you have variables in exponents:

Never refer to 3x as "three x. It’s called three to the x. Calling it three x" is likely to lead to a careless mistake, since three x typically means three times x.

An expression is something that ultimately represents a number; for example, x