The GMAT is known for reducing some students to tears. It’s a tough exam, no doubt, and it covers a wide range of topics and material. It is no surprise that some students become overwhelmed with the perceived enormity of the effort. They discover that it is normal to take 3-6 months to prepare for the GMAT. Thus, they sometimes resort to taking shortcuts so they won’t have to spend so much time studying. One of the more common shortcuts is to skip the material covering GMAT Focus math basics.
Students reason that this basic math stuff is a waste of time. How could they have earned an A in Financial Accounting or College Algebra unless they knew that boring stuff? Here’s the thing: basic math errors are second only to algebra errors in causing students to miss questions. Additionally, if your basic skills are rusty, you’ll take longer to answer more complicated GMAT math questions. Finally, you have no calculator access during the GMAT Focus Quantitative section. So, it’s imperative that you have all the basic skills down pat.
In this article, we’ll assess your quick and accurate recall of common basic GMAT math concepts and skills. Then, we’ll review these in greater detail through Problem Solving questions. Also, many of the concepts covered here can be used on your GMAT math cheat sheet.
Here are the topics we’ll cover:
- Quick Quiz: Basic GMAT Math Skills
- Math Skill #1: Percents
- Math Skill #2: Prime Numbers
- Math Skill #3: Factorial Arithmetic
- Math Skill #4: Square Roots
- Math Skill #5: Fractions
- Math Skill #6: Exponents
- Math Skill #7: PEMDAS
- Math Skill #8: Consecutive Integers
- Math Skill #9: Divisibility Rules
- In Summary: GMAT Math Basics
- What’s Next?
Before we begin our review, let’s assess your knowledge of basic skills.
Quick Quiz: Basic GMAT Math Skills
This quick quiz covers many of the basic skills needed for GMAT Focus mastery. You may use paper and pencil, but no calculator. While we are testing your problem-solving skills, many of these concepts are also tested in Data Sufficiency and Integrated Reasoning. These question types appear in the GMAT Focus Data Insights section.
- What is 30% of 7?
- What are the first 5 prime numbers?
- What is the value of 4! ?
- Evaluate √169
- Evaluate 2/7 + 4/3
- Evaluate 2/8^(-2)
- Evaluate 8 + 6 ➗2(3)
- What is the product of the consecutive integers from -3 to 5, inclusive?
- Is 3,171,942 evenly divisible by 3?
Before checking your answers, think about how you felt as you worked through these questions of basic GMAT math section knowledge. Did you breeze through? Or did you scratch your head on a few? If you’re like the average GMAT student, you had to stop and think about some of these. If so, a detailed review of math basics is in your future!
Here are the answers. In the remainder of this article, we’ll look at each solution in greater detail.
Answers:
- 2.1
- 2, 3, 5, 7, 11
- 24
- 13
- 34/21
- 128
- 17
- 0
- yes
So, how did you do? If you didn’t get 100%, you will definitely benefit from reviewing basic math skills. If you did get 100%, did it take a lot of energy or time to answer correctly? Almost everyone preparing for the GMAT Focus will benefit from a structured review of GMAT math essentials.
Next, we’ll look at the underlying math skills associated with each question. Note that we have provided broad coverage of each skill. For more detailed discussion of these topics and everything else tested on the GMAT, visit our GMAT Focus self-study course.
Math Skill #1: Percents
One of the most important percent formulas to remember for GMAT is that percent means “divide by 100.”
With this in mind, we see that 25% means 25/100, which is equivalent to the decimal 0.25. Here are some important facts about percents:
- To convert from a percent to a decimal, move the decimal point two places to the left. Thus, 56.2% becomes 0.562 when converted to a decimal.
- The phrase “percent of” means “multiply.” Thus, to find 8% of 50, we first convert 8% to the decimal 0.08 and then multiply it by 50, obtaining (0.08) x 50 = 4.
- There are other percent problem types that you should review, including markups, discounts, and percent change.
KEY FACT:
“Percent” means “divide by 100.”
Solution to Quiz Question #1: What is 30% of 7?
We first convert 30% to its decimal equivalent, so we have 30/100, or 0.30. Next, we recall that “percent of” means multiply, and so we have 0.30 x 7 = 2.1.
Example 1: GMAT Percent Question
What is (a% + b%) / c% ?
- ab / c
- (a + b) / c
- (a + b + c) / 10,000
- (ac + bc) / 100
- (ac + bc) / 10,000
Solution:
First, we re-express each percent in fraction form:
(a% + b%) / c% = (a/100 + b/100) / c/100 = [(a + b) / 100] / c/100
The next step is to recall that when we divide by a fraction, we invert the fraction and multiply.
[(a + b) / 100] x 100/c = (a + b) / c
Answer: B
Remember, this is just a foundational percent question. Percents can be used in linear equations and even inequalities when problems become more advanced.
Math Skill #2: Prime Numbers
This skill requires you to know the definition of a prime number. It also is helpful to commit the first 10 primes to memory.
Here are some facts about prime numbers:
- A prime number is any integer greater than 1 that has no factors other than 1 and itself.
- The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
- Note that 1 is not a prime number, and the only even prime number is 2.
- Prime factorization is the act of breaking a number into its prime factors. For example, the prime factorization of 60 is 2 x 2 x 3 x 5, or 2^2 x 3 x 5.
KEY FACT:
A prime number is divisible only by itself and 1.
Solution to Quiz Question #2: What are the first 5 prime numbers?
We have already listed the first 10 primes, noting that 1 is not a prime and 2 is. Thus, the first 5 prime numbers are 2, 3, 5, 7, and 11.
Example 2: GMAT Prime Number Question
How many prime factors do 30 and 35 share?
- 0
- 1
- 2
- 3
- 5
Solution:
While we have not covered prime factorization in detail, we see that 30 and 35 are straightforward to factor. Let’s look at each:
30 = 2 x 3 x 5
35 = 5 x 7
We see that the numbers 30 and 35 share only one prime factor in common: 5.
Answer: B
Math Skill #3: Factorial Arithmetic
The mathematical notation that has an exclamation mark, such as 5! or 13!, indicates multiplication. For example, 5! indicates multiplication of the integers from 5 to 1. Thus, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Here are some useful facts about factorials:
- 0! = 1 and 1! = 1
- Because factorials indicate “backward multiplication,” you can express factorials in a variety of equivalent ways. For example, 8! can be expressed as 8 x 7! or as 8 x 7 x 6 x 5!.
Keep in mind factorials get more challenging as they are used in equations and in combinations and permutations.
KEY FACT:
Factorial notation indicates “backward” multiplication.
Solution to Quiz Question #3: What is the value of 4! ?
Now that we know what factorial notation means, it’s easy to evaluate 4! as 4 x 3 x 2 x 1 = 24.
Example 3: GMAT Factorial Question
What is the value of 11! / 9! ?
- 2
- 21
- 99
- 110
- 5040
Solution:
You may suspect that a GMAT question would not make you do all the multiplication that is seemingly required. Instead, we use cancellation to simplifyz;
11! / 9! = (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
We see that we can cancel out 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 from both the numerator and the denominator. This leaves us with 11 x 10 = 110 in the numerator.
Alternate Solution:
We can re-express 11! as 11 x 10 x 9!:
11! / 9! = (11 x 10 x 9!) / 9!
We see that 9! can be canceled from the numerator and the denominator, leaving us with 11 x 10 = 110.
Answer: D
Math Skill #4: Square Roots
The square root of a number x is a value that, when multiplied by itself, gives x as the result. For example, the square root of 25 is either 5 or -5 because 5^2 = 25 and (-5)^2 = 25.
A source of confusion for many students is a number inside the radical (square root) symbol, such as √25. The only answer is 5. The rule is this: when the square root symbol is used, take only the positive square root. This is called the principal square root.
Here are some facts about square roots:
- When the square root symbol is used, there is only one result, called the principal square root. The principal square root is always positive unless the number is 0.
- √0 = 0
- Know the perfect square root numbers through at least 400.
√1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5
√36 = 6 √49 = 7 √64 = 8 √81 = 9 √100 = 10
√121 = 11 √144 = 12 √169 = 13 √196 = 14 √225 = 15
√256 = 16 √289 = 17 √324 = 18 √361 = 19 √400 = 20 - You can multiply two square roots by finding the product of the two expressions inside each square. Then, take the square root of that product. A similar rule applies to division.
KEY FACT:
When the radical (square root) symbol is used, take only the positive square root.
Solution to Quiz Question #4: What is the value of √169?
Did it take you longer than about 3 seconds to recall that √169 = 13? If so, you would be wasting valuable time on the actual GMAT. It is well worth it to memorize the first 20 perfect square roots.
Example 4: GMAT Square Root Question
What is the value of (√50 / √2) + (√75 x √3) ?
- 15
- 20
- 175
- 235
- 250
Solution:
In this question, we first simplify each term before adding:
√50 / √2 = √(50/2) = √25 = 5
√75 x √3 = √(75 x 3) = √225 = 15
Thus, we have:
(√50 / √2) + (√75 x √3) = 5 + 15 = 20
Answer: B
Math Skill #5: Fractions
If the thought of LCDs, LCMs, and complex fractions makes you shiver in your boots, you are not alone! Many of us have not seen fractions since grade school, so we are unsurprisingly rusty on solving fraction problems. However, fractions are very important because they appear in nearly every GMAT topic, including algebra, functions, coordinate geometry, and statistics.
Here are some facts and techniques you need to know about fractions:
- The number “on the top” is called the numerator, and the number “on the bottom” is called the denominator.
- To reduce a fraction, use a number that divides evenly into both the numerator and the denominator.
- To add or subtract two fractions, you must have a common denominator.
- To multiply two fractions, simply multiply the two numerators and then multiply the two denominators. Reduce the answer, if necessary.
- To divide two fractions, invert the second fraction and then multiply.
KEY FACT:
To add two fractions, you must have a common denominator.
Solution to Quiz Question #5: Evaluate 2/7 + 4/3.
To add two fractions, they must have a common denominator. The easiest way to get the common denominator for two fractions is to simply multiply the two denominators. Thus, for 2/7 and 4/3, the common denominator is 7 x 3 = 21.
We convert each fraction to have a denominator of 21:
2/7 x 3/3 = 6/21
4/3 x 7/7 = 28/21
We can now add the two fractions:
2/7 + 4/3 = 6/21 + 28/21 = 34/21
Example 5: GMAT Fraction Question
If b and d are positive integers, what is the value of a/b + c/d ?
- ac / bd
- ac / (b + d)
- (a + c) / bd
- (ad + bc) / bd
- (ad + bc) / b + d
Solution:
To add two fractions, we need a common denominator. The product of the two denominators of the fractions a/b and c/d is b x d = bd. We convert each fraction such that it has a denominator of bd:
a/b x d/d = ad /bd
c/d x b/b = bc/bd
The two fractions now have a common denominator, bd, so we can add them.
a/b + c/d = ad/bd + bc/bd = (ad + bc) / bd
Answer: D
Math Skill #6: Exponents
Questions containing exponents are pervasive on the GMAT. There are lots of rules, but it will beneficial to remember and use the following:
- Vocabulary is important. For example, if we have 4^3, we say that 4 is the base and 3 is the exponent.
- An exponential expression, such as 4^3, indicates repeated multiplication. We multiply the base (4) by itself as many times as is indicated by the exponent (3), Thus, 4^3 = 4 x 4 x 4 = 64.
- A negative exponent generally indicates moving the number to its opposite location in the fraction. Ater the number is moved, the exponent becomes positive. For example, 6^(-2) = 1 / (6^2) = 1/36, and 1 / (7^-3) = 7^3 = 343.
KEY FACT:
A negative exponent generally indicates making the exponent positive and moving the number to its opposite location in the fraction.
Solution to Quiz Question #6: Evaluate 2 / 8^(-2)
We have a number raised to a negative exponent. Therefore, we move the number 8^(-2) from the denominator to the numerator as 8^2:
2 / 8^(-2) = 2 x 8^2 = 2 x 64 = 128
Example 6: GMAT Exponent Question
Evaluate and simplify [2^(-4) x 10] / [5 x 4^(-3)]
- 1/2
- 1
- 8
- 32
- 64
Solution:
A base raised to a negative exponent is moved to its opposite location in the fraction. Therefore, 2^(-4) moves to the denominator as 2^4. Similarly, 4^(-3) moves to the numerator as 4^3. Thus, we have:
[ 2^(-4) x 10] / [5 x 4^(-3)] = (4^3 x 10) / (5 x 2^4) = (64 x 10) / (5 x 16) = 640 / 80 = 8
Answer: C
Math Skill #7: PEMDAS
Remember the mnemonic “Please Excuse My Dear Aunt Sally”? If so, you probably have a passing knowledge of the order of operations for mathematical expressions.
Here are some facts to keep in mind about PEMDAS:
- P = parentheses E = exponents [M = multiplication D = division] [A = addition S = subtraction]
- Make a note that multiplication is NOT necessarily done prior to division. Rather, when multiplication and division are in an expression, they are performed from left to right. For example, 5 x 6 ➗6 x 2 is equal to 10, not 30/12.
- A similar situation exists with addition and subtraction. When those operations are both in an expression, perform them from left to right.
KEY FACT:
Following PEMDAS ensures that your arithmetic is correct!
Solution to Quiz Question #7: Evaluate 8 + 6 ➗ 2(3)
Following PEMDAS, we first evaluate 6 ➗ 2(3). Because this expression contains only multiplication and division, we work from left to right:
8 + 6 ➗ 2(3) = 8 +3(3) = 8 + 9 = 17
Example 7: GMAT PEMDAS Question
Evaluate 4 x 7 – 30 ➗ 6 + 5(3+2)^2
- 110 ⅓
- 124 ⅔
- 136
- 148
- 193
Solution:
Using PEMDAS, we first add the numbers in the parentheses. Then we square the sum:
4 x 7 – 30 ➗ 6 + 5(3+2)^2 = 4 x 7 – 30 ➗ 6 + 5(5)^2 = 4 x 7 – 30 ➗ 6 + 5(25)
We then perform multiplication and division from left to right:
4 x 7 – 30 ➗ 6 + 5(25) = 28 – 5 + 125
Finally, we perform the subtraction and addition:
28 – 5 + 125 = 23 + 125 = 148
Answer: D
Math Skill #8: Consecutive Integers
This topic is more a trick than an actual skill, but it can be a time-saver. Let’s look at some useful facts about consecutive integers:
- An integer is a whole number and can be negative, positive, or zero.
- Consecutive integers are found by adding 1 to the preceding integer. For example, -2, -1, 0, and 1 are consecutive integers.
TTP PRO TIP:
Remember that 0 is an integer!
Solution to Quiz Question #8: What is the product of the consecutive integers from -3 to 5, inclusive?
The consecutive integers from -3 to 5, inclusive, include the integer 0. Remember, 0 times any number is 0. So, the product of (-3) x (-2) x (-1) x 0 x 1 x 2 x 3 x 4 x 5 is equal to 0.
Example 8: GMAT Consecutive Integers Question
The sum of the consecutive integers from -4 to 2, inclusive, is subtracted from the product of those same integers. What is the value of that difference?
- -55
- -48
- -7
- 7
- 48
Solution:
First, let’s calculate the sum of the consecutive integers from -4 to 2, inclusive:
(-4) + (-3) + (-2) + (-1) + 0 + 1 + 2 = -10 + 3 = -7
We see that the product of those same integers is 0:
(-4) x (-3) x (-2) x (-1) x 0 x 1 x 2 = 0
When we subtract the sum of -7 from the product of 0, we obtain:
0 – (-7) = 7
Answer: D
Math Skill #9: Divisibility Rules
Remember, you are not allowed to use a calculator on the GMAT Quant section. Accordingly, it is very useful to know the common divisibility rules. Let’s say you solve a fraction question and end up with 57/219. What if that is not an answer? You’re not sure if you’ve calculated incorrectly or if you have an answer that just needs to be reduced. Knowledge of the common divisibility rules can be a lifesaver! Let’s look at them now:
- Divisible by 0: Division by 0 is not defined.
- Divisible by 2: If the ones digit is even, then the number is divisible by 2.
- Divisible by 3: If the sum of the digits in the number is evenly divisible by 3, then the number is divisible by 3.
- Divisible by 4: If the last two digits of the number are divisible by 4, then the number is also divisible by 4.
- Divisible by 5: If the units digit is 5 or 0, then the number is divisible by 5.
- Divisible by 6: Combine the rules for divisibility by 2 and divisibility by 3. If the number is even and the sum of its digits is divisible by 3, then the number is divisible by 6.
- Divisible by 7: The rules are complicated. Just do the division.
- Divisible by 8: The number must be even. And if the last 3 digits of the number are evenly divisible by 8, then the number is divisible by 8.
- Divisible by 9: If the sum of the digits in the number is evenly divisible by 9, then the number is divisible by 9.
- Divisible by 10: If the units digit of the number is 0, then the number is divisible by 10.
TTP PRO TIP:
Know the divisibility rules!
Solution to Quiz Question #9: Is 3,171,942 evenly divisible by 3?
Let’s add the digits of the number 3,171,942.
3 + 1 + 7 + 1 + 9 + 4 + 2 = 27
Since the sum of the digits is divisible by 3, we know that 3,171,942 is divisible by 3.
Example 9: GMAT Divisibility Question
If W = 132,455,101, then W + 2 must be divisible by which of the following?
- 2
- 3
- 6
- I only
- II only
- III only
- I and II only
- I, II, and III
Solution:
The value of W + 2 is 132,455,103.
Because W + 2 is not an even number, it is not divisible by 2. Thus, statement I is false.
The sum of the digits of W + 2 is 1 + 3 + 2 + 4, + 5 + 5 + 1 + 0 + 3 = 24. Since 24 is divisible by 3, we know that 132,455,103 is also divisible by 3. Statement II is true.
In order for a number to be divisible by 6, it must be divisible by both 2 and 3. Since W + 2 is not divisible by 2, it is also not divisible by 6. Statement III is false.
Answer: B
In Summary: GMAT Math Basics
When you’re preparing for the GMAT Focus Quant section, it is critically important that you not skip the basics. You need a solid foundation in the basics before you approach higher-level math.
In this article, we covered 9 topics that are fundamental to GMAT Quant. They include percents, prime numbers, factorial arithmetic, square roots, fractions, exponents, PEMDAS, consecutive integers, and divisibility rules. Keep in mind that these are just a start. There are many more GMAT math formulas you must learn for GMAT Quant.
Know and practice these basic facts and until they are effortless. Doing so will save you time, energy, and anxiety when you are studying the more complex topics in your preparation. On test day, you’ll have a solid basic skillset to help you get that great score!
What’s Next?
After you master GMAT math basics, it’s time to move forward with more challenging math topics. If you’re uncomfortable with your GMAT math ability, learn why anyone can do well on the GMAT math section.
