Quantitative Study Guide for the GMAT

General Information

Because the GMAT is a computer adaptive test, you need to approach all sections, including this one, a little bit differently. You will not be able to skip harder questions and go back to them. Neither will you be able to go back and double-check an answer. Also, the penalty for not finishing a question is heavier than if you had gotten it wrong, so try to answer all you can.

Be sure you know basic math procedures very well and do not have to waste time trying to think of them during the test. Our flashcards can help with that! Be aware that there may be shortcuts to solutions, so look for them. Also, there is probably no such thing as too much practice on the types of analytical math puzzles on this test. The more experience you have with them, the more efficiently you will be able to complete them on test day.

The GMAT Quantitative section requires you to do much more than do simple procedures. You will need to reason and think critically, discarding irrelevant information, and finding shortcuts to the solution. Finding the answer to many questions will mean using multiple procedures and making judgments about which ones to use.

This section has two types of questions:

• Problem Solving (PS) questions simply ask you to take the given data and do mathematical calculations to solve a problem.

• Data Sufficiency (DS) questions require you to identify which parts of given information are necessary to answer a mathematical question about a subject.

You will have 62 minutes to complete the 31 questions contained in the Quantitative section of the GMAT.

No calculator is allowed for this test, but the testing center will provide a scratchboard for your use in computing problems.

Know the Basics

Most skills required on the GMAT Quantitative section will have been covered before the end of your 10th grade high school year. This doesn’t mean you can recall them, however, unless you happen to be a math major and for this test, you’ll need to be able to do them instinctively. Additionally, it may have been some time since you attempted basic math without a calculator and practice would be helpful. Take some time to review these concepts:

Basic Arithmetic Operations

Be sure you can perform all four basic operations (addition, subtraction, multiplication, and division with whole numbers, integers (positive and negative numbers), decimals, and fractions. There are a number of rules involved with these operations and they should be automatic for you. Remember that you will not have access to a calculator during this test, so your work needs to be quick and accurate.

Determining Rate

Determining rate in a math question may involve any of these formulas:

Distance = rate x time, or Distance = speed x time

Wage = rate x time

Work done = rate x time

While these formulas look simple enough, GMAT can add twists to make them more challenging, such as in combined work, 2 or more moving objects, or a car traveling at various speeds. The basic formulas don’t change, but they will need to be manipulated in many ways.

In a combined work problem, for instance, 2 persons can finish a task together in 3 hours. If A can complete the task alone 2 hours and 15 minutes, how long can B complete the same work alone?

Express the combined work with this equation:

*Total work done = Work done by A + Work done by B

\[1 = rate_a \cdot (3) + rate_b \cdot (3)\] \[\frac{1}{3} = rate_a + rate_b\] \[\frac{1}{3} = \frac{1}{time_a} + \frac{1}{time_b}\]

To solve a question of this type without going through that time-consuming procedure, we can generalize and come up with this formula:

\[\frac {1}{t} = \frac{1}{t_a} + \frac{1}{t_b}\]

Simple Geometry Formulas

Be familiar with the Pythagorean Theorem, right triangles, common area and volume formulas, and the slope and equations of a line. Memorize common integer ratios of right triangles, such as 3:4:5 and 5:12:13, and the ratio of the sides of 30-60-90 and 45-45-90 triangles, which are \(1:\sqrt{3}:2\) and \(1:1:\sqrt{2}\), respectively. Since you will not be allowed to use a calculator, knowing these values and formulas from memory will save you precious time.

When questions call for solving the area or volume of an irregular geometric figure, or a figure with seemingly inadequate information, try looking at the bigger picture. See if there’s a way to solve for the areas of the larger shapes, subtract the areas of smaller shapes, and be left with the area asked for.

Develop your reasoning skills and ability to visualize geometric concepts. The surface area of a cylinder that is open on both ends, for instance, is a rectangle. The perimeter of a square inscribed within a circle of radius r can be expressed in terms of r if you realize that the square is made up of eight 45-45-90 triangles with hypotenuse r.

Operations with Exponents

You’ll need to know the laws of exponents by heart to be able to answer many questions in GMAT Quantitative. This includes how to treat fractional and negative exponents. Exponents are often found in systems of equations and inequality questions.

Here are two interesting characteristics of GMAT questions: they don’t use very large numbers in the question or the given information (almost always less than 100), and the answers usually round up to whole numbers.

If GMAT had favorite numbers, they certainly include the squares of the integers from 2 through 13 and other unique characteristics of numbers that are raised to exponents. The square of 8 is 64, and the cube of 4 is 64. No other number below 100 is a square of an integer and the cube of another.

Once you’ve mastered the rules of exponents, practice on questions involving variables as exponents. This pattern can appear either as a DS or PS question:

What is a if \(5^{3a+1} = 25^{2a}\)?

Operations with Roots

Finding the roots (or radicals) involves the inverse operation of squaring or raising numbers to an exponent. Roots are simply numbers raised to fractional exponents, so the rules for operating exponents apply to operations with roots.

You may need to know the approximate values of commonly used roots in some quantitative questions. Examples of this are questions requiring you to solve for quantities involving radicals and the answer choices given don’t include radicals. Thus, you will need to know that \(\sqrt{2}\) is approximately 1.4 and \(\sqrt{3}\) is approximately 1.7.

As much as possible, however, manipulate radicals as they are, without converting them to their approximate value except when their value is a whole number. Asked for the value of \((\sqrt{6}x + 3\sqrt{3})^2\), you arrive at the answer \(x^2 + 3\sqrt{2} + 2\), an expression that still bears a radical term.

Ratios and Proportions

The topics of ratio and proportion are two of the most important subjects covered in GMAT Quantitative. They are seldom used as standalone questions; instead, they are combined with many topics, including triangles, slopes of lines, sequences, identifying multiples, and solving systems of equations.

In a 30-60-90 triangle, for example, a question that requires the computation of the longer side if the hypotenuse is given as \(6\sqrt{3}\), can be solved this way:

The ratio of the sides of a 30-60-90 triangle is \(1:\sqrt{3}:2\). We first set up the following proportion:

\[\frac{longer\;side}{hypotenuse} = \frac{\sqrt{3}}{2} = \frac{x}{6\sqrt{3}}\]

where x represents the longer side

From this proportion, we may then solve for the longer side of the triangle.

Combinations and Permutations

The concepts of permutation and combination are similar in the sense that they are used for counting the number of ways certain events can occur, things can be done, or elements can be combined or paired.

Combination counts the number of possibilities when the order of these elements does not matter. We count the combination or the number of ways a chef can mix 4 vegetables out of 10 kinds of vegetables to choose from. There is a formula for this:

\[Combination = \frac{n!}{r!(n-r)!}\]

where n is the number of elements to choose from and
r is the number of choices to be made

Permutation, on the other hand, takes the order of the elements into consideration and can be computed both when repetition of the elements is allowed and when repetition is not allowed. We compute for the permutation to determine the number of ways 3 numbers chosen from 1 through 9 can combine to form a 3-digit code because the correct order is important. The following are the formulas for permutation:

Permutation when repetition is allowed, \(P = n^r\)

Permutation when repetition is not allowed, \(P = \frac{n!}{(n-r)!}\)

Sequences of Numbers

Patterns or sequences encountered in GMAT are usually either arithmetic or geometric.

Here’s a DS question involving arithmetic sequence:

What is the 99th integer in “\(x_1, 8… x_99\)”?

\(x_{34} = 168\)
\(x_{33} = 163\)
\(x_1 = 3\)

The correct answer is (D) because either Statement 1 alone or Statement 2 alone is sufficient to answer the question.

The formula for the nth element in an arithmetic sequence is given as:

\[x_n = x_1 + d(n-1)\]

where:
\(x_n\) = the nth element
\(x_1\) = the 1st element
\(d\) = the common difference

The formula for the nth element in a geometric sequence is as follows:

\[x_n = x_1 \cdot r^{(n-1)}\]

where r = the common ratio

Estimation and Rounding

The word “closest” strongly suggests estimation and rounding up or down. Besides, GMAT challenges you to perform mathematical reasoning not extensive manual calculation. In this example, approximating will give you an answer that is close enough without wasting too much time:

\[\frac{\sqrt{99} \cdot 0.99^2}{0.49} = \frac{\sqrt{100}\cdot 1^2}{0.5} = \frac{10}{0.5} = 20\]

Answer choices that are so spread out tell the same thing—you can safely approximate. You may also increase your chance of getting a correct answer even with estimated answers, by eliminating highly unlikely answers. This technique will also be elaborated in information to follow.

Solving Basic Algebraic Equations

Algebraic equations are solved with the goal of isolating the unknown variable and solving for its value. This is done using elimination and substitution. When polynomials occur in fractions, it is best to factor them out. This cancels out common terms and reduces the equation to a simpler form.

Creating Algebraic Equations

Multiple math topics overlap in algebra, and these comprise the major content of most PS-type GMAT questions. The ability to create algebraic equations is definitely a vital tool. Another important skill is the ability to simplify verbose problems because GMAT PS questions are deliberately worded to confuse test-takers with unnecessary details.

(1) Read the problem to be solved. Simplify the question.
(2) Draw a sketch, labeling as you go along with the given information.
(3) Analyze the question and identify what is being asked. Assign it a name (usually a variable). Just giving an unknown a name already makes it less daunting.
(4) Relate the variable to the other elements of the problem. Find out the relationship between them.
(5) Treat these elements, including the variable, as terms in an equation.
(6) Solve for the unknown.

Systems of Equations

A system of equations involves two or more equations to solve a particular question. It may involve one or more variables. There must be at least two equations to be able to solve for two variables. The equations in a system are simultaneously true only at the point where they cross or meet. Take these given equations:

\[2r – 3s = 4\]

and

\[3r – 7s = 1\]

The two linear equations are both true at r = 5 and s = 2 which can be determined after simultaneously computing the two equations.

Linear and Quadratic Equations

Practice with a lot of linear and quadratic equations, familiarizing yourself with the standard and other forms of writing them.

Linear equations can be written as:

General form: Ax + By + C = 0; where A and B cannot be equal to zero at the same time

Slope-intercept form: y = mx + b; where m is the slope of the line and b is the y-intercept

Point-slope form: y – y1 = m(x-x1); where (x1 , y1) is a point on the line

Quadratic equations are equations of the second degree. The standard form of quadratic equation is:

\(Ax^2 + Bx + C = 0\); where \(A \ne 0\)

The ability to factor quadratics and linear equations is repeatedly tested in GMAT questions. Questions similar to the one below are often seen in GMAT PS:

\(\frac{3x^2-9x+12}{x^2-8x+16}\) is equal to which of the following expressions if \(x \gt 0\)?

Correct factoring will reduce this expression to \(\frac{3(x+1)}{(x-4)}\)

With a lot of practice, you will be able to identify on sight the square of a binomial, difference of squares, and the slope of a line given an equation.

Average, Range, Mean, Median, Mode, and Standard Deviation

The arithmetic mean (or average), median, and mode are measures used to represent the central value of a set or group of numbers.

The mean is computed by taking the sum of all terms in a set or group and dividing this sum by the number of terms.

To determine the median, the numbers must be sorted in ascending or descending order first. The number in the middle of the list is the median. In the case of two middle numbers (called middle pair), the average of the pair is computed and this becomes the median.

The mode is the number that occurs most often in the set.

The range is the difference between the smallest and largest number in a set. Standard deviation is computed as the square root of the variance and it gives the measure of how scattered the numbers are.

These are some of the common statistical concepts that are often combined with GMAT questions. The following DS question is an example:

Alyssa is the eldest child in a family with 6 children. How old is she?

(1) The other children are younger than Alyssa by 4, 5, 8, 9 and 10 years.
(2) The average age of the 6 children is 28.

Lines and Angles

Learn about lines, angles, parallel lines, and perpendicular lines.

Perpendicular lines intersect each other and form a 90-degree angle at the point where they meet.

When two or more angles form a straight line, the sum of their angles is 180-degrees.

Parallel lines will never intersect each other.

When parallel lines are intersected by another line, here are important things to remember about the angles formed.

When parallel lines are intersected by another line, such as shown in the figure, the resulting angles a, d, e and h will be equal, and the angles b, c, f and g will be equal.

Factors, Multiples, and Remainders

The ability to find the factors and multiples of commonly used integers in GMAT speeds up computation, such as in this question:

\[\frac{(81)(128)(48)(49)}{(84^3)(36)} = \;?\]

Recognizing that many of these numbers are multiples of smaller primes simplifies computation:

\[\frac{(3^4)(2^7)(3)(2^4)(7^2)}{(7^3)(2^6)(3^3)(2^2)(3^2)} = \frac {2^3}{7} = \frac {8}{7}\]

Remainders may be written in decimal form, as a fraction with the remainder as the numerator and the divisor as the denominator. It’s also quite common to write the remainder after the notation r. which denotes that the next single-digit number is a remainder. This very basic concept is often the subject of some challenging GMAT math questions, such as this DS sample below:

What is the remainder in this expression if x > 0

\[\frac{x^2 – 2}{6}\]

(1) \(x^2\) is a multiple of 12.
(2) \(\frac{x}{6}\) is a whole number.

Basic Probability

There will only be one or two probability questions in GMAT, but this is important for those aiming for the 720 – 800 score. Probability is often mixed with combination and permutation concepts. The following are the most basic things to remember:

Probability is the number of times a desired outcome can occur divided by the total number of times any possible outcome can occur. It is a number between 0 and 1; an event with a probability that is closer to 1 denotes a higher likelihood that said event will occur or happen.

The probability of pulling out a stuffed rabbit toy from a bag containing 10 different stuffed toys is:

\[P = \frac{1}{10}\]

The probability of two or more separate events occurring is the product of their individual probabilities. The probability of pulling out a rabbit and a bear from a bag in two separate events (first toy is returned back to the bag after it is taken out) is:

\[P = \frac{1}{10} \cdot \frac{1}{10} = \frac{1}{100}\]

Take note that this probability changes if the first toy is not returned after pulling it out of the bag:

\[P = \frac{1}{10} \cdot \frac{1}{9} = \frac{1}{90}\]

The probability of either of two events occurring is the sum of their individual probabilities.

The probability of an event not occurring and the probability that it will occur is equal to 1.

Simple and Compound Interest

The simple interest formula is stated as:

\[Interest = P \cdot r \cdot t\]

where P is the principal amount,
r is the annual interest rate, and
t is the time expressed in years

Compound interest is expressed as:

Compound Interest = \(P(1 + \frac{r}{x})^{xt}\)

where x is the number of times the interest compounds in a year

Expect GMAT questions to be wordy and confusing. Convert time to years if given in other units. Make sure that the rate is the annual rate; convert accordingly if given as monthly rate or other units.

Profit and Loss

Profit and loss are the results of doing business. The total sales or money generated by a business is called revenue. When cost is subtracted from revenue, the result is called profit if it is a positive number, and it is called loss if the result is negative. This concept is often combined with concepts on percent, percent change, and ratios to come up with puzzling questions, such as this one:

“A company generated a 14% profit of its revenue in 2014. The following year, revenue increased by 5% but the company gained only 2% of this revenue. What is the percent change in profit between 2014 and 2015?”

The task at hand is to find the percent change in profit from 2014 to 2015, which can be solved with this formula:

Percent Change = \(\frac{(P_2 – P_1)}{P_1 \cdot 100\%}\)

where:

\(P_1\) = Profit for 2014
\(P_2\) = Profit for 2015

From the given information, we can derive the following:

\[P_1 = R \frac{14}{100}\] \[P2 = (R + 0.05R)\left(\frac{2}{100}\right) = 1.05R\left(\frac{2}{100}\right)\]

Revenue (R) is unknown, but we don’t really need to solve its exact value or carry the variable to the percent phange formula. By picking a random number for R, the calculation can be simplified. When working with percents, 100 is a good “random” number, as you will see.

\[P_1 = 100 \left(\frac{14}{100}\right) = 14\] \[P_2 = 1.05(100)\left(\frac{2}{100}\right) = 2.1\] \[Percent\;Change = \frac{(2.1 – 14)}{2.1} \cdot 100\% = -85\% = 85\% \;decrease\]

Order of Operations

“Please Excuse My Dear Aunt Sally” is the mnemonic for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction (or PEMDAS). This is the order that must be followed when performing a series of mathematical operations. It means that operations inside parentheses must be performed first. Terms raised to exponents are next. Operations involving multiplication and division follow, performed from left to right. Addition and subtraction are the last, performed from left to right.

Specific Skills to Practice

Just knowing basic math will not enable you to ace the GMAT Quantitative Test. You’ll also need to use given information to reason and make decisions. Here are some problem-solving skills to practice:

Locating Pertinent Information

Analyze questions and reduce them to the simplest form. Questions are not framed in a way that they can be easily understood; in fact, they’re deliberately written to be complicated. Learn to discard unimportant details, but be careful not to throw away important clues. Take this question, for instance:

“In a particular month, a sporting goods store sold 6 times more Brand A tennis balls than the other brands combined. What was the ratio of the sales of Brand A balls to the total sales of tennis balls?”

It’s easy to see that the ratio of Brand A to the other brands is “6:1”, but don’t be too quick to click this answer from the choices. The question started by comparing Brand A with the other brands, but notice that the question suddenly shifted by asking the ratio of Brand A to the total sales, which is “6:7”, the correct answer.

Logical Reasoning

PS and DS often pose questions based on very basic math concepts but which demand careful logical reasoning. The following sample question illustrates this:

“If g > 0, c > f, e > g, d > f, f > e, which of the following statements is correct:”

I. d < 0
II. g < d
III. c > e

The concept being tested is basic numeracy, and the relative location of values on a number line but letters are used instead of numbers. Questions of this kind can be confusing, but with the aid of some techniques, they are actually the easiest ones to answer since there is practically no math or formulas involved.

Try this technique: Take one letter at a time, writing each letter on your scratchpad and positioning it relative to the others as it is given in the question. Write the smallest value on the rightmost part of your paper, with larger values positioned toward the left.

Take the first given “g > 0”. Write 0 on the rightmost location.
Write g to its left, because it is given that g is bigger.
Pick “e > g” next.
Since e is bigger than g, write it to the left of g. Take “f > e” next.
Write f to the left of e.
Take “d > f”, write d to the left of f.
Take “c > f”, and write c beside d but after f.
The relationship between c and d is not very clear at this point, but we’re sure about the relative positions of all the other letters.

With this, it can be inferred that Statement I is incorrect, Statement II is correct, and Statement III is also correct.

If the answer choices are given as follows, automatically eliminate choices that contain the incorrect answer.

(A) I
(B) II
(C) III
(D) I and II
(E) II and III

Choices (A) and (D) are eliminated right away, leaving you with only 3 choices among which to decide. This narrows down the choices and increases your chance of getting the correct answer.

Deciding if Data is Sufficient

An item of the DS type contains a question followed by two statements. The given statements are always true and correct. The test-taker’s task is to determine the sufficiency or insufficiency of the information contained in the statements provided in answering the question.

The DS question is either a Yes/No question or a Value question. Answer choices will always be written in the following format and order:

(A) Statement (1) alone is sufficient, but statement (2) is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

It is necessary to memorize the meaning of each choice and to develop strategies in answering questions of this type.

After inspecting a statement, the answer choices can be narrowed down to three or even two. Here’s why.

If you find Statement 1 to be sufficient in solving the question, you can eliminate choices (B), (C) and (E) right away. The only possible choices now are (A) and (D).

Evaluate whether Statement 2 is sufficient to solve the question. If it is sufficient, the correct answer is (D). If it is insufficient, the correct answer is (A). With narrowed-down choices, your chance of getting a correct answer is higher.

Finding Multiple Correct Answers

In the example given under the subtopic on Logical Reasoning, two of the given statements are correct – Statement II and Statement III. Immediately eliminate all answer choices that include the incorrect statement (I) and all answer choices that have only one correct statement. Choices (A), (B), (C) and (D) are easily eliminated this way, leaving only the correct choice which is (E).