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GMAT® QUANTITATIVE REVIEW. 2ND EDITION. • Actual questions from past GMAT tests, including 75 questions new to this edition. • past Problem Solving. the specific skills required to take the GMAT CAT, download the free organic material from microbes but does Th The Official Guide for GMAT Review . 7 The Official Guide for GMAJ41' Quantitative Review 2nd Edition What Is the Content of the Test Like? It is important to recognize that the GMAT test.

Exponents A positive integer exponent of a num ber or a variable indicates a product, and the positive integer is the number of times that the numb er or variable is a factor in the product. Set s 4. Members of a social club met to address real numbers x? For every even positive integer m, f m represents the product of all even integers from 2 to m, inclusive. Equations 8.

There are many benefits to timing your practice , including:.

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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Control Number: The test was designed to help admissions officers evaluate how suitable individual applicants are for their graduate business and management programs.

It measures basic verbal, mathematical, and analytical writing skills that a test taker has developed over a long period of time through education and work. The GMAT test does not measure a person's knowledge of specific fields of study. Graduate business and management programs enroll people from many different undergraduate and work backgrounds, so rather than test your mastery of any particular subject area, the GMAT test will assess your acquired skills.

Your GMAT score will give admissions officers a statistically reliable measure of how well you are likely to perform academically in the core curriculum of a graduate business program. Of course, there are many other qualifications that can help people succeed in business school and in their careers-for instance, job experience, leadership ability, motivation, and interpersonal skills.

The GMAT test does not gauge these qualities. That is why your GMAT score is intended to be used as one standard admissions criterion among other, more subjective, criteria, such as admissions essays and interviews.

GMAT scores are used by admissions officers in roughly 1, graduate business and management programs worldwide. Schools that require prospective students to submit GMAT scores in the application process are generally interested in admitting the best-qualified applicants for their programs, which means that you may find a more: Thus, while which you apply.

Furthermore, the percentile table you may be exceptionally capable, the odds are against your achieving a perfect score. Admissions officers performance of other test takers, giving you one use GMAT scores in conjunction with undergraduate records, application essays, way to gauge your competition for admission to interviews, letters of recommendation, and business school.

MAT scores and other admissions criteria such as your undergraduate grades, essays, and letters of recommendation to evaluate candidates for admission. School admissions offices, school Web sites, and materials published by the school are the best sources for you to tap when you are doing research about where you might want to go to business school. For more information about how schools should use G. MAT scores in admissions decisions, please read Appendix A of this book. For more information on the GMAT, registering to take the test, sending your scores to schools, and applying to business school, please visit our Web site at www.

You start the test with two minute Analytical Writing Assessment AWA questions that require you to type your responses using the computer keyboard. The writing section is followed by two minute, multiple-choice sections: These questions are drawn from a huge wrong. So, although we talk about the GMAT as one test, the GMAT test you take may be F - Getting an easier question completely different from the test of the person sitting next does not necessarily mean to you.

Here's how it works. At the start of each GMAT multiple- To ensure that everyone receives the same choice section Verbal and Qyantitative , you will be content, the test selects a specific number presented with a question of moderate difficulty. The of questions of each type. The test may call for your next question to be a relatively computer uses your response to that first question to hard problem-solving item involving determine which question to present next.

If you respond arithmetic operations. But, if there are no correctly, the test usually will give you questions of more relatively difficult problem-solving items involving arithmetic, you might be increasing difficulty. If you respond incorrectly, the next given an easier item. As you continue to respond to the Most people are not skilled at estimating item difficulty, so don't worry when taking questions presented, the computer will narrow your score the test or waste valuable time trying to to the number that best characterizes your ability.

When determine the difficulty of the questions you are answering. You may not skip, return to, or change your responses to previous questions. Random guessing can significantly lower your scores. If you do not know the answer to a question, you should try to eliminate as many choices as possible, then select the answer you think is best.

If you answer a question incorrectly by mistake-or correctly by lucky guess- your answers to subsequent questions will lead you back to questions that are at the appropriate level of difficulty for you.

Each multiple-choice question used in the GMAT test has been thoroughly reviewed by professional test developers. New multiple-choice questions are tested each time the test is administered. Answers to trial questions are not counted in the scoring of your test, but the trial questions are not identified and could appear anywhere in the test. Therefore, you should try to do your best on every question. The test includes the types of questions found in this guide, but the format and presentation of the questions are different on the computer.

When you take the test: Analysis of an Argument 1 30 min. Optional break Qyantitative 37 75 min. It is important to recognize that the GMAT test evaluates skills and abilities developed over a relatively long period of time. Although the sections contain questions that are basically verbal and mathematical, the complete test provides one method of measuring overall ability.

Keep in mind that although the questions in this guide are arranged by question type and ordered from easy to difficult, the test is organized differently. When you take the test, you may see different types of questions in any order. Two types of multiple-choice questions are used in the Qyantitative section: Both types of questions require basic knowledge of: For test-taking tips specific to the question types in the Qyantitative section of the GMAT test, sample questions, and answer explanations, see chapters 4 and 5.

Because the Verbal section includes reading sections from several different content areas, you may be generally familiar with some of the material; however, neither the reading passages nor the questions assume detailed knowledge of the topics discussed.

Three types of multiple-choice questions are used in the Verbal section: What Computer Skills Willi Need? You will be required to type your essays on the computer keyboard using standard word-processing keystrokes. In the multiple-choice sections, you will select your responses using either your mouse or the keyboard.

The GMAT test is administered at a test center providing the quiet and privacy of individual computer workstations. You will have the opportunity to take two optional breaks-one after completing the essays and another between the Qyantitative and Verbal sections.

An erasable notepad will be provided for your use during the test. Your GMAT scores are determined by: When you answer the easier questions correctly, you get a chance to answer harder questions-making it possible to earn a higher score.

After you have completed all the questions on the test-or when your time is up-the computer will calculate your scores. Your scores on the Verbal and Qyantitative sections are combined to produce your Total score. If you have not responded to all the questions in a section 37 Qyantitative questions or 41 Verbal questions , your score is adjusted, using the proportion of questions answered.

Analysis of an Issue and Analysis of an Argument. The responses to each of these tasks are scored on a 6-point scale, with 6 being the highest score and 1, the lowest.

A score of zero O is given to responses that are off-topic, are in a foreign language, merely attempt to copy the topic, consist only of keystroke characters, or are blank. The readers who evaluate the responses are college and university faculty members from various subject matter areas, including management education. These readers read holistically-that is, they respond to the overall quality of your critical thinking and writing.

For details on how readers are qualified, visit www. In addition, responses may be scored by an automated scoring program designed to reflect the judgment of expert readers.

Each response is given two independent ratings. If the ratings differ by more than a point, a third reader adjudicates. Because of ongoing training and monitoring, discrepant ratings are rare. Your final score is the average rounded to the nearest half point of the four scores independently assigned to your responses-two scores for the Analysis of an Issue and two for the Analysis of an Argument.

For example, if you earned scores of 6 and 5 on the Analysis of an Issue and 4 and 4 on the Analysis of an Argument, your final score would be 5: Your Analytical Writing Assessment scores are computed and reported separately from the multiple- choice sections of the test and have no effect on your Verbal, Qyantitative, or Total scores. The schools that you have designated to receive your scores may receive your responses to the Analytical Writing Assessment with your score report.

Your own copy of your score report will not include copies of your responses. All questions are subjected to independent reviews and are revised or discarded as necessary.

Multiple-choice questions are tested during GMAT test administrations. Analytical Writing Assessment tasks are tried out on first-year business school students and then assessed for their fairness and reliability. For more information on test development, see www.

By answering questions that have appeared on the GMAT test before, you will gain experience with the types of questions you may see on the test when you take it. As you practice with this guide, you will develop confidence in your ability to reason through the test questions. No additional techniques or strategies are needed to do well on the standardized test if you develop a practical familiarity with the abilities it requires.

Simply by practicing and understanding the concepts that are assessed on the test, you will learn what you need to know to answer the questions correctly. The software is available for download at no charge for those who have created a user profile on www.

It is also provided on a disk, by request, to anyone who has registered for the GMAT test. We recommend that you download the software as you start to prepare for the test.

You should test to familiarize yourself with the test and to get review the math skills algebra, geometry, an idea of how you might score. The difficulty of determine whether you need to shift your focus to GMAT Quantitative questions stems from other areas you need to strengthen.

The following are general suggestions to help you perform your best on the test. Use your time wisely. Although the GMAT test stresses accuracy more than speed, it is important to use your time wisely.

Once you start the test, an onscreen clock will continuously count the time you have left. You can hide this display if you want, but it is a good idea to check the clock periodically to monitor your progress. The clock will automatically alert you when 5 minutes remain in the allotted time for the section you are working on. Answer practice questions ahead of time.

After you become generally familiar with all question types, use the sample questions in this book to prepare for the actual test. It may be useful to time yourself as you answer the practice questions to get an idea of how long you will have for each question during the actual GMAT test as well as to determine whether you are answering quickly enough to complete the test in the time allotted.

Read all test directions carefully. The directions explain exactly what is required to answer each question type. If you read hastily, you may miss important instructions and lower your scores. To review directions during the test, click on the Help icon.

But be aware that the time you spend reviewing directions will count against the time allotted for that section of the test. Read each question carefully and thoroughly. Before you answer a multiple-choice question, determine exactly what is being asked, then eliminate the wrong answers and select the best choice. Never skim a question or the possible answers; skimming may cause you to miss important information or nuances. Do not spend too much time on any one question.

Myth -vs- FACT lf you do not know the correct answer, or if the: Try not to worry about the impact on your score-guessing may lower your score, but not F - There is a severe penalty finishing the section will lower your score more. Bear in mind that if vou do not finish a section in If you are stumped by a question, give it the allotted time, you will still receive a score.

If you guess incorrectly, the computer program will likely give you an easier question, which you are 6. Confirm your answers ONLY when you likely to answer correctly, and the computer will rapidly return to giving you questions are ready to move on. If you don't finish the test, your score will be reduced greatly.

Once you have selected your answer to a multiple- Failing to answer five verbal questions, for choice question, you will be asked to confirm it. Pacing is important.

You may not skip questions, because the computer selects each question on the basis of your responses to preceding questions. Myth -vs- FACT 7. Plan your essay answers before you begin to write. V- The first 10 questions are critical and you should invest The best way to approach the two writing tasks the most time on those.

Take care to It is true that the computer-adaptive testing algorithm uses the first 10 questions to organize your ideas and develop them fully, but obtain an initial estimate of your ability; leave time to reread your response and make any however, that is only an initial estimate.

As revisions that you think would improve it. Your final score is based on all your responses and considers the difficulty of all the questions you answered. Taking additional time on the first 10 questions will not game the system and can hurt your ability to finish the test. You may wish to consult an arithmetic, algebra, or geometry book for a more detailed discussion of some of the topics. Section 3. Properties of Integers 7.

Powers and Roots ofNumbers 2. Fractions 8. Descriptive Statistics 3. Decimals 9. Sets 4. Real Numbers Counting Methods 5.

Ratio and Proportion Discrete Probability 6. Percents Section 3. The topics included are as follows: Simplifying Algebraic Expressions 7. Exponents 2. Equations 8. Inequalities 3. Solving Linear Equations with One Unknown 9. Absolute Value 4. Solving Two Linear Equations with Functions Two Unknowns 5. Solving Equations by Factoring 6.

Solving Q! Extensive knowledge of theorems and the ability to construct proofs, skills that are usually developed in a formal geometry course, are not tested.

The topics included in this section are the following: Lines 6. Triangles 2. Intersecting Lines and Angles 7. Perpendicular Lines 8.

Circles 4. Parallel Lines 9. Rectangular Solids and Cylinders 5. Polygons Convex Coordinate Geometry Section 3. Rate Problems 6. Profit 2. Work Problems 7. Sets 3. Mixture Problems 8. Geometry Problems 4. Interest Problems 9. Measurement Problems 5. Discount Properties of Integers An integer is any number in the set[ In this case, y is also said to be divisible by x or to be a multiple of x. Note that y is divisible by x if and only if the remainder r is 0; for example, 32 has a remainder of 0 when divided by 8 because 32 is divisible by 8.

Also, note that when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. Any integer that is divisible by 2 is an even integer; the set of even integers is [. Integers that are not divisible by 2 are odd integers; [.

If at least one factor of a product of integers is even, then the product is even; otherwise the product is odd. If two integers are both even or both odd, then their sum and their difference are even. Otherwise, their sum and their difference are odd. A prime number is a positive integer that has exactly two different positive divisors, 1 and itself. For example, 2, 3, 5, 7, 11, and 13 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and The number 1 is not a prime number since it has only one positive divisor.

Every integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors. The numbers -2, -1, 0, 1, 2, 3, 4, 5 are consecutive integers. The numbers 0, 2, 4, 6, 8 are consecutive even integers, and 1, 3, 5, 7, 9 are consecutive odd integers. Properties ofthe integer 1. Properties ofthe integer 0.

The integer 0 is neither positive nor negative. Division by 0 is not defined. Fractions In a fraction ;. The denominator of a fraction can never be 0, because division by 0 is not defined. Two fractions are said to be equivalent if they represent the same number. For example, and 14 are equivalent since thev both represent the number l. In each case, the fraction is reduced to The gcd of 8 and 36 is 4 and the gcd of 14 and 63 is 7.

Addition and subtraction of fractions. Two fractions with the same denominator can be added or subtracted by performing the required operation with the numerators, leaving the denominators the same.

If two fractions do not have the same denominator, express them as 5 77 7 7 3 4 equivalent fractions with the same denominator. For example, to addS and 7' multiply the numerator and denominator of the first fraction by 7 and the numerator and denominator of the second flract1on.

To multiply two fractions, simply multiply the two numerators and multiply the two denominators. In general, the reciprocal of a fraction! A number that consists of a whole number and a fraction, for example, 71, is a mixed number: Decimals In the decimal system, the position of the period or decimal point determines the place value of the digits. For example, the digits in the number 7, F 7 6 5 4 3 2 1 Some examples of decimals follow.

This is called scientific notation. For example, can be written as 2. The decimal point is moved to the right if the exponent is positive and to the left if the exponent is negative. For example, 2. To add or subtract two decimals, the decimal points of both numbers should be lined up.

If one of the numbers has fewer digits to the right of the decimal point than the other, zeros may be inserted to the right of the last digit. For example, to add To multiply decimals, multiply the numbers as if they were whole numbers and then insert the decimal point in the product so that the number of digits to the right of the decimal point is equal to the sum of the numbers of digits to the right of the decimal points in the numbers being multiplied.

For example: To divide a number the dividend by a decimal the divisor , move the decimal point of the divisor to the right until the divisor is a whole number. Then move the decimal point of the dividend the same number of places to the right, and divide as you would by a whole number.

The decimal point in the quotient will be directly above the decimal point in the new dividend. For example, to divide Real Numbers All real numbers correspond to points on the number line and all points on the number line correspond to real numbers. All real numbers except zero are either positive or negative. If n is "between 1 and 4, inclusive," then 1 s n s 4. The distance between a number and zero on the number line is called the absolute value of the number.

Thus 3 and -3 have the same absolute value, 3, since they are both three units from zero. Note that the absolute value of any nonzero number is positive. Here are some properties of real numbers that are used frequently. Ratio and Proportion The ratio of the number a to the number b b.

A ratio may be expressed or represented in several ways. For example, the ratio of 2 to 3 can be written as 2 to 3, 2: The order of the terms of a ratio is important. For example, the ratio of the number of months with exactly 30 days to the number with exactly 31 days is. Percents Percent means per hundred or number out of A percent can be represented as a fraction with a denominator of , or as a decimal. The percent 0. For example, 0. Percent change.

Often a problem will ask for the percent increase or decrease from one quantity to another quantity. In the example above the percent increase would be found in the following way: Note that the percent increase from 24 to 30 is not the same as the percent decrease from 30 to In the following example, the increase is greater than percent: If the cost of a certain house in was percent of its cost in , by what percent did the cost increase?

Powers and Roots of Numbers When a number k is to be used n times as a factor in a product, it can be expressed as kn, which means the nth power of k. Squaring a number that is greater than 1, or raising it to a higher power, results in a larger number; squaring a number between 0 and 1 results in a smaller number.

The square root of a negative number is not a real number. Every positive number n has two square roots, one positive and the other negative, but fn denotes thsrositive number whose square is n. For example, J9 denotes 3. The two square roots of 9 are J9 Every real number r has exactly one real cube root, which is the number s such that s 3 r.

The real cube root of r is denoted by. Descriptive Statistics A list of numbers, or numerical data, can be described by various statistical measures. One of the most common of these measures is the average, or arithmetic mean, which locates a type of"center" for the data. The average of n numbers is defined as the sum of the n numbers divided by n. The median is another type of center for a list of numbers.

To calculate the median of n numbers, first order the numbers from least to greatest; if n is odd, the median is defined as the middle number, whereas if n is even, the median is defined as the average of the two middle numbers.

In the example above, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number. Note that the mean of these numbers is 7. The median of a set of data can be less than, equal to, or greater than the mean.

Note that for a large set of data for example, the salaries of company employees , it is often true that about half of the data is less than the median and about half of the data is greater than the median; but this is not always the case, as the following data show. The mode of a list of numbers is the number that occurs most frequently in the list.

For example, the mode of 1, 3, 6, 4, 3, 5 is 3. A list of numbers may have more than one mode. For example, the list 1, 2, 3, 3, 3, 5, 7, 10, 10, 10, 20 has two modes, 3 and The degree to which numerical data are spread out or dispersed can be measured in many ways.

The simplest measure of dispersion is the range, which is defined as the greatest value in the numerical data minus the least value. Note how the range depends on only two values in the data.

One of the most common measures of dispersion is the standard deviation. Generally speaking, the more the data are spread away from the mean, the greater the standard deviation. The standard deviation of n numbers can be calculated as follows: Shown below is this calculation for the data 0, 7, 8, 10, 10, which have arithmetic mean 7. This is why a distribution with data grouped closely around the mean will have a smaller standard deviation than will data spread far from the mean.

To illustrate this, compare the data 6, 6, 6. Note that the numbers in the second set of data seem to be grouped more closely around the mean of 7 than the numbers in the first set. This is reflected in the standard deviation, which is less for the second set approximately 1. There are many ways to display numerical data that show how the data are distributed.

One simple way is with a frequency distribution, which is useful for data that have values occurring with varying frequencies. For example, the 20 numbers -4 0 0 -3 -2 -1 -1 0 -1 -4 -1 -5 0 -2 0 -5 -2 0 0 -1 are displayed on the next page in a frequency distribution by listing each different value x and the frequency f with which x occurs.

Sets In mathematics a set is a collection of numbers or other objects. The objects are called the elements of the set. If S is a set having a finite number of elements, then the number of elements is denoted by lsi. The intersection of A and B is the set of all elements that are both in A and in B. Two sets that have no elements in common are said to be disjoint or mutually exclusive. The relationship between sets is often illustrated with a Venn diagram in which sets are represented by regions in a plane.

For two sets Sand T that are not disjoint and neither is a subset of the other, the intersection S n T is represented by the shaded region of the diagram below. S T This diagram illustrates a fact about any two finite sets S and T: This counting method is called the general addition rule for two sets. Counting Methods There are some useful methods for counting objects and sets of objects without actually listing the elements to be counted.

The following principle of multiplication is fundamental to these methods. If an object is to be chosen from a set of m objects and a second object is to be chosen from a different set of n objects, then there are mn ways of choosing both objects simultaneously.

As an example, suppose the objects are items on a menu. As another example, each time a coin is flipped, there are two possible outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, then the experiment has 2 8 possible outcomes, where each of these outcomes is a list of heads and tails in some order.

A symbol that is often used with the multiplication principle is the factorial. If n is an integer greater than 1, then n factorial, denoted by the symbol n! Therefore, 2! Also, by definition, 0! The factorial is useful for counting the number of ways that a set of objects can be ordered. If a set of n objects is to be ordered from 1st to nth, then there are n choices for the 1st object, n - 1 choices for the 2nd object, n - 2 choices for the 3rd object, and so on, until there is only 1 choice for the nth object.

For example, the number of ways of ordering the letters A, B, and C is 3!

These orderings are called the permutations of the letters A, B, and C. A permutation can be thought of as a selection process in which objects are selected one by one in a certain order. If the order of selection is not relevant and only k objects are to be selected from a larger set of n objects, a different counting method is employed. Then the number of possible complete selections of k objects is called the number of combinations of n objects taken kat a time and is denoted by: Note that: In general,: Discrete Probability Many of the ideas discussed in the preceding three topics are important to the study of discrete probability.

Discrete probability is concerned with experiments that have a finite number of outcomes. Given such an experiment, an event is a particular set of outcomes. For example, rolling a number cube with faces numbered 1 to 6 similar to a 6-sided die is an experiment with 6 possible outcomes: The probability that an event E occurs, denoted by P E , is a number between 0 and 1, inclusive. Two events A and Bare said to be independent if the occurrence of either event d?

Therefore, A and B are independent. The following multiplication rule holds for any independent events E and F: Also, suppose that events Aand Bare mutually exclusive and events B and Care independent. Thus, one can conclude that 0. Letters such as x or n are used to represent unknown quantities.

For example, suppose Pam has 5 more pencils than Fred. Such an expression is called a second degree or quadratic polynomial in x since the highest power of xis 2. The expression 3x 2 is not a polynomial because it is not a sum of terms that are each powers of x multiplied 2x- 5 by coefficients. Simplifying Algebraic Expressions Often when working with algebraic expressions, it is necessary to simplify them by factoring or combining like terms. In the expression 9x- 3 y, 3 is a factor common to both terms: If there are common factors in the numerator and denominator of an expression, they can be divided out, provided that they are not equal to zero.

Equations A major focus of algebra is to solve equations involving algebraic expressions. An equation may have no solution or one or more solutions. If two or more equations are to be solved together, the solutions must satisfy all the equations simultaneously. Two equations having the same solution s are equivalent equations. Note that the second equation is the first equation multiplied by 2. Solving Linear Equations with One Unknown To solve a linear equation with one unknown that is, to find the value of the unknown that satisfies the equation , the unknown should be isolated on one side of the equation.

This can be done by performing the same mathematical operations on both sides of the equation. Remember that if the same number is added to or subtracted from both sides of the equation, this does not change the equality; likewise, multiplying or dividing both sides by the same nonzero number does not change the equality. Solving Two Linear Equations with Two Unknowns For two linear equations with two unknowns, if the equations are equivalent, then there are infinitely many solutions to the equations, as illustrated at the end of section 3.

If the equations are not equivalent, then they have either one unique solution or no solution. The latter case is illustrated by the two equations: Thus, no values of x andy can simultaneously satisfy both equations. There are several methods of solving two linear equations with two unknowns.

Otherwise, a unique solution can be found. One way to solve for the two unknowns is to express one of the unknowns in terms of the other using one of the equations, and then substitute the expression into the remaining equation to obtain an equation with one unknown.

This equation can be solved and the value of the unknown substituted into either of the original equations to find the value of the other unknown.

For example, the following two equations can be solved for x andy. There is another way to solve for x andy by eliminating one of the unknowns. This can be done by making the coefficients of one of the unknowns the same disregarding the sign in both equations and either adding the equations or subtracting one equation from the other. These answers can be checked by substituting both values into both of the original equations.

Solving Equations by Factoring Some equations can be solved by factoring. To do this, first add or subtract expressions to bring all the expressions to one side of the equation, with 0 on the other side.

Then try to factor the nonzero side into a product of expressions. If this is possible, then using property 7 in section 3. The solutions of the simpler equations will be solutions of the factored equation. A fraction equals 0 if and only if its numerator x-4 equals 0. Thus, the solutions are 0 and 3.

The solutions of an equation are also called the roots of the equation. These roots can be checked by substituting them into the original equation to determine whether they satisfy the equation. Jb 2 - 4ac and -b-. Jb 2 - 4ac is not a 2a real number and the equation has no real roots. Exponents A positive integer exponent of a number or a variable indicates a product, and the positive integer is the number of times that the number or variable is a factor in the product.

For example, x 5 means x x x x x ; that is, x is a factor in the product 5 times. Some rules about exponents follow. Let x andy be any positive numbers, and let rand s be any positive integers. It can be shown that rules also apply when rands are not integers and are not positive, that is, when r and s are any real numbers. Inequalities An inequality is a statement that uses one of the following symbols: As in solving an equation, the same number can be added to or subtracted from both sides of the inequality, or both sides of an inequality can be multiplied or divided by a positive number without changing the truth of the inequality.

However, multiplying or dividing an inequality by a negative number reverses the order of the inequality.

Functions An algebraic expression in one variable can be used to define a function of that variable. A function is denoted by a letter such as for g along with the variable in the expression. In any function there can be no more than one output for any given input. The set of all allowable inputs for a function is called the domain of the function. Forf and g defined above, the domain off is the set of all real numbers and the domain of g is the set of all numbers greater than The domain of a function can consist of only the positive integers and possibly 0.

Lines In geometry, the word "line" refers to a straight line that extends without end in both directions. The part of the line from P to Q is called a line segment. P and Q are the endpoints of the segment. Intersecting Lines and Angles If two lines intersect, the opposite angles are called vertical angles and have the same measure.

Perpendicular Lines An angle that has a measure of90 o is a right angle. If two lines intersect at right angles, the lines are perpendicular.

Parallel Lines If two lines that are in the same plane do not intersect, the two lines are parallel. Polygons Convex A polygon is a closed plane figure formed by three or more line segments, called the sides of the polygon. Each side intersects exactly two other sides at their endpoints.

The points of intersection of the sides are vertices. The following figures are polygons: Note that a pentagon can be partitioned into three triangles and therefore the sum of the angle measures can be found by adding the sum of the angle measures of three triangles.

The perimeter of a polygon is the sum of the lengths of its sides. The commonly used phrase "area of a triangle" or any other plane figure is used to mean the area of the region enclosed by that figure. Triangles There are several special types of triangles with important properties.

But one property that all triangles share is that the sum of the lengths of any two of the sides is greater than the length of the third side, as illustrated below. All angles of an equilateral triangle have equal measure. An isosceles triangle has at least two sides of the same length. If two sides of a triangle have the same length, then the two angles opposite those sides have the same measure. Conversely, if two angles of a triangle have the same measure, then the sides opposite those angles have the same length.

In a right triangle, the side opposite the right angle is the hypotenuse, and the other two sides are the legs. An important theorem concerning right triangles is the Pythagorean theorem, which states: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Any triangle in which the lengths of the sides are in the ratio 3: For example, in tl. The altitude of a triangle is the segment drawn from a vertex perpendicular to the side opposite that vertex. Relative to that vertex and altitude, the opposite side is called the base. The area of a triangle is equal to: The area of tl. Similarly, any altitude of an equilateral triangle bisects the side to which it is drawn.

The area of 6. Quadrilaterals A polygon with four sides is a quadrilateral. A quadrilateral in which both pairs of opposite sides are parallel is a parallelogram. The opposite sides of a parallelogram also have equal length.

A parallelogram with right angles is a rectangle, and a rectangle with all sides of equal length is a square. The area of trapezoid PQRS may be calculated as follows: Circles A circle is a set of points in a plane that are all located the same distance from a fixed point the center of the circle. A chord of a circle is a line segment that has its endpoints on the circle. A chord that passes through the center of the circle is a diameter of the circle. A radius of a circle is a segment from the center of the circle to a point on the circle.

The words "diameter" and "radius" are also used to refer to the lengths of these segments. The circumftrence of a circle is the distance around the circle. If r is the radius of the circle, then the circumference is equal to 2nr, where: The area of a circle of radius r is 7 equa1 to nr. PR is a diameter and OR is a radius. The number of degrees of arc in a circle or the number of degrees in a complete revolution is A line that has exactly one point in common with a circle is said to be tangent to the circle, and that common point is called the point oftangency.

A radius or diameter with an endpoint at the point of tangency is perpendicular to the tangent line, and, conversely, a line that is perpendicular to a radius or diameter at one of its endpoints is tangent to the circle at that endpoint. If each vertex of a polygon lies on a circle, then the polygon is inscribed in the circle and the circle is circumscribed about the polygon.

If each side of a polygon is tangent to a circle, then the polygon is circumscribed about the circle and the circle is inscribed in the polygon. If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is a right triangle.

Rectangular Solids and Cylinders A rectangular solid is a three-dimensional figure formed by 6 rectangular surfaces, as shown below. Each rectangular surface is a foce. Each solid or dotted line segment is an edge, and each point at which the edges meet is a vertex. A rectangular solid has 6 faces, 12 edges, and 8 vertices.

Opposite faces are parallel rectangles that have the same dimensions. A rectangular solid in which all edges are of equal length is a cube. The surfoce area of a rectangular solid is equal to the sum of the areas of all the faces. The volume is equal to length x width x height ; in other words, area of base x height.

In the rectangular solid above, the dimensions are 3, 4, and 8. The figure above is a right circular cylinder. The two bases are circles of the same size with centers 0 and P, respectively, and altitude height OP is perpendicular to the bases.

The surface area of a right circular cylinder with a base of radius rand height his equal to 2. The volume of a cylinder is equal to nr 2 h, that is, area of base x height. The horizontal line is called the x-axis and the perpendicular vertical line is called they-axis.

The point at which these two axes intersect, designated 0, is called the origin. Each point in the plane has an x-coordinate and a y-coordinate. A point is identified by an ordered pair x,y of numbers in which the x-coordinate is the first number and they-coordinate is the second number. Similarly, the xV' coordinates of point Q are -4, The origin 0 has coordinates 0,0. One way to find the distance between two points in the coordinate plane is to use the Pythagorean theorem.

One can verify this for the points -2,2 , 2,0 , and 0,1 by substituting the respective coordinates for x andy in the equation.

For any two points on the line, the slope is defined to be the ratio of the difference in they-coordinates to the difference in the x-coordinates. They-intercept is they-coordinate of the point at which the line intersects they-axis. The x-intercept is the x-coordinate of the point at which the line intersects the x-axis. Thus, the x-intercept is 2. For example, consider the points -2,4 and 3,-3 on the line below.

They-intercept is lThe x-intercept can be found as follows: If the slope of a line is negative, the line slants downward from left to right; if the slope is positive, the line slants upward. There is a connection between graphs of lines in the coordinate plane and solutions of two linear equations with two unknowns.

If two linear equations with unknowns x andy have a unique solution, then the graphs of the equations are two lines that intersect in one point, which is the solution. If the equations are equivalent, then they represent the same line with infinitely many points or solutions.

If the equations have no solution, then they represent parallel lines, which do not intersect. There is also a connection between functions see section 3. If a function is graphed in the coordinate plane, the function can be understood in different and useful ways.

Similarly, any function f x can be graphed by equating y with the value of the function: One can plot several points x,f x on the graph to understand the connection between a function and its graph: The following discussion of word problems illustrates some of the techniques and concepts used in solving such problems. Example 1: If a car travels at an average speed of 70 kilometers per hour for 4 hours, how many kilometers does it travel?

Thus, the car travels kilometers in 4 hours. To determine the average rate at which an object travels, divide the total distance traveled by the total amount of traveling time. Example 2: On a mile trip, Car X traveled half the distance at 40 miles per hour mph and the other half at 50 mph.

What was the average speed of Car X? First it is necessary to determine the amount of traveling time. Some rate problems can be solved by using ratios. Example 3: Cross multiplication results in the equation Solution: Work Problems In a work problem, the rates at which certain persons or machines work alone are usually given, and it is necessary to compute the rate at which they work together or vice versa. The reasoning is that in 1 hour Rae does 1 of the job, Sam does 1 of the job, and Rae and Sam together do 1 of r s h the job.

If Machine X can produce 1, bolts in 4 hours and Machine Y can produce 1, bolts in 5 hours, in how many hours can Machines X andY, working together at these constant rates, produce 1, bolts? If Art and Rita can do a job in 4 hours when working together at their respective constant rates and Art can do the job alone in 6 hours, in how many hours can Rita do the job alone?

Mixture Problems In mixture problems, substances with different characteristics are combined, and it is necessary to determine the characteristics of the resulting mixture. How many liters of a solution that is 15 percent salt must be added to 5 liters of a solution that is 8 percent salt so that the resulting solution is 10 percent salt? Interest Problems Interest can be computed in two basic ways.

If interest is compounded, then interest is computed on the principal as well as on any interest already earned. Discount If a price is discounted by npercent, then the price becomes - n percent of the original price.

If that price represented a 25 percent discount on the original price of the dress, what was the original price of the dress? If pis the original price of the dress, then 0. The price of an item is discounted by 20 percent and then this reduced price is discounted by an additional 30 percent. These two discounts are equal to an overall discount of what percent? If p is the original price of the item, then 0.

The price after the second discount is 0. Profit Gross profit is equal to revenues minus expenses, or selling price minus cost. At what price should the merchant sell the appliance in order to make a gross profit of 50 percent of the cost of the appliance?

Sets can also be represented by Venn diagrams. That is, the relationship among the members of sets can be represented by circles. Each of 25 people is enrolled in history, mathematics, or both. If 20 are enrolled in history and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics? The 25 people can be divided into three sets: Thus a Venn diagram may be drawn as follows, where n is the number of people enrolled in both courses, 20 - n is the number enrolled in history only, and n is the number enrolled in mathematics only.