Cliffs math review for standardized tests download

Orwell's vision of the future may be grim, but your understanding of his novel can be bright thanks to detailed summaries and commentaries for every chapter. Other features that help you study include Character analyses of major players A character map that graphically illustrates the relationships among the characters Critical essays A review section that tests your knowledge A Resource Center full of books, articles, films, and Internet sites Classic literature or modern-day treasure—you'll understand it all with expert information and insight from CliffsNotes study guides.

Author : Denis M. The latest generation of titles in this series also feature glossaries and visual elements that complement the classic, familiar format.

CliffsNotes on The Crucible takes you into Arthur Miller's play about good and evil, self-identity and morality. Following the atmosphere and action of the Salem witch trials of the s, this study guide looks into Puritan culture with critical commentaries about each act and scene.

Other features that help you figure out this important work include Life and background of the author Introduction to the play Character web and in-depth analyses of the major roles Summaries and glossaries related to each act Essays that explore the author's narrative technique and the play's historical setting A review section that tests your knowledge and suggests essay topics and practice projects A Resource Center for checking out details on books, publications, and Internet resources Classic literature or modern-day treasure—you'll understand it all with expert information and insight from CliffsNotes study guides.

The latest generation of titles in this series also features glossaries and visual elements that complement the familiar format. Scott Fitzgerald's novel of triumph, tragedy, and a classic love triangle in the s.

Following the story of a young Midwesterner who's fascinated by the mysterious past and opulent lifestyle of his landlord, this study guide provides summaries and critical commentaries for each chapter within the novel.

Other features that help you figure out this important work include Personal background on the author Introduction to and synopsis of the book In-depth character analyses Critical essays on topics of interest Review section that features interactive questions and suggested essay topics and practice projects Resource Center with books, videos, and websites that can help round out your knowledge Classic literature or modern-day treasure—you'll understand it all with expert information and insight from CliffsNotes study guides.

Is divisible by 6? Because ends in 6, it is divisible by 2. Because 15 is divisible by 3, is divisible by 3. Is 2, divisible by 8? Because is divisible by 8, you know that 2, is divisible by 8. Is 2, divisible by 9? Because 18 is divisible by 9, you know that 2, is divisible by 9. The numerator tells you how many of these equal parts are being considered.

Thus, if the fraction is of a pie, then the denominator, 5, tells you that the pie has been divided into 5 equal parts, of which 3 the numerator are in the fraction.

All rules for signed numbers also apply to fractions. Negative Fractions Fractions may be negative as well as positive see the number line on p. However, negative fractions are typically written not or although they are all equal : Proper Fractions and Improper Fractions A fraction like , where the numerator is smaller than the denominator, is less than one.

This kind of fraction is called a proper fraction. But sometimes a fraction may be more than or equal to one. This is when the numerator is larger than or equal to the denominator. Thus, is more than one.

This is called an improper fraction. Here are some examples of proper fractions: , , , Here are some examples of improper fractions: , , , , 27 Part I: Basic Skills Review Mixed Numbers When a term contains both a whole number and a fraction, it is called a , are both mixed numbers.

To and mixed number. For instance, change an improper fraction to a mixed number, you divide the denominator into the numerator. For example: To change a mixed number to an improper fraction, you multiply the denominator times the whole number, add in the numerator, and put the total over the original denominator. Change the following mixed numbers to improper fractions: 6.

Answers: Mixed Numbers 1. Equivalent Fractions Reducing Fractions A fraction must be reduced to lowest terms. This is done by dividing both the numerator and denominator by the largest number that will divide evenly into both. For example, is reduced to by dividing both numerator and denominator by 5.

Answers: Reducing Fractions 1. Enlarging Denominators The denominator of a fraction may be enlarged by multiplying both the numerator and the denominator by the same number. For example: Practice: Enlarging Denominators 1. Change to tenths.

Express as eighths. Change the fraction of Factors Factors of a number are those whole numbers that, when multiplied together, yield the number. For example: What are the factors of 8? What are the factors of 24? Practice: Factors Find the factors of the following: 1. For example: What are the common factors of 6 and 8? Note: Some numbers may have many common factors.

For example: What are the common factors of 24 and 36? Practice: Common Factors Find the common factors of the following: 1. For example: What is the greatest common factor of 12 and 30? Practice: Greatest Common Factor Find the greatest common factor of the following: 1. Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, and so on. Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, and so on. Practice: Multiples Name the first seven multiples of the following: 1.

For example: What are the common multiples of 2 and 3? Notice that common multiples may go on indefinitely. Practice: Common Multiples Find the first three common multiples of the following: 1. For example: What is the least common multiple of 2 and 3? Arithmetic and Data Analysis 6 is the smallest multiple common to both 2 and 3. Another example: What is the least common multiple of 2, 3, and 4? Practice: Least Common Multiple Find the least common multiple of the following: 1.

When you have all the denominators the same, you may add fractions by simply adding the numerators the denominator remains the same.

For example: 3 8 1 2 3 8 4 8 7 8 one-half is changed to four-eighths 1 4 1 3 3 12 4 12 7 12 change both fractions to LCD of 12 35 Part I: Basic Skills Review In the first example, we changed the to because 8 is the lowest com- mon denominator, and then we added the numerators 3 and 4 to get.

In the second example, we had to change both fractions to get the lowest common denominator of 12, and then we added the numerators to get. Of course, if the denominators are already the same, just add the numerators. For example: Note that fractions may be added across, as well. For example: Practice: Adding Fractions 1. Answers: Adding Fractions 1. Practice: Adding Positive and Negative Fractions 1.

Answers: Adding Positive and Negative Fractions 1. For example: Again, a subtraction problem may be done across, as well as down: Practice: Subtracting Fractions 1. Subtracting Positive and Negative Fractions The rule for subtracting signed numbers p. Practice: Subtracting Positive and Negative Fractions 1. Answers: Subtracting Positive and Negative Fractions 1. For example: Sometimes you may end up with a mixed number that includes an improper fraction.

In that case, you must change the improper fraction to a mixed number and combine it with the sum of the integers. Practice: Adding Mixed Numbers 1. Answers Adding Mixed Numbers Problems 1.

For example: Remember that the rules for signed numbers p. Practice: Subtracting Mixed Numbers 1. Answers: Subtracting Mixed Numbers 1. Multiplying Fractions and Mixed Numbers Multiplying Fractions To multiply fractions, simply multiply the numerators, and then multiply the denominators. Reduce to the lowest terms if necessary. Because whole numbers can also be written as fractions , , and so on , the problem would be worked by changing 3 to. To cancel, find a number that divides evenly into one numerator and one denominator.

In this case, 2 will divide evenly into 2 in the numerator it goes in one time and 12 in the denominator it goes in 6 times. Thus: Remember: You may cancel only when multiplying fractions. The rules for multiplying signed numbers hold here, too p. For example: and Practice: Multiplying Fractions Problems 1. Answers: Multiplying Fractions Problems 1. Multiplying Mixed Numbers To multiply mixed numbers, first change any mixed number to an improper fraction. Then multiply as previously shown p.

Remember: The rules for multiplication of signed numbers apply here as well p. Practice: Multiplying Mixed Numbers 1. Answers: Multiplying Mixed Numbers 1. Then reduce, if necessary. For example: Here, too, the rules for division of signed numbers apply p. Practice: Dividing Fractions 1. Answers: Dividing Fractions 1. Dividing Complex Fractions Sometimes a division-of-fractions problem may appear in this form; these are called complex fractions.

Practice: Dividing Complex Fractions 1. Answers: Dividing Complex Fractions 1. Dividing Mixed Numbers To divide mixed numbers, first change them to improper fractions p. Then follow the rule for dividing fractions p. Notice that after you invert and have a multiplication-of-fractions problem, you may then cancel tops with bottoms when appropriate. Practice: Dividing Mixed Numbers 1. Answers: Dividing Mixed Numbers 1. Simplifying Fractions and Complex Fractions If either numerator or denominator consists of several numbers, these numbers must be combined into one number.

Then reduce if necessary. Practice: Simplifying Fractions and Complex Fractions 1. Answers: Simplifying Fractions and Complex Fractions 1. All numbers to the left of the decimal point are whole numbers.

All numbers to the right of the decimal point are fractions with denominators of only 10, , 1,, 10,, and so on. For example: Read it: 0. Practice: Changing Decimals to Fractions Change the following decimals to fractions. Reduce if necessary. For example: For example: 17 — 8.

Then count the total number of digits above the line which are to the right of all decimal points. Place your decimal point in your answer so the same number of digits are to the right of the decimal point as there are above the line. For example: Practice: Multiplying Decimals 1. Then move the decimal point in the dividend the number being divided into the same number of places. Sometimes you may have to add zeros to the dividend the number inside the division sign. For example: or Practice: Dividing Decimals 1.

Divide 8 by 0. Divide Answers: Dividing Decimals 1. In other words, means 13 divided by So do just that insert decimal points and zeros accordingly. Answers: Changing Fractions to Decimals 1. The word percent means hundredths per hundred. For example: Changing Decimals to Percents To change decimals to percents: 1. Move the decimal point two places to the right. Insert a percent sign.

For example: 0. Eliminate the percent sign. Move the decimal point two places to the left sometimes adding zeros will be necessary. Change the decimal to a percent. Drop the percent sign. Write over Remember, the word of means multiply. Answers: Finding Percent of a Number 1. For what, substitute the letter x; for is, substitute an equal sign; for of substitute a multiplication sign.

Change percents to decimals or fractions, whichever you find easier. Then solve the equation. Answers: Other Applications of Percent 1. Percent—Proportion Method A proportion is a statement that says that two values expressed in fraction form are equal.

Since and both have values of , it can be stated that. In the example of 60 Arithmetic and Data Analysis You can use this cross-products fact in order to solve a proportion. Because the percent is the unknown, put an x over the The number 30 is next to the word is, so it goes on top of the next fraction, and 50 is next to the word of, so it goes on the bottom of the next fraction. Practice: Percent—Proportion Method 1. What percent of 25 is 10?

Answers: Percent—Proportion Method 1. Answer: 4. Note that the terms percentage rise, percentage difference, and percentage change are the same as percent change. Find the percent decrease from to What is the percent increase in rainfall from January 2. What is the percent change from 2, to 1,? Powers and Exponents An exponent is a positive or negative number or zero placed above and to the right of a quantity.

It expresses the power to which the quantity is to be raised or lowered. In 43, 3 is the exponent. The number can be simplified as follows: A few more examples: 64 Arithmetic and Data Analysis Operations with Powers and Exponents To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents.

To multiply or divide numbers with exponents, if the base numbers are different, you must simplify each number with an exponent first and then perform the operation.

To add or subtract numbers with exponents, whether the base numbers are the same or different, you must simplify each number with an exponent first and then perform the indicated operation. A number written in scientific notation is a number between 1 and 10 and multiplied by a power of Simply place the decimal point to get a number between 1 and 10 and then count the digits to the right of the decimal to get the power of Simply place the decimal point to get a number between 1 and 10 and then count the digits from the original decimal point to the new one.

That is, if a number expressed in scientific notation has a positive exponent, then its value is greater than 1, and if it has a negative exponent, then it is a positive number but is less than 1. Practice: Scientific Notation Change the following to scientific notation: 1. To square a number, just multiply it by itself the exponent would be 2.

Here is a list of the first 13 perfect squares. Here is a list of the first 8 perfect cubes. Square Roots and Cube Roots Note that square and cube roots and operations with them are often included in algebra sections, and the following will be discussed further in the algebra section. Square Roots To find the square root of a number, you want to find some number that, when multiplied by itself, gives you the original number. In other words, to find the square root of 25, you want to find the number that, when multiplied by itself, gives you The square root of 25, then, is 5.

The symbol for square root is. Here is a list of the first 11 perfect whole number square roots. Other roots are similarly defined and identified by the index given. Special note: If no sign or a positive sign is placed in front of the square root, then the positive answer is required. Only if a negative sign is in front of 72 Arithmetic and Data Analysis the square root is a negative answer required.

This notation is used on most standardized exams and will be adhered to in this book. Cube Roots To find the cube root of a number, you want to find some number that, when multiplied by itself twice, gives you the original number.

In other words, to find the cube root of 8, you want to find the number that, when multiplied by itself twice, gives you 8. Notice that the symbol for cube root is the square root sign with a small three called the index above and to the left. In square root, an index of two is understood and usually not written. Following is a list of the first five perfect whole number cube roots: Notice that the cube root of a negative number is a real number, but that the square root of a negative number is not a real number: , which is a real number Approximating Square Roots To find the square root of a number that is not an exact square, you will need to find an approximate answer by using the procedure explained here: Approximate The is between.

To check, multiply: and 6. Square roots of nonperfect squares can be approximated or looked up in tables. Answers: Approximating Square Root Problems 1. In fractions, can be reduced to. In square roots,. To simplify a square root, first factor the number under the plified to into a counting number times the largest perfect square number that will divide into the number without leaving a remainder. Perfect square numbers are 1, 4, 9, 16, 25, 36, For example: Then take the square root of the perfect square number: and finally write as a single expression:.

Remember that most square roots cannot be simplified, as they are , or. Answers: Simplifying Square Roots 1. When all outcomes are equally likely to occur, the probability of the occurrence of a given outcome can be found by using the following formula: Examples: 1. Using the spinner below, what is the probability of spinning a 6 in one spin?

Using the spinner above, what is the probability of spinning either a 3 or a 5 in one spin? Since there are two favorable outcomes out of ten possible outcomes, the probability is , or. What is the probability that both spinners below will stop on a 3 on the first spin?

What is the probability that on two consecutive rolls of a die the numbers will be 2 and then 3? Since the probability of getting a 2 on the first roll is and the probability of getting a 3 on the second roll is , and since the rolls are independent of each other, simply multiply: 5.

What is the probability of tossing heads three consecutive times with a two-sided fair coin? Since each toss is independent and the probability is the probability would be: for each toss, 6. What is the probability of rolling two dice in one toss so that they total 5?

These are all the ways of tossing a total of 5 on two dice. Thus, there are four favorable outcomes, which gives the probability of throwing a five as: 7. Three green marbles, two blue marbles, and five yellow marbles are placed in a jar.

What is the probability of selecting at random a green marble on the first draw? Since there are ten marbles total possible outcomes and three green marbles favorable outcomes , the probability is. Using the equally spaced spinner above, what is the probability of spinning a 4 or greater in one spin? Using the equally spaced spinner above, what is the probability of spinning either a 2 or a 5 on one spin?

What is the probability of rolling two dice in one toss so that they total 7? What is the probability of tossing tails four consecutive times with a two-sided fair coin? What is the probability that each equally spaced spinner above will stop on a 2 on its first spin? In a regular deck of 52 cards, what is the probability of drawing a heart on the first draw?

There are 13 hearts in a deck. Answers: Probability 1. Since there are five numbers that are 4 or greater out of the eight numbers and all the numbers are equally spaced, the probability is. Since there are two favorable outcomes out of eight possible outcomes, the probability is , or. Since each toss is independent and the probability is probability would be for each toss, the. Since the probability that the first spinner will stop on the number 2 is , and the probability that the second spinner will stop on the number 2 is , and since each event is independent of the other, simply multiply: 6.

Since there are 13 favorable outcomes out of 52 possible outcomes, the probability is , or. Combinations and Permutations If there are a number of successive choices to make and the choices are independent of each other order makes no difference , the total number of possible choices combinations is the product of each of the choices at each stage.

How many possible combinations of shirts and ties are there if there are five different color shirts and three different color ties? How many ways can you arrange the letters S, T, O, P in a row?

Thus, there are 24 different ways to arrange four different letters. Following is a more difficult type of combination involving permutations. In how many ways can four out of seven books be arranged on a shelf?

Notice that the order in which the books are displayed makes a difference. The symbol to denote this is P n, r , which is read as the permutations of n things taken r at a time.

If, from among five people, three executives are to be selected, how many possible combinations of executives are there? Notice that the order of selection makes no difference. The symbol used to denote this situation is C n, r , which is read as the number of combinations of n things taken r at a time. Practice: Combinations and Permutations Problems 1. How many possible outfits could Tim wear if he has three different color shirts, four different types of slacks, and two pairs of shoes?

A three-digit PIN requires the use of the numbers from 0 to 9. How many different possible PINs exist? How many different ways are there to arrange three jars in a row on a shelf? There are nine horses in a race. How many different 1st-, 2nd-, and 3rd-place finishes are possible?

A coach is selecting a starting lineup for her basketball team. She must select from among nine players to get her starting lineup of five. How many possible starting lineups could she have? How many possible combinations of a, b, c, and d taken two at a time are there? Answers: Combinations and Permutations Problems 1. To find the total number of possible combinations, simply multiply the numbers together.

Since the order of the items is affected by the previous choice s , the number of different ways equals 3! The order in which players is selected does not matter; thus, use the combinations formula. Statistics Some Basics: Measures of Central Tendencies Any measure indicating a center of a distribution is called a measure of central tendency.

The arithmetic mean is the most frequently used measure of central tendency. It is generally reliable, is easy to use, and is more stable than the median. To determine the arithmetic mean, simply total the items and then divide by the number of items.

What is the arithmetic mean of 0, 12, 18, 20, 31, and 45? What is the arithmetic mean of 25, 27, 27, and 27? What is the arithmetic mean of 20 and —10? Practice: Arithmetic Mean Problems 1. Find the arithmetic mean of 3, 6, and Find the arithmetic mean of 2, 8, 15, and Find the arithmetic mean of 26, 28, 36, and Find the arithmetic mean of 3, 7, —5, and — Answers: Arithmetic Mean Problems 1. The weighted mean is, thus, For the first nine months of the year, the average monthly rainfall was 2 inches.

For the last three months of that year, rainfall averaged 4 inches per month. What was the mean monthly rainfall for the entire year? What was the mean score of all ten students?

Answers: Weighted Mean Problems 1. If there is an even number of items in the set, their median is the arithmetic mean of the middle two numbers. The median is easy to calculate and is not influenced by extreme measurements.

Find the median of 3, 4, 6, 9, 21, 24, Find the median of 4, 5, 6, Practice: Median Problems Find the median of each group of numbers. Mode The set, class, or classes that appear most, or whose frequency is the greatest is the mode or modal class. In order to have a mode, there must be a repetition of a data value. Mode is not greatly influenced by extreme cases but is probably the least important or least used of the three types.

For example: Find the mode of 3, 4, 8, 9, 9, 2, 6, 11 The mode is 9 because it appears more often than any other number. Practice: Mode Problems Find the mode of each group of numbers. The range depends solely on the extreme values. For example: Find the range of the following numbers. Practice: Range Problems 1. Find the range of 2, 45, , 99 2. Find the range of 6, , , —5 Answers: Range Problems 1. A small standard deviation indicates that the data values tend to be very close to the mean value.

Each colored band 88 Arithmetic and Data Analysis has a width of one standard deviation. You will find approximately At three standard deviations from the mean, approximately The basic method for calculating the standard deviation for a population is lengthy and time consuming.

It involves five steps: 1. Find the mean value for the set of data. For each data value, find the difference between it and the mean value; then square that difference. Find the sum of the squares found in Step 2. Divide the sum found in Step 3 by how many data values there are. Find the square root of the value found in Step 4. The result found in Step 4 is referred to as the variance. The square root of the variance is the standard deviation.

For example: Find the variance and standard deviation for the following set of data. Find the mean value: 2. Find the squares of the differences between the data values and the mean. Find the sum of the squares from Step 2: Divide the sum from Step 3 by how many data values there are:. Find the square root of the value found in Step 4: standard deviation. The statistical values that would be affected are the mean, median, and mode.

The statistical values that would not be affected are the range, variance, and standard deviation. The mean, median, and mode would each increase by the amount that each data value was increased.

For example: A scientist discovered that the instrument used for an experiment was off by 2 milligrams. If each weight in his experiment needed to be increased by 2 milligrams, then which of the following statistical measures would not be affected? Only range and standard deviation would not be affected. Number Sequences Progressions of numbers are sequences with some patterns.

Unless the sequence has a simple repeat pattern 1, 2, 4, 1, 2, 4,. Practice: Number Sequence Problems Find the next number in each sequence. Weight 16 ounces oz. A kilometer is about 0. A kilogram is about 2. A liter is slightly more than a quart. Converting Units of Measure Examples: 1. If 36 inches equals 1 yard, then 3 yards equals how many inches? Change 3 decades into weeks. Since 1 decade equals 10 years and 1 year equals 52 weeks, then 3 decades equal 30 years.

If 1, yards equal 1 mile, how many yards are in 5 miles? If 1 kilometer equals approximately 0. How many cups are in 3 gallons? How many ounces are in 6 pounds? If 1 kilometer equals 1, meters and 1 decameter equals 10 meters, how many decameters are in a.

The numbers 1, 2, 3, 4,. The numbers 0, 1, 2, 3,. The numbers. Give the symbol or symbols for each of the following. Show three of them. List the properties that are represented by each of the following.

In the number ,, which digit is in the ten thousands place? Express in expanded notation. Round off 7. Complete the number line below: A The number 8, is divisible by which numbers between 1 and 10? Change to a mixed number. Change to an improper fraction.

Change to twelfths. List all the factors of Why CliffsNotes? Go with the name you know and trust Get the information you need—fast! Each stand-alone plan includes: Diagnostic test—helps you pinpoint your strengths and weaknesses so you can focus your review on the topics in which you need the most help Subject reviews—cover everything you can expect on the actual exam: text completions, sentence equivalences, vocabulary, reading comprehension, analytical writing, arithmetic, algebra, geometry, and applications Full-length practice test with answers and detailed explanations—a simulated GRE exam gives you an authentic test-taking experience Test-prep essentials from the experts at CliffsNotes Reflects changes to the latest GRE General Test Make the most of the time you have left!

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