Exam success vocabulary_9 pps

For this question, use the definition of the operation as the formula and substitute the values 3 and 2 for a and b, respectively. a 2 – 2b = 3 2 – 2(2) = 9 – 4 = 5. The correct answer is d.  Factors, Multiples, and Divisibility In the following section, the principles of factors, multipliers, and divisibility are covered. Factors A whole number is a factor of a number if it divides into the number without a remainder. For example, 5 is a factor of 30 because without a remainder left over. On the GMAT exam, a factor question could look like this: If x is a factor of y, which of the following may not represent a whole number? a. xy b. c. d. e. This is a good example of where substituting may make a problem simpler. Suppose x = 2 and y = 10 (2 is a factor of 10). Then choice a is 20, and choice c is 5. Choice d reduces to just y and choice e reduces to just x, so they will also be whole numbers. Choice b would be ᎏᎏ 1 2 0 ᎏ , which equals ᎏ 1 5 ᎏ , which is not a whole number. Prime Factoring To prime factor a number, write it as the product of its prime factors. For example, the prime factorization of 24 is 24 = 2 × 2 × 2 × 3 = 2 3 × 3 24 12 6 2 2 3 2 xy y yx x y x x y 30 Ϭ 5 ϭ 6 – ARITHMETIC– 325 Greatest Common Factor (GCF) The greatest common factor (GCF) of two numbers is the largest whole number that will divide into either number without a remainder. The GCF is often found when reducing fractions, reducing radicals, and fac- toring. One of the ways to find the GCF is to list all of the factors of each of the numbers and select the largest one. For example, to find the GCF of 18 and 48, list all of the factors of each: 18: 1, 2, 3, 6, 9, 18 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Although a few numbers appear in both lists, the largest number that appears in both lists is 6; there- fore, 6 is the greatest common factor of 18 and 48. You can also use prime factoring to find the GCF by listing the prime factors of each number and mul- tiplying the common prime factors together: The prime factors of 18 are 2 × 3 × 3. The prime factors of 48 are 2 × 2 × 2 × 2 × 3. They both have at least one factor of 2 and one factor of 3. Thus, the GCF is 2 × 3 = 6. Multiples One number is a multiple of another if it is the result of multiplying one number by a positive integer. For example, multiples of three are generated as follows: 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12, . . . There- fore, multiples of three can be listed as {3, 6, 9, 12, 15, 18, 21, } Least Common Multiple (LCM) The least common multiple (LCM) of two numbers is the smallest number that both numbers divide into without a remainder. The LCM is used when finding a common denominator when adding or subtracting fractions. To find the LCM of two numbers such as 6 and 15, list the multiples of each number until a com- mon number is found in both lists. 6: 6, 12, 18, 24, 30, 36, 42, . . . 15: 15, 30, 45, . . . As you can see, both lists could have stopped at 30; 30 is the LCM of 6 and 15. Sometimes it may be faster to list out the multiples of the larger number first and see if the smaller number divides evenly into any of those multiples. In this case, we would have realized that 6 does not divide into 15 evenly, but it does divide into 30 evenly; therefore, we found our LCM. Divisibility Rules To aid in locating factors and multiples, some commonly known divisibility rules make finding them a little quicker, especially without the use of a calculator. – ARITHMETIC– 326 ■ Divisibility by 2. If the number is even (the last digit, or units digit, is 0, 2, 4, 6, 8), the number is divisible by 2. ■ Divisibility by 3. If the sum of the digits adds to a multiple of 3, the entire number is divisible by 3. ■ Divisibility by 4. If the last two digits of the number form a number that is divisible by 4, then the entire number is divisible by 4. ■ Divisibility by 5. If the units digit is 0 or 5, the number is divisible by 5. ■ Divisibility by 6. If the number is divisible by both 2 and 3, the entire number is divisible by 6. ■ Divisibility by 9. If the sum of the digits adds to a multiple of 9, the entire number is divisible by 9. ■ Divisibility by 10. If the units digit is 0, the number is divisible by 10.  Prime and Composite Numbers In the following section, the principles of prime and composite numbers are covered. Prime Numbers These are natural numbers whose only factors are 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Two is the smallest and the only even prime number. The number 1 is neither prime nor composite. Composite Numbers These are natural numbers that are not prime; in other words, these numbers have more than just two fac- tors. The number 1 is neither prime nor composite. Relatively Prime Two numbers are relatively prime if the GCF of the two numbers is 1. For example, if two numbers that are relatively prime are contained in a fraction, that fraction is in its simplest form. If 3 and 10 are relatively prime, then is in simplest form.  Even and Odd Numbers An even number is a number whose units digit is 0, 2, 4, 6, or 8. An odd number is a number ending in 1, 3, 5, 7, or 9.You can identify a few helpful patterns about even and odd numbers that often arise on the Quan- titative section: odd + odd = even odd × odd = odd even + even = even even × even = even even + odd = odd even × odd = even 3 10 – ARITHMETIC– 327 When problems arise that involve even and odd numbers, you can use substitution to help remember the patterns and make the problems easier to solve.  Consecutive Integers Consecutive integers are integers listed in numerical order that differ by 1. An example of three consecutive integers is 3, 4, and 5, or –11, –10, and –9. Consecutive even integers are numbers like 10, 12, and 14 or –22, –20, and –18. Consecutive odd integers are numbers like 7, 9, and 11. When they are used in word problems, it is often useful to define them as x, x + 1, x + 2, and so on for regular consecutive integers and x, x + 2, and x + 4 for even or odd consecutive integers. Note that both even and odd consecutive integers have the same algebraic representation.  Absolute Value The absolute value of a number is the distance a number is away from zero on a number line. The symbol for absolute value is two bars surrounding the number or expression. Absolute value is always positive because it is a measure of distance. |4| = 4 because 4 is four units from zero on a number line. |–3| = 3 because –3 is three units from zero on a number line.  Operations with Real Numbers For the quantitative exam, you will need to know how to perform basic operations with real numbers. Integers This is the set of whole numbers and their opposites, also known as signed numbers. Since negatives are involved, here are some helpful rules to follow. A DDING AND SUBTRACTING INTEGERS 1. If you are adding and the signs are the same, add the absolute value of the numbers and keep the sign. a. 3 + 4 = 7 b. –2 + –13 = –15 2. If you are adding and the signs are different, subtract the absolute value of the numbers and take the sign of the number with the larger absolute value. a. –5 + 8 = 3 b. 10 + –14 = –4 – ARITHMETIC– 328 3. If you are subtracting, change the subtraction sign to addition, and change the sign of the number fol- lowing to its opposite. Then follow the rules for addition: a. –5 + –6 = –11 b. –12 + (+7) = –5 Remember: When you subtract, you add the opposite. M ULTIPLYING AND DIVIDING INTEGERS 1. If an even number of negatives is used, multiply or divide as usual, and the answer is positive. a. –3 × –4 = 12 b. (–12 Ϭ –6) × 3 = 6 2. If an odd number of negatives is used, multiply or divide as usual, and the answer is negative. a. –15 Ϭ 5 = –3 b. (–2 × –4) × –5 = –40 This is helpful to remember when working with powers of a negative number. If the power is even, the answer is positive. If the power is odd, the answer is negative. Fractions A fraction is a ratio of two numbers, where the top number is the numerator and the bottom number is the denominator. R EDUCING FRACTIONS To reduce fractions to their lowest terms, or simplest form, find the GCF of both numerator and denominator. Divide each part of the fraction by this common factor and the result is a reduced fraction. When a fraction is in reduced form, the two remaining numbers in the fraction are relatively prime. a. b. When performing operations with fractions, the important thing to remember is when you need a com- mon denominator and when one is not necessary. ADDING AND SUBTRACTING FRACTIONS It is very important to remember to find the least common denominator (LCD) when adding or subtract- ing fractions. After this is done, you will be only adding or subtracting the numerators and keeping the com- mon denominator as the bottom number in your answer. a. b. 6 15 ϩ 10 15 ϭ 16 15 3 ϫ x y ϫ x ϩ 4 xy ϭ 3x ϩ 4 xy 2 ϫ 3 5 ϫ 3 ϩ 2 ϫ 5 3 ϫ 5 LCD ϭ xyLCD ϭ 15 3 y ϩ 4 xy 2 5 ϩ 2 3 32x 4xy ϭ 8 y 6 9 ϭ 2 3 – ARITHMETIC– 329 MULTIPLYING FRACTIONS It is not necessary to get a common denominator when multiplying fractions. To perform this operation, you can simply multiply across the numerators and then the denominators. If possible, you can also cross-can- cel common factors if they are present, as in example b. a. b. DIVIDING F RACTIONS A common denominator is also not needed when dividing fractions, and the procedure is similar to multi- plying. Since dividing by a fraction is the same as multiplying by its reciprocal, leave the first fraction alone, change the division to multiplication, and change the number being divided by to its reciprocal. a. b. Decimals The following chart reviews the place value names used with decimals. Here are the decimal place names for the number 6384.2957. It is also helpful to know of the fractional equivalents to some commonly used decimals and percents, especially because you will not be able to use a calculator. 0.4 ϭ 40% ϭ 2 5 0.3 ϭ 33 1 3 % ϭ 1 3 0.1 ϭ 10% ϭ 1 10 T H O U S A N D S H U N D R E D S T E N S O N E S D E C I M A L P O I N T T E N T H S H U N D R E D T H S T H O U S A N D T H S T E N T H O U S A N D T H S 638 42 95 7 . 3x y Ϭ 12x 5xy ϭ 3 1 x 1 y 1 ϫ 5xy 1 12 4 x 1 ϭ 5x 4 4 5 Ϭ 4 3 ϭ 4 1 5 ϫ 3 4 1 ϭ 3 5 12 25 ϫ 5 3 ϭ 12 4 25 5 ϫ 5 1 3 ϭ 4 5 1 3 ϫ 2 3 ϭ 2 9 – ARITHMETIC– 330 ADDING AND SUBTRACTING DECIMALS The important thing to remember about adding and subtracting decimals is that the decimal places must be lined up. a. 3.6 b. 5.984 +5.61 –2.34 9.21 3.644 MULTIPLYING DECIMALS Multiply as usual, and count the total number of decimal places in the original numbers. That total will be the amount of decimal places to count over from the right in the final answer. 34.5 × 5.4 1,380 + 17,250 18,630 Since the original numbers have two decimal places, the final answer is 186.30 or 186.3 by counting over two places from the right in the answer. DIVIDING DECIMALS Start by moving any decimal in the number being divided by to change the number into a whole number. Then move the decimal in the number being divided into the same number of places. Divide as usual and keep track of the decimal place. .3 5.1ͤෆෆෆෆෆ1.53 ⇒ 51ͤෆෆෆෆෆ15.3 Ϫ15.3 0 1.53 Ϭ 5.1 0.75 ϭ 75% ϭ 3 4 0.6 ϭ 66 2 3 % ϭ 2 3 0.5 ϭ 50% ϭ 1 2 – ARITHMETIC– 331 Ratios A ratio is a comparison of two or more numbers with the same unit label. A ratio can be written in three ways: a: b a to b or A rate is similar to a ratio except that the unit labels are different. For example, the expression 50 miles per hour is a rate — 50 miles/1 hour. Proportion Two ratios set equal to each other is called a proportion. To solve a proportion, cross-multiply. Cross multiply to get: Percent A ratio that compares a number to 100 is called a percent. To change a decimal to a percent, move the decimal two places to the right. .25 = 25% .105 = 10.5% .3 = 30% To change a percent to a decimal, move the decimal two places to the left. 36% = .36 125% = 1.25 8% = .08 Some word problems that use percents are commission and rate-of-change problems, which include sales and interest problems. The general proportion that can be set up to solve this type of word problem is , although more specific proportions will also be shown. Part Whole ϭ % 100 x ϭ 12 1 2 4x 4 ϭ 50 4 4x ϭ 50 4 5 ϭ 10 x a b – ARITHMETIC– 332 COMMISSION John earns 4.5% commission on all of his sales. What is his commission if his sales total $235.12? To find the part of the sales John earns, set up a proportion: Cross multiply. RATE OF CHANGE If a pair of shoes is marked down from $24 to $18, what is the percent of decrease? To solve the percent, set up the following proportion: Cross multiply. Note that the number 6 in the proportion setup represents the discount, not the sale price. SIMPLE INTEREST Pat deposited $650 into her bank account. If the interest rate is 3% annually, how much money will she have in the bank after 10 years? x ϭ 25% decrease in price 24x 24 ϭ 600 24 24x ϭ 600 6 24 ϭ x 100 24 Ϫ 18 24 ϭ x 100 part whole ϭ change original cost ϭ % 100 x ϭ 10.5804 Ϸ $10.58 100x 100 ϭ 1058.04 100 100x ϭ 1058.04 x 235.12 ϭ 4.5 100 part whole ϭ change original cost ϭ % 100 – ARITHMETIC– 333 Interest = Principal (amount invested) × Interest rate (as a decimal) × Time (years) or I = PRT. Substitute the values from the problem into the formula I = (650)(.03)(10). Multiply I = 195 Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her account. Exponents The exponent of a number tells how many times to use that number as a factor. For example, in the expres- sion 4 3 , 4 is the base number and 3 is the exponent,or power. Four should be used as a factor three times: 4 3 = 4 × 4 × 4 = 64. Any number raised to a negative exponent is the reciprocal of that number raised to the positive expo- nent: Any number to a fractional exponent is the root of the number: Any nonzero number with zero as the exponent is equal to one: 140° = 1. Square Roots and Perfect Squares Any number that is the product of two of the same factors is a perfect square. 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, Knowing the first 20 perfect squares by heart may be helpful. You probably already know at least the first ten. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 256 1 4 ϭ 4 2 256 ϭ 4 27 1 3 ϭ 3 2 27 ϭ 3 25 1 2 ϭ 2 25 ϭ 5 3 Ϫ2 ϭ 1 1 3 2 2 ϭ 1 9 – ARITHMETIC– 334 [...]... choices equals the total number of possibilities For example, if you have two choices for an appetizer, four choices for a main course, and five choices for dessert, you can choose from a total of 2 × 4 × 5 = 40 possible meals The symbol n! represents n factorial and is often used in probability and counting problems n! = (n) × (n – 1) × (n – 2) × × 1 For example, 5! = 5 × 4 × 3 × 2 × 1 = 120 Permutations... you must reverse the direction of the inequality symbol For example, solve the inequality –3x + 6 Յ 18: 1 First subtract 6 from both sides: –3x ϩ 6 Ϫ 6 Յ 18 Ϫ 6 2 Then divide both sides by –3: –3x –3 3 The inequality symbol now changes: x Ն Ϫ4 12 Յ –3 Solving Compound Inequalities A compound inequality is a combination of two inequalities For example, take the compound inequality –3 Ͻ x + 1 Ͻ 4 To solve... parentheses Multiply the inner terms in the parentheses Multiply the last terms in the parentheses Examples 1 (x – 1)(x + 2) = x 2 + 2x – 1x – 2 = x 2 + x – 2 F O I L 2 (a – b)2 = (a – b)(a – b) = a2 – ab – ab – b2 F O I L Factoring Polynomials Factoring polynomials is the reverse of multiplying them together Examples Factor the following: 1 2 3 4 2x 3 + 2 = 2 (x 3 + 1) Take out the common factor of 2 x... undefined The fraction 5 x Ϫ 1 is undefined when the denominator x – 1 = 0; therefore, x = 1 347 – ALGEBRA – You may be asked to perform various operations on rational expressions See the following examples Examples 2 1 Simplify x b x3b2 2 Simplify x Ϫ 9 3x Ϫ 9 3 Multiply 4x x 2 – 16 2 ϫ 2 4 Divide 5 Add 1 xy a ϩ 2a a2 ϩ 3a ϩ 2 x ϩ 4 2x 2 2 a Ϭ 2a – 3a ϩ 2 3 ϩy 6 Subtract x ϩ 6 x – x – 2 3x 7 Solve... angles are formed 1 4 2 3 358 – GEOMETRY – Vertical angles are the nonadjacent angles formed, or the opposite angles These angles have the same measure For example, m ∠ 1 = m ∠ 3 and m ∠ 2 = m ∠ 4 The sum of any two adjacent angles is 180 degrees For example, m ∠ 1 ϩ m ∠ 2 = 180 The sum of all four of the angles formed is 360 degrees If the two lines intersect and form four right angles, then the lines... important concept to recognize is like terms In algebra, like terms are expressions that have exactly the same variable(s) to the same power and can be combined easily by adding or subtracting the coefficients Examples 3x + 5x 4x 2y + –10x 2y 2xy 2 + 9x 2y These terms are like terms, and the sum is 8x These terms are also like terms, and the sum is –6x2y These terms are not like terms because the variables,... raising a power to another power, multiply the exponents: 1 2 2 ϭ x 2ϫ3 ϭ x 6 ■ Remember that a fractional exponent means the root: ͙ෆ = x ᎏ2ᎏ and ͙ෆ = x ᎏ3ᎏ x x x5 x 2= x5 – 2= x3 1 3 1 The following is an example of a question involving exponents: Solve for x: 2x + 2 = 83 a 1 b 3 c 5 d 7 e 9 The correct answer is d To solve this type of equation, each side must have the same base Since 8 can be expressed... property Combine like terms on the same side of the equal sign Move the variables to one side of the equation Solve the one- or two-step equation that remains, remembering the two previous properties Examples Solve for x in each of the following equations: a 3x – 5 = 10 Add 5 to both sides of the equation: 3x – 5 + 5 = 10 + 5 Divide both sides by 3: 3x 3 ϭ 15 3 x=5 b 3 (x – 1) + x = 1 Use distributive... may be in the form of a formula You may be asked to solve a literal equation for one variable in terms of the other variables Use the same steps that you used to solve linear equations 342 – ALGEBRA – Example Solve for x in terms of a and b: Subtract b from both sides of the equation: 2x + b = a 2x + b – b = a – b Divide both sides of the equation by 2: 2x 2 ϭaϪb 2 xϭaϪb 2 Solving Inequalities Solving... 2 To add or subtract square roots, you must have like terms In other words, the radicand must be the same If you have like terms, simply add or subtract the coefficients and keep the radicand the same Examples 1 3͙ෆ + 2͙ෆ = 5͙ෆ 2 2 2 2 4͙ෆ – ͙ෆ = 3͙ෆ 2 2 2 3 6͙ෆ + 3͙ෆ cannot be combined because they are not like terms 2 5 Here are some rules to remember when multiplying and dividing radicals: Multiplying: . = 9, 4 × 4 = 16, 5 × 5 = 25, Knowing the first 20 perfect squares by heart may be helpful. You probably already know at least the first ten. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 1 69, . the entire number is divisible by 6. ■ Divisibility by 9. If the sum of the digits adds to a multiple of 9, the entire number is divisible by 9. ■ Divisibility by 10. If the units digit is 0, the. problem into the formula I = (650)(.03)(10). Multiply I = 195 Since she will make $ 195 in interest over 10 years, she will have a total of $ 195 + $650 = $845 in her account. Exponents The exponent