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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 . 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. 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– 3 29 MULTIPLYING FRACTIONS It is not necessary to get a common denominator when multiplying. 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