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Basic Math Review NEWCOLORs_basic_math_rev 3/31/08 3:52 PM Page 1 Basic Math Review Key Words and Symbols Integers (continued) Fractions (continued) Rates, Ratios, Proportions, The following words and symbols are used for the and Percents MULTIPLYING AND DIVIDING WITH NEGATIVES operations listed. # Equivalent fractions are found by multiplying the numerator -a b =-ab and denominator of the fraction by the same number. In the RATES AND RATIOS Addition previous example, - # - = A rate is a comparison of two quantities with different units. Numbers Important Properties Sum, total, increase, plus a b ab # # - 2 2 4 8 1 1 3 3 For example, a car that travels 110 miles in 2 hours is mov- a = a = = and = = . addend addend = sum # # ing at a rate of 110 miles/2 hours or 55 mph. NATURAL NUMBERS PROPERTIES OF ADDITION -b b 3 3 4 12 4 4 3 12 Subtraction a A ratio is a comparison of two quantities with the same {1, 2, 3, 4, 5, …} Identity Property of Zero: a + 0 = a -a , b =- Difference, decrease, minus b MULTIPLYING AND DIVIDING FRACTIONS units. For example, a class with 23 students has a WHOLE NUMBERS Inverse Property: a + -a = 0 When multiplying and dividing fractions, a common student–teacher ratio of 23:1 or23 . 1 2 minuend subtrahend = difference Some examples: 1 denominator is not needed. To multiply, take the product {0, 1, 2, 3, 4, …} + = + Multiplication - # =- Commutative Property: a b b a 3 5 15 of the numerators and the product of the denominators: PROPORTIONS Product, of, times -7 -6 = 42 # INTEGERS Associative Property: a + b + c = a + b + c 1 21 2 a # c a c ac A proportion is a statement in which two ratios or rates are 1 2 1 2 * # - - = = = {…, Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, …} a b, a b, a b , ab 24 8 3 # equal. PROPERTIES OF MULTIPLICATION 1 21 2 1 2>1 2 b d b d bd factor factor = product ϪϪ18 36 An example of a proportion is the following statement: The Number Line # or 18 To divide fractions, invert the second fraction and then Property of Zero: a 0 = 0 2 36 30 dollars is to 5 hours as 60 dollars is to 10 hours. # Division 2 multiply the numerators and denominators: Identity Property of One: a 1 = a, when a Z 0. Quotient, per, divided by a c a d ad This is written –5–5 – 4–4 –3–3 –2–2 –1–1 0 1 2 3 4 5 # 1 , = = # = Z a b d b c bc $30 = $60 Negative integers Positive integers Inverse Property: a 1, when a 0. a Ϭ b ϭ ϭ a b ϭ bͤa . a # # b > Fractions 5 hr 10 hr Commutative Property: a b = b a Some examples: Zero A typical proportion problem will have one unknown # # # # 3 2 6 Associative Property: a b c = a b c dividend divisor = quotient LEAST COMMON MULTIPLE # = quantity, such as 1 2 1 2 5 7 35 The LCM of a set of numbers is the smallest number that is a 1 mile x miles RATIONAL NUMBERS PROPERTIES OF DIVISION = . multiple of all the given numbers. 5 1 5 2 10 5 20 min 60 min All numbers that can be written in the form a b, where a 0 , = # = = Z > Property of Zero: = 0, when a Z 0. Order of Operations For example, the LCM of 5 and 6 is 30, since 5 and 6 have no 12 2 12 1 12 6 and b are integers andb 0 . a We can solve this equation by cross multiplying as shown: 1st: Parentheses factors in common. IRRATIONAL NUMBERS a = # Property of One: = 1, when a Z 0. Simplify any expressions inside parentheses. REDUCING FRACTIONS 20x 60 1 Real numbers that cannot be written as the quotient of two a GREATEST COMMON FACTOR 2nd: Exponents To reduce a fraction, divide both the numerator and denom- 60 integers but can be represented on the number line. a # x = = 3. Identity Property of One: = a 1 The GCF of a set of numbers is the largest number that can inator by common factors. In the last example, 20 1 Work out any exponents. be evenly divided into each of the given numbers. 10 10 , 2 5 REAL NUMBERS 3rd: Multiplication and Division = = . So, it takes 60 minutes to walk 3 miles. For example, the GCF of 24 and 27 is 3, since both 24 and 12 12 , 2 6 Include all numbers that can be represented on the number Solve all multiplication and division, working from 27 are divisible by 3, but they are not both divisible by any line, that is, all rational and irrational numbers. PERCENTS Absolute Value left to right. numbers larger than 3. 4th: Addition and Subtraction MIXED NUMBERS A percent is the number of parts out of 100. To write a per- cent as a fraction, divide by 100 and drop the percent sign. Real Numbers The absolute value of a number is always ≥ 0. These are done last, from left to right. FRACTIONS A mixed number has two parts: a whole number part and a ƒ ƒ 3 _4 2 If a 7 0, a = a. fractional part. An example of a mixed number is 5 . This For example, Rational Numbers 23, 22.4, 21 5 , 0, 0.6, 1, etc. For example, Fractions are another way to express division. The top num- 8 Irrational 6 ƒ - ƒ = 57 Numbers If a 0, a a. - # + - , 2 ber of a fraction is called the numerator, and the bottom really represents = p 23, 22, 21, 0, 1, 2, 3, p 15 2 3 30 3 3 3 57% . 25VN3, Integers ƒ - ƒ = ƒ ƒ = # 1 2 number is called the denominator. 5 + , 100 For example, 5 5 and 5 5. In each case, the = 15 - 2 3 + 27 , 9 VN2, p, etc. answer is positive. 8 Whole Numbers 0, 1, 2, 3, p = 15 - 6 + 3 ADDING AND SUBTRACTING FRACTIONS To write a fraction as a percent, first check to see if the which can be written as denominator is 100. If it is not, write the fraction as an = Natural Numbers 1, 2, 3, p 12. Fractions must have the same denominator before they can 40 3 43 equivalent fraction with 100 in the denominator. Then the + = . be added or subtracted. 8 8 8 numerator becomes the percent. For example, a b a + b 4 80 + = , with d Z 0. Similarly, an improper fraction can be written as a mixed = = 80%. Integers d d d 5 100 PRIME NUMBERS number. For example, a b a - b - = Z 20 2 To find a percent of a quantity, multiply the percent by the A prime number is a number greater than 1 that has only ADDING AND SUBTRACTING WITH NEGATIVES , with d 0. can be written as 6 , ISBN-13: 978-0-321-39476-7 d d d 3 3 quantity. itself and 1 as factors. - - = - + - ISBN-10: 0-321-39476-3 a b a b If the fractions have different denominators, rewrite them as Some examples: 1 2 1 2 since 20 divided by 3 equals 6 with a remainder of 2. For example, 30% of 5 is 90000 -a + b = b - a equivalent fractions with a common denominator. Then add 2, 3, and 7 are prime numbers. 30 # 150 3 a - -b = a + b or subtract the numerators, keeping the denominators the 5 = = . COMPOSITE NUMBERS 1 2 same. For example, 100 100 2 Some examples: A composite number is a number that is not prime. For 2 1 8 3 11 -3 - 17 = -3 + -17 =-20 + = + = example, 8 is a composite number since 1 2 1 2 . # # -19 + 4 = 4 - 19 =-15 3 4 12 12 12 8 = 2 2 2 = 23. 9 780321 394767 more➤ more➤ 1 2 3 NEWCOLORs_basic_math_rev 3/31/08 3:52 PM Page 1 Basic Math Review Key Words and Symbols Integers (continued) Fractions (continued) Rates, Ratios, Proportions, The following words and symbols are used for the and Percents MULTIPLYING AND DIVIDING WITH NEGATIVES operations listed. # Equivalent fractions are found by multiplying the numerator -a b =-ab and denominator of the fraction by the same number. In the RATES AND RATIOS Addition previous example, - # - = A rate is a comparison of two quantities with different units. Numbers Important Properties Sum, total, increase, plus a b ab # # - 2 2 4 8 1 1 3 3 For example, a car that travels 110 miles in 2 hours is mov- a = a = = and = = . addend addend = sum # # ing at a rate of 110 miles/2 hours or 55 mph. NATURAL NUMBERS PROPERTIES OF ADDITION -b b 3 3 4 12 4 4 3 12 Subtraction a A ratio is a comparison of two quantities with the same {1, 2, 3, 4, 5, …} Identity Property of Zero: a + 0 = a -a , b =- Difference, decrease, minus b MULTIPLYING AND DIVIDING FRACTIONS units. For example, a class with 23 students has a WHOLE NUMBERS Inverse Property: a + -a = 0 When multiplying and dividing fractions, a common student–teacher ratio of 23:1 or23 . 1 2 minuend subtrahend = difference Some examples: 1 denominator is not needed. To multiply, take the product {0, 1, 2, 3, 4, …} + = + Multiplication - # =- Commutative Property: a b b a 3 5 15 of the numerators and the product of the denominators: PROPORTIONS Product, of, times -7 -6 = 42 # INTEGERS Associative Property: a + b + c = a + b + c 1 21 2 a # c a c ac A proportion is a statement in which two ratios or rates are 1 2 1 2 * # - - = = = {…, Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, …} a b, a b, a b , ab 24 8 3 # equal.
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