Number and operations review1
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This document provides an overview of several topics in number and operations that may be covered on the SAT, including properties of integers, operations with integers, rational numbers and fractions, number lines, squares and square roots, scientific notation, elementary number theory, ratios, proportions, percents, sequences, and arithmetic and geometric sequences. Sample problems are included to illustrate key concepts.
Number and operations review1
- 1. Number and Operations Review The SAT doesn’t include •tedious or long computations •matrix operations
- 2. Properties of Integers Integers • Whole numbers and their negatives (including zero) ..., -3, -2, -1, 0, 1, 2, 3, 4,... • Extend infinitely in both negative and positive directions • Do not include fractions or decimals
- 3. Integers • Negative integers -4, -3, -2, -1 • Positive integers 1, 2, 3, 4 • The integer zero (0) is neither positive nor negative.
- 4. • Odd Numbers ..., -5, -3, -1, 1, 3, 5,... • Even Numbers ..., -4, -2, 0, 2, 4,... • The integer zero (0) is an even number.
- 5. Consecutive Integers • Integers that follow in sequence, where the difference between two successive integers is 1, are consecutive integers. • Here are three examples of some consecutive integers: -1, 0, 1, 2, 3 1001, 1002, 1003, 1004 -14, -13, -12, -11 • The following is an expression representing consecutive integers: n, n + 1, n + 2, n + 3,..., where n is any integer.
- 6. Addition of Integers • even + even = even • odd + odd = even • odd + even = odd • Adding zero (0) to any number doesn’t change the value: 8 + 0 = 8 -1 + 0 = -1
- 7. Multiplication of Integers • even x even = even • odd x odd = odd • even x odd = even • Multiplying any number by one (1) doesn’t change the value: 3 x 1 = 3 10 x -1 = -10
- 8. Arithmetic Word Problems • The SAT includes questions which test your ability to correctly apply arithmetic operations in a problem situation. • To solve the problem, 1. you need to identify which quantities are being given 2. what is being asked for 3. which arithmetic operations must be applied to the given quantities to get the answer
- 9. Example 1 Ms. Hall is preparing boxes of books for children in the hospital. If she puts 3 books in each box, she will make 25 boxes of books with no books left over. If instead she puts 5 books in each box, how many boxes of books will she have? • Multiplying 3 times 25, there are 75 books. • If she puts 5 boxes in each box, she can make 15 boxes of books. (75 divided by 5) • The correct answer is 15 boxes of books.
- 10. Example 2 Jack bought 6 doughnuts at the bakery with a five-dollar bill and got back three quarters in change. He realized he had gotten too much change, so he gave one of the quarters back to the cashier. What was the price of each doughnut? • Jack originally paid $5.00 - 0.75 = $4.25. • When he gave back the quarter, he actually paid $4.25 + 0.25 = $4.50 for the 6 doughnuts. • Each doughnut cost $4.50/6 = $ . 75.
- 11. Number Lines Number line is used to graphically represent the relationships between numbers like integers, fractions, or decimals. • Numbers increase as you move to the right and tick marks are always equally spaced. • Number lines are drawn to scale. You will be expected to make reasonable approximations of positions between labeled points on the line.
- 12. Number line questions may ask: • Where a number should be placed in relation to other numbers; • The difference or product of two numbers; • The lengths and the ratios of the lengths of line segments represented on the number line.
- 13. Example 1 • The ratio of AC to AG is equal to the ratio of CD to which of the following? (A)AD (B)BD (C)CG (D)DF (E)EG • From the number line, AC = 2, AG = 6, CD = 1 • AC is to AG as CD is to what? AC CD • AG x 2 = = 1 2 x =6 x =3 6 x • Of the answer choices, only AD = 3 so the answer is A.
- 14. Example 2 Find the coordinate of P. • Subtract the points we know and divide by the number of intervals to find the distance of each interval. • - = .430 .425 .001 5 • Add .001 x 3 (number of intervals from lowest point to point P) • .425 + .003 = .428 so Point P is at 0.428.
- 15. Squares and Square Roots • Squares of Integers
- 16. Your knowledge of common squares and square roots may speed up your solution to some math problems. The most common types of problems for which this knowledge will help you will be those involving: • Factoring and/or simplifying expressions • Problems involving the Pythagorean theorem (a2 + b2 = c2 ) • Areas of circles or squares
- 17. Squares of Fractions • Remember that if a positive fraction with a value less than 1 is squared, the result is always smaller than the original fraction: If 0 < n < 1 Then n 2 < n • What are the values of the following fractions? 1 64 4 9 • The answers are and respectively. Each of these is less than the original fraction. • For example, . 4 < 2 9 3
- 18. Fractions and Rational Numbers Basic operations with fractions: • Adding, subtracting, multiplying, and dividing fractions • Reducing to lowest terms • Finding the least common denominator • Expressing a value as a mixed number and as an improper fraction 2 1 3 æ ö çè ÷ø 7 3 æ ö çè ÷ø • Working with complex fractions—ones that have fractions in their numerators or denominators
- 19. Rational Number • A number that can be represented by a fraction whose numerator and denominator are both integers (and the denominator must be nonzero). • The following are rational numbers: ⎛ ⎞ ⎜⎝ ⎟⎠ 3 7 or 3 1 − 5 3 (or 3) 4 2 2 7 1 • Every integer is a rational number.
- 20. Decimal Fraction Equivalents • Recognize common fractions as decimals and vice versa. • To change any fraction to a decimal, divide the numerator by the denominator.
- 21. You can figure out the decimal equivalent of any fraction with a calculator, however you’ll save time if you know the following:
- 22. Reciprocals • The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 5 1 is 5 . Note that 5 ´ 1 = . 1 5 • The product of a number and its reciprocal is always 1. • You can find the reciprocal of any nonzero fraction by switching its numerator and denominator.
- 23. Examples of reciprocals: 3 11 1 2 4 4 4 1 -2® - 1 = ® 2 The reciprocal of 1 is 1. The number zero has no reciprocal.
- 24. Scientific Notation 2,300,000,000 = 2.3´109 .0000000000000076 = 7.6´10-15
- 25. Elementary Number Theory Factors The factors of a number are positive integers that can evenly be divided into the number —that is, there is no remainder. • Example: 24 - The numbers 24, 12, 8, 6, 4, 3, 2, and 1 are all factors of the number 24.
- 26. Divisible By Divisible without any remainder or with a remainder of zero. • 15 is divisible by 5 because 15 divided by 5 is 3 with a remainder of 0. • 15 is not divisible by 7, because 15 divided by 7 is 2 with a remainder of 1.
- 27. Common factors Factors that two (or more) numbers have in common. • 3 is a common factor of 12 and 18. • The largest common factor of two (or more numbers) is called their greatest common factor (GCF). • 6 is the GCF of 12 and 18.
- 28. Multiples Those numbers that can be divided by a given number without a remainder. • You can find the multiples of a number by multiplying it by 1, 2, 3, 4, and so on. • 8, 16, 24, 32, 40, and 48 are some of the multiples of 8.
- 29. • The multiples of any number will always be multiples of all the factors of that number. • The numbers 30, 45, 60, and 75 are all multiples of the number 15. • Two factors of 15 are the numbers 3 and 5. • That means that 30, 45, 60, and 75 are all multiples of 3 and 5.
- 30. Common multiples Any number that is a multiple of all the given numbers. • 48 and 96 are both common multiples of 8 and 12. • The smallest multiple of two (or more) numbers is called their least common multiple (LCM). • 24 is the LCM of 8 and 12.
- 31. Example 1 What is the least positive integer divisible by the numbers 2, 3, 4, and 5? • One possible answer is to multiply those numbers together. You could multiply and the result would be divisible by all those factors. 2g3g4g5 • However, to find the least positive number divisible by all four (in other words, the LCM of the four numbers), you have to eliminate any extra factors.
- 32. Example 1 (continued) What is the least positive integer divisible by the numbers 2, 3, 4, and 5? • Any number divisible by 4 will also be divisible by 2, so you can eliminate 2 from your initial multiplication. So if you multiply , you will get a smaller number and the number will still be divisible by 2. 3 g 4 g 5 • Answer: is the Least Common Multiple (LCM). 3 g 4 g 5 = 60
- 33. Example 2 Which of the following could be the remainders when four consecutive positive integers are each divided by 3? (A) 1, 2, 3, 1 (B) 1, 2, 3, 4 (C) 0, 1, 2, 3 (D) 0, 1, 2, 0 (E) 0, 2, 3, 0
- 34. • When you divide any positive integer by 3, the remainder must be less than or equal to 2. • All the choices except (D) include remainders greater than 2, so (D) is the only possible correct choice. • If the first and fourth of the consecutive integers are multiples of 3, the remainders will be 0, 1, 2, and 0. example - 3,4,5,6
- 35. Example 3 Does the equation 3x + 6y = 47 have a solution in which x and y are both positive integers? • Notice that 3x + 6y = 3(x + 2y). • Therefore, the sum 3x + 6y = 3(x + 2y) is a multiple of 3. But 47 is not a multiple of 3. • No matter what positive integers you choose for x and y, the sum will not be 47. Thus, the equation 3x + 6y = 47 does not have a solution in which x and y are both positive integers.
- 36. Prime Numbers A prime number is a positive integer greater than 1 that has exactly two whole-number factors—itself and the number 1. • The number 1 itself is not prime. • Prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19 • 2 is the only even prime number.
- 37. Prime factors Prime factors are the factors of a number that are prime numbers. • Prime factors of a number cannot be further divided into factors. • The prime factors of 24 are 2 and 3.
- 38. Ratios, Proportions, and Percents Ratio • A mathematical relationship between two quantities. • Ratio is a quotient of those quantities. • Ratios can be expressed in several ways: 2 2 : 3 2 to 3 3
- 39. Percent (%) • A ratio in which the second quantity is 100. 85 100 • 85 percent is equivalent to .
- 40. Proportion • An equation in which two ratios are set equal to each other.
- 41. Example 1 The weight of the tea in a box of 100 identical tea bags is 8 ounces. What is the weight, in ounces, of the tea in 3 tea bags? • The ratio of the tea in 3 tea bags to the tea in all of the tea bags is 3 to 100 æ 3 ö çè ÷ø . 100 • Let x equal the weight, in ounces, of the tea in 3 tea bags. • The ratio of the weight of the tea in 3 tea bags to the total weight of the tea in 100 tea bags is x ounces to 8 ounces . æ x ö çè 8 ÷ø
- 42. x = 3 8 100 100x = 24 x = .24 The answer is .24 ounces in 3 bags of tea.
- 43. Example 2 The ratio of the length of a rectangular floor to its width is 3:2. If the length of the floor is 12 meters, what is the perimeter of the floor, in meters? • Set 3:2 equal to the ratio of the actual measures of the sides of the rectangle. length x x width x 3 = , 3 = 12 , 3 = 24, = 8 ( width) 2 2
- 44. • Find the perimeter: 2(length + width) = 2(12 + 8). • The perimeter of the floor is 40 meters.
- 45. Sequences A sequence is an ordered list of numbers. • Infinite sequence goes on forever and indicated by … 2,5,8,11,14… • Finite sequence terminates. 3,6,9,…21,24
- 46. Arithmetic Sequences • Sequences for which there is a constant difference (d) between consecutive terms. • 4, 7, 10, 13,... is an arithmetic sequence in which the first term is 4 and the constant difference (d) is 3. • Notice that 7-4=3 10-7=3 13-10=3 so d =3. • To find the nth term use: 1 1 ( 1) a + n - d where a = first term
- 47. Geometic Sequences Sequences for which there is a constant ratio (r) between consecutive terms. • 4, 12, 36, 108,... is a geometric sequence in which the first term is 4 and the constant ratio (r) is 3. • Notice that 12 = 3 36 = 3 108 = 3 4 12 36 so r = 3. • To find the nth term use: a grn- 1 where a = first term 1 1
- 48. • The mathematics section does not ask you to figure out the rule for determining the numbers in a sequence without giving the information in some way. • For instance, in the preceding example, if you were given that the nth term of the sequence was 4g3n-, 1 you may be asked to determine that each term after the first was 3 times the term before it. • Or if you were given that the first term in the sequence was 4 and that each term after the first was 3 times the term before it, you might be asked to find that the nth term of the sequence was 4 g 3 n - 1 .
- 49. Number sequence questions might ask you for the following: • The sum of certain terms in a sequence • The average of certain terms in a sequence • The value of a specific term in a sequence
- 50. Sets (Union, Intersection, Elements) Set • A collection of things, and the things are called elements or members of the set. Union • Union of two sets is the set consisting of the elements that are in either set or both sets. Intersection • Intersection of two sets is the set consisting of the common elements.
- 51. Given: set A = {1, 3, 7, 10, 12} and set B = {8, 10, 12, 15} • the union of sets A and B would be {1, 3, 7, 8, 10, 12, 15} • the intersection of sets A and B would be {10, 12}
- 52. Counting Problems Counting problems involve figuring out how many ways you can select or arrange members of groups, such as letters of the alphabet, numbers, or menu selections. Fundamental counting principle • The principle by which you figure out how many possibilities there are for selecting members of different groups. • If one event can happen in n ways, and a second, independent event can happen in m ways, the total ways in which the two events can happen is n times m.
- 53. Example On a restaurant menu, there are three appetizers and four main courses, five beverages and three desserts. How many different dinners can be ordered if each dinner consists of one appetizer, one main course, one beverage and one dessert? 3g4g5g3 = 180 • There are 180 possible dinners.
- 54. Permutations • A way to arrange things in which order is important. Example A security system uses a four-digit password, but no number can be used more than once. How many possible passwords are there? 10 9 8 7 5,040 or use your calculator and enter P 10 4 g g g =
- 55. Combinations • A way to arrange things in which order is not important.
- 56. Example There are 12 students in a statistics class. Three students will be selected to attend a conference on statistical research. How many different groups of three can be selected from the class? • Order does not matter • or = = g g g 12 11 10 permutation of 12 things 3 at a time 220 3 2 permutation of 3 things 3 at a time 12C3 • Use the calculator = 220 different groupings
- 57. Logical Reasoning Logical thinking questions • You have to figure out how to draw conclusions from a set of facts.
- 58. Example • In the figure above, circular region A represents the set of all numbers of the form 2m, circular region B represents the set of all numbers of the form n 2, and circular region C represents the set of all numbers of the form 10k , where m, n, and k are positive integers. • Which of the following numbers belongs in the set represented by the shaded region? • (A)2 (B)4 (C)10 (D)25 (E)100
- 59. • The shaded region is part of all of the circles. Therefore, any numbers in the shaded region have to obey the rules for all the circles. • Circle A represents even numbers (2m). • Circle B represents perfect squares (n 2). • Circle C represents numbers with a whole-number power of 10 (10k) like 10, 100, 1000, 10000. • The only choice that represents all the circles is 100. Thus the answer is E (100).