Level 1 Algebra Geometry Placement Test Study Answers

1 ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math teacher or someone else who understands the topic. TOPIC 1: INTEGERS A. What is an integer? Any natural number (1, 2, 3, 4, 5,…), its opposite (-1,-2, -3, -4, -5,…), or zero (0). (Integers are useful for problems involving “below normal,” debts, “below sea level,” etc.) Problems 1-10: Identify each number as an integer (I) or not an integer (NI): 1. 367 2. !4.4 6. 0 7. " 23 3. 2 12 8. 0.027 4. –1010 9. 5. 100 ! 3 ! Problems 11-14: Write the opposite of each integer: ! 11. 42 12. !3 ! 13. 0 3 14. !4 Problems 15-19: Choose the greater: 15. 5, –10 16. 5,–5 17. 5, 0 18. –5, 0 19. –5,–10 20. What is the result of adding an integer and its opposite? 21. What number is its own opposite? B. Absolute Value: Absolute value is used for finding distance, explaining addition of integers, etc. The absolute value of a positive number or zero is itself. The absolute value of a negative number is its opposite. Problems 22-26: Choose the integer with the greater absolute value: 22. 4 or –3 23. –4 or 3 24. 3 or –3 One positive, one negative: treat each as positive, subtract, make the answer sign of the one with the greater absolute value: example: Add –4 and 3: 4 ! 3 = 1 ; the answer is –1 because –4 has the greater absolute value. example: Add 4 and –3: 4 ! 3 = 1 ; the answer is 1 because 4 has the greater absolute value. Problems 27-33: Add the two integers: 1 2 10. 2 example: Add –4 and –3: Treat as positive and add: 4 + 3 = 7 . The answer is –7 because it must be negative. 25. 3 or 0 26. – 3 or 0 C. Adding, subtracting, multiplying and dividing integers: To add two integers: Both positive: add as natural numbers: example: Add 4 and 3: 4 + 3 = 7 Both negative: add as though positive; make the result negative: 27. 4 and –3 (This means (4 ) + (!3) ) 28. 4 and 3 29. –4 and –3 30. 4 and 0 31. –4 and 3 32. 16 and –7 33. –3 and 0 To subtract two integers: add the opposite of the one to be subtracted: example: 3 subtract –4, or (3) ! (!4) : The opposite of –4 is 4, so we add 4 (rather than subtract –4). We change the problem from (3) ! (!4) to (3) + (4) , which we know how to do: (3) + (4) = 3 + 4 = 7 example: (!4 ) ! (3) : Add the opposite of 3, namely –3: (!4 ) ! (3) = (!4) + ( !3) = !7 example: (4 ) ! (3) = ( 4) + (!3) = 1 example: !5 ! 8 = (!5) ! (8) = (!5 ) + (!8 ) = !13 Problems 34-43: Calculate: 34. (12) ! (3) = 35. !12 ! 3 = (Hint: this means "12 + ("3) ) 36. 37. 38. 39. !12 ! ( !3) = 3 ! 12 = !3 ! 12 = (!7) ! (!7) = 40. 41. 42. ! 43. 0 !3= 0+4= !12 + 3 = (!3) + (!12) = To multiply two integers: 1stinteger ! 2ndinteger = + – + – + + – – Answer + – – + 2 Both positive: multiply as two natural numbers. example: (3) ! (4) = 3 ! 4 = 12 Both negative: multiply as if positive; and make the answer positive. and remember, two negatives make a positive. When multiplying two negative numbers, you always get a positive answer. example: (!3 )( !4) so 3 ! 4 = 12 ; make it positive, and the answer is 12. One positive, one negative: When multiplying a negative number and a positive number, the answer is always negative. example: (3)( !4 ) so 3 ! 4 = 12 ; make the answer negative; answer –12. Problems 44-55: Multiply: 44. 3 ! ("4) = ! 50. ("4) • 0 = (3) • ("4) = 51. 0 = 46. 47. 48. 49. (3)( !4 ) = 3( !4 ) = ! (!3)( !4) = !3( !4 ) = 52. (!3 ) = 2 53. (4 ) = 54. ("3) • 4 = 55. 3• 4 = 2 ! example: 6 • = 1, so the reciprocal of 6 is (And the reciprocal of ( example: ("4 ) " 1 4 ! is " 1 . 4 1 6 1 . 6 is 6.) ! ! 57. 1 58. 10 59. –1 60. What number is its own reciprocal? (Can you find “more than one”?) 61. Using the reciprocal definition, explain why there is no reciprocal of zero. To divide two integers: multiply by the reciprocal of the one to be divided by: example: 20 divided by !5 = 20 ÷ (!5) . The reciprocal of –5 is " 15 so we multiply by " 15 : 20 ÷ (!5) = 20 " (! 15 ) = example: ! "3 "6 = "3 ÷ ("6) = "3• (" 16 ) = 63 = (Note negative times negative is positive.) ! ! ! 62. (!14 ) ÷ ( !2) = 65. 63. 2 ÷ 3 = 66. 64. 3 ÷ 2 = 67. ! "15 = 3 "5 = (careful)* 0 0 = 7 * Problem 61 says!10 has no value (you cannot divide by zero). ! 68. From the rule! for division, why is it impossible to divide by zero? ! To “sum” it all up: Positive+positive = larger positive Negative+negative = more negative Positive+negative = in between both 69. Given the statement “Two negatives make a positive.” Provide an example of a situation where the statement would be true and another when it would be false. 70. Write “18 divided by 30” in three ways: , and using a fraction bar ––. using ÷, ) 71. 4 ! 10 + 3 ! 2 = ! 72. 4 ! (10 + 3 ! 2) = 73. 4 + 3 ! (10 ! 2) = 74. 6(8 ! 3) = 75. (!6 )(8) + 9 = 76. 77. 78. 79. 80. !2 [( !6 )(8) + 9] = 5 + (3 ! 7) = 5 ! (3 ! 7) = 5!3+7 = !1 + 2 ! 3 + 4 = 81. What is the meaning of “sum”, “product”, quotient”, and “difference”? D. Factoring: If a number is the product of two (or more) integers, then the integers are factors of the number. example: 40 = 4 ! 10, 2 ! 20, 1 ! 40 , and 8 ! 5 . So 1, 2, 4, 5, 8, 10, 20, 40 are all factors of 40. (So are all their negatives.) Problems 82-86: Find all positive factors of: " (# 15 ) = # 20•1 = # 20 = #4 1•5 5 "5 ! 1 = " 41 example: 20 = "5 ÷ 20 = "5 • 20 20 1 ! Problems 62-67: Calculate: Problems 71- 80: Calculate: ) = 1, so the reciprocal of –4 ! Problems 56-59: Find the reciprocal: !56. –5 = 0 ÷ 3 = 0 • 13 = 0 To subtract add the opposite. To divide, multiply by the reciprocal. Reciprocals are used for dividing. Every integer except zero has a reciprocal. The reciprocal is ! the number that multiplies the integer to give 1. 1 6 0 3 Positive ! positive = positive Negative ! negative = positive Positive ! negative = negative 2 45. example: 82. 10 1 2 83. 7 84. 24 85. 9 86. 1 If a positive integer has exactly two positive factors, it is a prime number. Prime numbers are 3 used to find the greatest common factor (GCF) and least common multiple (LCM), which are used to reduce fractions and find common denominators, which in turn are often needed for adding and subtracting fractions. example: The only positive factors of 7 are 1 and 7, so 7 is a prime number. example: 6 is not prime, as it has 4 positive factors: 1, 2, 3, 6. 87. From the prime number definition, why is 1 not a prime? 88. Write the 25 prime numbers from 1 to 100. Every positive integer has one way it can be factored into primes, called its prime factorization. example: Find the prime factorization (PF) of 30: 30 = 3! 10 = 3 ! 2 ! 5, so the PF of 30 is 2 • 3• 5 . example: 72 = 2 • 2 • 2 • 3• 3 = 2 3 • 32 , the PF. (The PF can be found by making a “factor tree.”) Problems 89-91: Find the PF: ! ! 89. 36 90. 10 91. 7 Greatest common factor (GCF) and least common multiple (LCM). If you need to review GCF or LCM, see the worksheet in this series: “Topic 2: Fractions”. Problems 92-95: Find the GCF and the LCM of: 92. 4 and 6 Answers: 1. I 2. NI 3. NI 4. I 5. I 6. I 7. NI 8. NI 9. NI 10. I 11. –42 12. 3 13. 0 14. 64 15. 5 16. 5 17. 5 18. 0 19. –5 20. zero 21. zero 22. 4 93. 4 and 7 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. –4 both same 3 –3 1 7 –7 4 –1 9 –3 9 –15 –9 –9 –15 0 –3 4 –9 –15 –12 94. 4 and 8 95. 3 and 5 E. Word problems: 96. The temperature goes from !14° to 28° C. How many degrees Celsius does it change? 97. 28 ! ( !14) = 98. Derek owes $43, has $95, so “is worth”…? 99. If you hike in Death Valley from 282 feet below sea level to 1000 feet above sea level, how many feet of elevation have you gained? 100. 1000 ! (!282) = 101. A hike from 243 feet below sea level (FBSL) to 85 FBSL means a gain in elevation of how many feet? 102. !85 ! (!243 ) = 103. What number added to –14 gives –24? 104. What does “an integral number” mean? 105. Jim wrote a check for $318. His balance is then $2126. What was the balance before he wrote the check? 106. What number multiplied by 6 gives –18? 107. If you hike downhill and lose 1700 feet of elevation and end at 3985 feet above sea level (FASL), what was your starting elevation? 108. Anne was 38 miles south of her home. She drove 56 miles north. How far from home was she at that time and in what direction? 109. 5 subtracted from what number gives –12? 110. What number minus negative four gives ten? ! ! 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. ! ! –12 –12 –12 12 12 0 0 9 16 –12 12 " 15 1 1 10 –1 1; also –1 no number times 0 = 1 7 2 3 3 2 65. 66. 67. 68. 69. 70. ! –5 no value (not defined) 0 zero has no reciprocal true if ! , false if + . 18 ÷ 30 , 30 18 , 18 30 ) 71. –5 72. –7 ! 73. –1 ! 74.!30 75. –39 76. 78 77. 1 78. 9 79. 9 80. 2 ! ! 4 +, !, ÷, " 81. 82. 83. 84. 85. 86. 87. 88. 1, 2, 5, 10 1, 7 1, 2, 3, 4, 6, 8, 12, 24 1, 3, 9 1 1 has one factor 89. 90. 91. 92. 93. 94. 2 2 • 32 2•5 7 2, 12 1, 28 4, 8 95. 1, 15 96. 42 97. 42 98. $52 99. 1282 100. 1282 101. 158 102. 158 103. –10 104. an integer 105. $2444 106. –3 107. 5685 FASL 108. 18 mi. N 109. –7 110. 6 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 TOPIC 2: FRACTIONS A. Greatest Common Factor (GCF): The GCF of two integers is used to simplify (reduce, rename) a fraction to an equivalent fraction. A factor is an integer multiplier. A prime number is a positive whole number with exactly two positive factors. Problems 5-7: Find the GCF of: 5. 18 and 36 example: Reduce 27 36 ! ! example: Find the factors of 42. Factor into primes: 42 = 2 • 3• 7 1 is always a factor 2 is a prime factor 3 is a prime factor 4 is!a prime factor 7 is a prime factor 2 • 3 = 6 is a factor 2 • 7 = 14 is a factor 3• 7 = 21is a factor 2 • 3• 7 = 42 is a factor ! Thus 42 has 8 factors. ! 9•3 9•4 = 99 • 43 = 1• 43 = 3 4 Problems 8-13: Reduce: 8. 9. 10. 13 = 52 26 = 65 3+6 3+9 11. 12. = 13. ! ! C. Equivalent Fractions: ! example: ! ( ! 3 4 3 4 3 4 16 64 24 42 24 18 = = = ! is equivalent to how many eighths? = 8 ) ! = 1• 43 = 22 • 43 = 2•3 2•4 = 6 8 Problems 14-17: Complete: ! ! Problems 3-4: Find all positive factors: ! 3. 18 4. 24 To find the GCF: example: Looking at the factors of 42 and 24, we see that the common factors of both are 1, 2, 3, and 6, of which the greatest it 6; so: the GCF of 42 and 24 is 6. (Notice that “common factor” means “shared factor.”) = 27 : 36 (Note that you must be able to find a common factor, in this case 9, in both the top and bottom ! in order to reduce.) Problems 1-2: Find the prime factorization: 2. 42 7. 8 and 15 B. Simplifying fractions: example: the prime factorization of 18 is 2 • 3• 3, or 2 • 32 . 1. 24 6. 27 and 36 ! 14. ! 15. 16. ! ! ! 4 = 72 9 3 is how 5 56 = 50 100 many twentieths? 17. How many halves are in 3? (Hint: think 3 = D. Ratio: If the ratio of boys to girls in a class is 2 to 3, it means that for every 2 boys, there are 3 girls. A ! 3 1 = 2) 5 ratio is like a fraction: think of the ratio 2 to 3 as the fraction 23 . 31. Which is larger, 57 or 43 ? (Hint: find and compare LCD fractions) 32. Which is larger, 83 or 13 ? example: If the class had 12 boys, how many girls are there? Write the fraction ratio: number of boys of girls !number G. Adding!and!subtracting fractions: = 23 = 12 Complete the equivalent fraction: So there are 18 girls. 2 3 = 2•6 3•6 18. If the class had 21 girls and the ratio of boys to girls was 2 to 3, how many boys would be ! in the class? 19. If the ratio of X to Y is 4 to 3, and there are 462 Y’s, how many X’s are there? 20. If the ratio of games won to games played is 6 to 7 and 18 games were won, how many games were played? ! If denominators are the same, combine the ! ! numerators: 7 example: 10 " 101 = 7"1 = 106 = 53 10 = 12 18 Problems 33-37: Find the sum or difference (reduce if possible): !33. 34. 35. ! E. Least common multiple (LCM): The LCM of two or more integers is used to find ! the lowest common denominator of fractions in ! order to add or subtract them. 38. ! 39. Make the LCM by taking each prime factor to its greatest power: ! LCM = 2 2 • 33 = 4 • 27 = 108 ! 24. 8 and 12 25. 8, 12, and 15 ! ! To find LCD fractions for two or more given fractions: 8 example: Given 56 and 15 First find LCM of 6 and 15: 6 = 2• 3 15 = 3• 5 = 2 • 3• !5 = 30 = LCD LCM ! 5 25 8 = 16 So 6 = 30 and 15 30 ! 27. 28. 2 ! and 3 3 and 8 4 and 5 40. " 58 = 1 2 " 23 = 63 " 46 = 3"4 6 = "1 6 3 5 5 8 5 2 " 23 = 41. 2 43 + 5 78 = + 41 = 42. 3 41 " ( + 45 = ! ) + 12 = 43. 4 13 " ( 3 12 " 3) = 3 4 H. Multiplying and dividing fractions: ! multiply the tops, To multiply fractions, multiply the bottoms, ! and reduce if possible: 3 2 3•2 6 example: 4 • 5 = 4•5 = 20 = 103 2 9 29. 7 12 2 3 30. ! ! !! ! ! ! ! ! 44. !45. 46. ! ! ! ! Problems 26-32: Find equivalent fractions with the LCD: ! 26. 37. 1" Problems 44-52: Calculate: F. Lowest common denominator (LCD): ! ! + 16 = 1 = 2 2 = 3 ! are different, first find If the denominators equivalent fractions ! with common denominators (preferably the LCD): 22 + 10 = 15 = 1 157 example: 45 + 23 = 12 15 15 ! 3 27 = 3 36 = 2 2 • 32 21. 6 and 15 22. 4 and 8 23. 3 and 5 36. 3 + Problems 38-43: Calculate: First factor into primes: Problems 21-25: Find the LCM: + 27 = example: To find the LCM: example: Find the LCM of 27 and 36. ! 4 7 5 6 7 8 1 2 , , and 43 2 3 7 and 58 8 ! ! 2 3 • = 3 8 1 2 • = 2 3 4 "5 = 5 3 2 = 4 2 3 = 2 (2 12 ) 2 = 50. • 30 = 51. 8 • 43 = 49. 4 5 () () " 14 = 52. 15 ! 21 25 ! 48. !making a compound fraction Divide fractions by 47. and then multiply ! the top and bottom (of the larger fraction) by the lowest common denominator (LCD) of both. example: 43 ÷ 2 3 = 3 4 2 3 The LCD is 12, so multiply by 12: ! ! ! 3 •12 4 2 •12 3 = 9 8 6 example: 68. Three fourths is equal to how many twelfths? 69. What is 43 of a dozen? 70. Joe and Mae are decorating the gym for a dance. Joe has done 13 of the work and Mae has done 25 . What fraction of the work still must be done? ! The ratio of winning tickets to tickets sold is 2 71. to 5. If 3,500,000 are sold, how many tickets are ! ! winners? 3 72. An 11 8 -inch wide board can be cut into how 7 7•6 = 2 1 1 " 2 ( 3 " 2) • 6 42 42 42 = 2 = = = 42 1 • 6 " • 6 4 " 3 1 3 2 2 3 !Problems 53-62: Calculate: !53. 3 2 ÷ 41 = 54. 11 83 ÷ ! 55. 56. ! ! 57. ! 3 ÷2 4 3 4 = 2 3 1+ 12 1" 43 3 4 58. = = 59. ! ! ! = ! 2 3 4 2 3 4 = = 60. 4 5 ÷5 = 61. 3 8 ÷3= ! 74. If Lee works 2 43 hours, what will he be paid? 75. Together, what is the total time they work? 76. What is their total pay? I. Comparing fractions: ! ! 7 5 , , 9 7 and ! 3 4 Visual Problems: ! Problems 77-80: What fraction of the figure is shaded? LCD is 2 2 • 32 • 7 = 252 7 9 5 7 3 4 ! = = = ! So the order 7•28 = 196 9•28 252 5•36 180 = 7•36 252 ! 3•63 = 189 4•63 252 is 57 , 43 , 79 Problems 73-76: Inga and Lee each work for ! ! $4.60 per hour: 73. If Inga works 3 12 hours, what will her pay be? 2 13 " 13 62. = 3 23 + 1 12 example: Arrange small to large: many strips of width 58 inch, if each cut takes 18 inch of the width? (Must the answer be a whole number?) 77. 79. 78. 80. ! ! Fractions can also be compared by writing in ! decimal from and comparing the decimals. Problems 63-65: Arrange small to large: ! 63. 64. 15 11 , 6 8 7 5 11 , , 8 6 12 65. Problems 81-83: What letter best locates the given number? P QRST 2 , 7 , 5 , 25 3 12 6 36 0 Word Problems: 66. How many 2’s!are in 8? 67. How many 12 ’s are in 8? ! ! Answers: 1. 2 3 • 3 2.! 2• 3• 7 3. 1, 2, 3, 6. 9, 18 4. 1, 2, 3, 4, 6, 8, 12, 24 ! 5. 18 ! 6. 9 7. 1 8. 14 9. 10. ! 11. 2 5 3 4 1 4 81. 5 9 1 7 2 7 82. 3 7 4 7 5 7 3 4 6 7 1 83. 2 3 ! ! ! ! ! ! 12. ! ! 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 4 7 4 3 32 12 28 6 14 616 21 30 8 ! ! 23. 24. 25. 26. 15! 24 120 6 , 29 9 27. 9 , 14 24 24 12 10 , 15 15 6 , 8, 9 12 12 12 7 , 58 8 3 (because 4 3 (because 8 28. ! 29. ! 30. ! 31. ! 32. ! ! ! ! ! ! ! ! 20 28 9 24 < > 21 ) 28 8 ) 24 7 50. 51. 52. 53. 54. 24 6 ! 55. 3 8 9 8 ! ! 56. 57. 6 58. 83 ! 59. 6 7 33. 34. 1 35. 14 ! 36. 3 12 37. 1 3 ! 38. " 151 ! 39. ! 40. ! ! 41. 8 58 42. 3 43. 3 56 ! 44. ! ! 45. 46. 4 47. 169 ! 48. ! 7 8 15 4 60. 1 4 1 3 61. ! 62. ! 63. ! ! 9 4 49. 6 14 or ! ! 25 4 64. 2 5 6 15 16 1 6 4 25 1 8 12 31 11 , 6 5 , 6 7 , 12 65. 66. 4 ! ! ! 15 8 7 , 8 2 , 3 1,400,000 18; yes $16.10 $12.65 6 14 $28.75 78. 2 3 2 5 11 35 79. 80. 125 81. Q 82. T 83. S B. Fraction-decimal conversion: Each digit position has a value ten times the place to its right. The part to the left of the point is the whole number part. Fraction to decimal: divide the top by the bottom: example: 43 = 3 ÷ 4 = 0.75 20 3 example: 3 25 example: example: 324.519 = ( 3 "100) + (2 "10) + (4 "1) !Problems 11-14: Write each as a decimal. If the decimal repeats, show the repeating block: Problems 1-5: Which is larger? ! 4. 1.9 or 1.09 5. .5 or .49 Problems 6-8: Arrange in order of size from smallest to largest: 8. 4.5, 5.4, 4.49, 5.41 Repeating decimals are shown with a bar over the repeating block of digits: example: .3 means .333333333… example: .43 means .4343434343… example: .43 means .4333333333… Problems 9-10: Arrange in order, large to small: = 20 ÷ 3 = 6.6 = 3 + 25 = 3 + (2 ÷ 5) = 3+ .4 = 3.4 ! 1 1 +(5 " 101 ) + (1" 100 ) + (9 " 1000 ) 9. .3 , .3, .34 10. .6, .67 , .67, .67 , .6 4 15 ! !A. Meaning ! of Places: 6. .02, .2, .19, .0085 7. .45, .449, .451, .5 ! ! 16 9 9 ! ! !TOPIC 3: DECIMALS 1. .59 or .7 !2. .02 or .03 3. .2 or .03 ! 11 12 25 , 56 36 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 11. 12. ! ! 5 8 3 7 = 13. 4 13 = = 14. 3 100 = Non-repeating decimals to fractions: say the number as a fraction, ! write the fraction you say; reduce if possible: ! 4 = 25 example: .4 = four tenths = 10 example: 3.76 = three and seventy six 76 = 3 19 hundredths = 3 100 25 Problems 15-18: ! Write as a fraction: 15. .01 = 16.! .38 = 17. 4.9 = 18. 1.25 = Comparison of fractions and decimals: usually it is easiest to convert fractions to decimals, then compare: example: Arrange from small to large: .3, 25 , .3 , 27 ! ! 8 As decimals these are: .3, .4, .33333…, .285714 … So the order is: .285714 , .3. .3 , .4, or 27 , .3, .3, 25 Problems 19-21: Arrange in order, small to large: ! 1 5 19. !23 , .6, .67, .67 21. 100 , .01, .009, 500 , 0.88 ! 20. 78 , 0.87, 13 16 ! ! Adding and subtracting decimals: like places must be combined!(line up the points): example: 4 + .3 = 4.3 example: 3.43 + .791+ 12 : example: 8 ! 4.96 : 8. 00 "4.96 !3.04 ! 3.430 .791 12.000 16.221 example: 6.04 ! ( 2 ! 1.4 ) = 6.04 ! .6 = 5.44 Problems 22-30: Calculate: 22. 23. 24. 25. 26. 27. 28. 29. 30. ! 5.4 + .78 = 1.36 ! 0.63 = 4 ! .3 + .001 ! .01 + .1 = $ 3.54 ! $1.68 = $ 17 ! $10.50 = 17.5 ! 10 = 4 + .3 + .02 + .001 = 8.3 ! 0.92 = 4.7 + 47 + 0.47 = C. Multiplying and dividing decimals: Multiplying decimals example: .3 ! .5 = .15 example: .3 ! .2 = .06 2 example: (.03) = .0009 31. 3.24 ! 10 = 32. .01! .2 = 34. 5 ! 0.4 = 2 35. (.51) = 2 33. (.04) = Dividing decimals: Change the problem to an equivalent whole number problem by multiplying both numbers by the same power of 10: example: .3 ÷ .03 Multiply both by 100 to get 30 ÷ 3 = 10 example: .014 .07 Multiply both by 1000 to get 14 ÷ 70 = .2 = 40. 7.20 2.4 41. 1.44 ÷ 2.4 = = 42. 36.8 10 36. .013 ÷ 100 = !37. .053 ÷ .2 = = 38. 340 3.4 39. ! ! 8.4 0.07 = ! ! D. Percent: Meaning: translate percent as hundredths: example: 8% means 8 hundredths or .08 or 8 100 = 2 25 Percent-decimal conversion: To change a decimal to percent form, multiply by 100 (move the point 2 places right), write the percent symbol (%): example: .075 = 7.5% example: 1 41 = 1.25 = 125% Problems 43-45: Write as a percent: 43. .3 = 44. 4 = 45. .085 = ! To change a percent to decimal form, move the point 2 places left (divide by 100) and drop the % symbol: example: 8.76% = .0876 example: 67% = .67 Problems 46-49: Write as a decimal: 46. 10% = 47. 136% = 48. .03% = 49. 4% = Solving percent problems: Step 1: Without changing the meaning, write the problems so it says “__ of __ is __”, and from this, identify a, b, and c: a % of b is c Problems 50-52: Write in the form a% of b is c, and tell the values of a, b, and c: 50. 3% of 40 is 1.2 51. 600 is 150% of 400 52. 3 out of 12 is 25% Step 2: Given a and b, change a% to a decimal and multiply (“of” can be translated “multiply”). Or, given c and one of the others, divide c by the other (first change percent to decimal); if answer is a, write it as a percent: example: What is 9.4% of $5000? Compare a% of b is c: 9.4% of $5000 is __? Given a and b: multiply: .094 ! $5000 = $470 example: 56 problems correct out of 80 is what percent? Compare a% of b is c: __% of 80 is 56? Given c and other (b): 56 ÷ 80 = .7 = 70% example: 5610 people, which is 60% of the registered voters, vote in an election. How many are registered? Compare a% of b is c: 60% of __ is 5610? Given c and other (a): 5610 ÷ .6 = 9350 9 53. 54. 55. 56. 57. 63. .0042210398 ÷ .01904982 = (.02, .2, .5, 5, 20, 50) 64. 101.7283507 + 3.14159265 = (2, 4, 98, 105, 400) 3 65. (4.36285903) = (12, 64, 640, 5000, 12000) 66. 1.147 ! 114.7 = (–100, –10, 0, 10, 100) 4% of 9 is what? What percent of 70 is 56? 15% of what is 60? What is 43% of 500? 10 is what percent of 40? E. Estimation and approximation: Rounding to one significant digit: example: 3.67 rounds to 4 example: .0449 rounds to .04 example: 850 rounds to either 800 or 900 example: .4 = .44444... rounds to .4 Word Problems: Problems 67-69: A cassette which cost $9.50 last year costs $11 now. Problems 58-61: Round to one significant digit: 58. 45.01 59. 1.09 60. .00083 61. 0.5 Problems 70-71: Jodi’s weekly pay is $89.20. She gets a 5% raise. To estimate an answer, it is often sufficient to round each given number to one significant digit, then compute: example: .0298 ! .000513 Round and compute: .03 ! .0005 = .000015 .000015 is the estimate Problems 62-66: Select the best approximation of the answer: 62. 1.2346825 ! 367.003246 = (4, 40, 400, 4000, 40000) Answers: 1. .7 2. .03 3. .2 4. 1.9 5. .5 6. .0085, .02, .19, .2 7. .449, .45, .451, .5 8. 4.49, 4.5, 5.4, 5.41 9. .34, .3,.3 10. .67, .67 , .67, .6 , .6 11. .625 !12. .428571 13. 4.3 ! ! 14. .03 1 15. 100 ! 16. 19 ! 50 17. 4 109 = 49 50 ! 18. 1 14 = 45 ! 19. .6, 23 , .67, .67 ! 20. 13 , 16 ! ! ! ! ! .87, 78 , .88 67. What is the amount of the increase? 68. What percent of the original price is the increase? 69. What is the percent increase? 21. all equal 22. 6.18 23. .73 24. 3.791 25. !$1.86 26. $6.50 27. 7.5 28. 4.321 29. 7.38 30. 52.17 31. 32.4 32. .002 33. .0016 34. 2 35. .2601 36. .00013 37. .265 38. 100 39. 120 40. 3 41. .6 42. 3.68 70. What will be her new weekly pay? 71. How much more will she get? Problems 72-74: Sixty percent of those registered voted in the last election. 72. What fraction voted? 73. If there was 45,000 registered, how many voted? 74. If 33,000 voted, how many were registered? 75. A person weighs 125 pounds. Their ideal weight is 130 pounds. Their actual weight is what percent of their ideal weight? 1 100 43. 44. 45. 46. 47. 48. 49. 50. 30% 400% 8.5% .1 1.36 .0003 .04 3% of 40 is 1.2; a = 3%, b = 40,c = 1.2 51. 150% of 400 is 600; a = 150% , b = 400,c = 600 52. 25% of 12 is 3; a = 25% , b = 12 , c = 3 53. .36 54. 80% 55. 400 56. 215 57. 25% 58. 50 59. 1 60. .0008 61. .6 10 62. 63. 64. 65. 66. 400 .2 105 64 –100 ! ! TOPIC 4: EXPONENTS 67. 68. 69. 70. 71. $1.50 "15.8% "15.8% $93.66 $4.46 Meaning of exponents: 4 example: 3 = 3 ! 3 ! 3 ! 3 31. 32. 33. 34. = 3• 3 • 3 • 3 = 81 example: 4 = 4 • 4 • 4 = 64 Problems 1-12: Find the value: 7. (!2 ) = 2 8. 100 = 2 9. (2.1) = 2 5. "3 = " 3 = 3 6. !2 = ! 2 3 11. () 12. (" 23 ) 2 3 = 3 = ! b a means use a as ! a factor b times. (b is the exponent or power of a) 5 example: 2 means !2• 2•2•2• 2 5 2 has a value 32 5 is the exponent or power 2 is the factor 2 example: 5 • 5 can be written 5 . Its value is 25. 1 example: 4 = 4 Problems 13-24: Write the meaning and find the value: 4 3 19. (0.1) = 13. 6 = 2 14. (!4 ) = 20. 4 21. 15. 0 = 1 16. 7 = 4 17. 1 = 3 18. (!1) = example: example: 8 24 63 62 ! = 168 = 12 = 216 36 25. ! 26. 27. Problems 47-52: Write each expression as a power of the same factor: 2 22. 2 = 2 23. (.03) = 2 3 24. 3 • 2 = 1 ! ! ! 4 50. (!1) • (!1) = 4 51. 10 • 10 = 52. 10 • 10 = 6 4 54. 3 = 56. 729 ÷ 81 = 55. 3 = note: 36 ÷ 34 = 36 34 = 3•3•3•3•3•3 3•3•3•3 57. Circle the exponents: 30. ! 5 2 47. 4 • 4 = 3 3 48. 5 • 5 = 3 49. 3 • 3 = 2 29. ! 6 = 33 • 33 • 33 • 33 • 3 • 3 = 1• 1•1•1 • 3• 3 = 3 28. ! 4 example: 3 • 3 = 3 Problems 54-56: Find the value: =6 6 = 32 5 2 = 8 4 •5 = 10 40. 3 = 53. Make a formula by filling in the brackets: [ ] a b • a c = a . This is an exponent rule. 4 ( 23 ) = 2 (1 12 ) = 36 34 = 32 ! 58. How ! are the circled numbers related? Problems 25-30: Simplify: ! 33 + 43 + 53 = 63 = 41. Write as a power of 3: 3 • 3• 3 • 3 • 3• 3 = 2 4 42. Write the meaning: 3 • 3 = 43. Write your answer to 42 as a power of 3, then find the value. 2 4 44. Now find each value and solve: 3 • 3 = 2 4 6 45. So 3 • 3 = 3 . Circle each of the powers. Note how the circled numbers are related. 46. How are they related? 10 ! (3.1)2 ! (.03)2 = (3.1)2 + (.03)2 = 4 39. 3 = 10. (".1) = ( ) 2 35. 36. 37. 38. Problems 39-40: Write the meaning (not the value): 3 2 4. ! (3) = 32 + 42 = 52 = 32 + 42 + 12 2 = 132 = B. Integer exponent laws: 3 2 1. 3 = 3 2. 2 = 2 3. (!3 ) = ! ! Problems 31-38: Find the value: A. Positive integer exponents: 3 72. 35 73. 27,000 74. 55,000 75. " 96% 10 = 4 2 •5 3 4 2 •2 = 2 5 •2 5•12 = 6 2 •10 ! Problems 59-63: Write each expression as a power: example: 4 36 34 =!32 4 59. 2 ÷ 2 = 60. ! ! 25 2 = 11 61. 62. ! ! 52 = 5 7 ("4 ) ("4 ) 2 63. = Problems 81-84: Write in scientific notation: 64. Make a formula by filling in the brackets: ab ac ! = a[ ] . This is another exponent rule. Problems 85-87: Write in standard notation: 3 85. 1.4030 ! 10 = "2 86. 9.11 ! 10 = Problems 65-67: Find each value: 3 2 ( ) 3 67. 4 2 = ( 64) = 87. 4 ! 10 "6 = 66. 4 = To compute with numbers written in scientific form, separate the parts, compute, then recombine: Problems 68-69: Write the meaning of each expression: example: 3.14 ! 10 6 3 2 ( ) example: 4 3 ( 2 4 ( ) = 4 •4 = 4• 4•4 • 4• 4•4 Problems 88-95: Write answer in scientific notation: 1 3 69. 5 = 88. 10 3 2 ! 6 70. =4 71. What is the relation of the circled numbers? b c ( ) 72. Make a rule: a = a[ ! b c ( ) III. a 6 74. 3 • 3 = 75. 210 25 77. 5 2 ( ) (3 ) 76. 2 4 4 4 3 3 = 1 2 ( ) 79. 5 4 ! ! 3 ! C. Scientific notation: Note that scientific form always looks like a ! 10n , where 1 ! a < 10 , and n is an integer power of 10. 4 example: 32800 = 3.2800 ! 10 if the zeros in the ten’s and one’s places are significant. If the one’s zero is not significant, write: 3.280 ! 104 ; if neither is significant: 3.28 ! 10 4 . "3 example: .0040301 = 4.031! 10 2 example: 2 ! 10 = 200 "1 example: 9.9 ! 10 = .99 ! 10 40 1010 2 ! 10 = 92. = 1.86"10 3"10 91. 4 3.6"10 5 1.8"10 3 1.8"10 8 3.6"10 5 ( = 3 2 93. 4 ! 10 = ! 2 3 94. 1.5 "10 " 5 "10 = ! 2 #2 95. 1.25 "10 4 "10 = ( ( = ) ( )( ) = ) ) ! a = 100 , because a = 10 , then 10 = a = 100 example: If ! = 40 D. Square roots or perfect squares: a = b means b 2 = a , where b ! 0 . Thus 49 = 7 , because 72 = 49 . Also, " 49 = "7 . 49 does not equal –7, (even though Note: 2 (!7) does = 49 ) because –7 is not ! 0 . ! 80. 10 •10 = = = ! ! 78. = 90. ! = Problems 74-80: Use the rules to write each expression as a power of the factor, and tell ! which rule you’re using: 4 89. ] 73. Write your three exponent rules below: b c I. a • a = b II. a c = a )(2 ) 5 = (3.14)(2) ! 10 = 6.28 ! 10 6 4 4.28 10 6 • 10 2 = 2.00 ! 10 example: 4.28"10 2 = 2.14 2.14"10 3 ( )= Circle the three exponents: (4 ) 68. 3 5 5 = 4 • 4 • 4 • 4 • 4 • 4 = 46 ! 83. 5.07 = 84. !32 = 81. 93,000,000 = 82. .000042 = = 65. 4 = ! 15 13 2 Problems 96-99: Find the value and tell why: 96.! If 97. If 98. If ! 99. If a = 5 then a = x = 4 , then x = 36 = b , then b = 169 = y , then y = ! Problems 100-110: Find the value: ! 2 2 2 106. 3 + 4 + 12 = ! 100. 81 = 2 2 2 101. 8 = 107. 17 "15 = 2 2 2 102. 8 = 108. 13 "12 = 2 ! 103. ("7) = ! 109. 4 3 = ! 2 2 4 104. 6 + 8 = 110. 3 = ! ! 32 + 4 2 = 105. ! ! ! ! ! 12 Answers: 1. 9 2. 8 3. 9 4. –9 5. –9 6. –8 7. –8 8. 10,000 9. 4.41 10. –.001 11. 278 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 12. " 278 ! ! 13. 14. 15. 16. 17. 18. 19. 20. 21. 6 • 6 • 6 = 216 (!4)(!4) = 16 0• 0•0•0 = 0 7=7 1• 1•1•1 = 1 (!1)(!1)(!1) = !1 (.1)(.1)(.1)(.1) = .0001 2 3 3 2 • 23 • 23 • 23 = 16 81 • 32 = 49 = 2 14 22. 2 • 2 • 2 • 2 • 2 ! ! ! ! ! 57. ! •2 • 2 • 2 • 2 • 2 = 1024 23. (.03)(.03) = .0009 24. 3 • 3• 2 • 2 • 2 = 72 25. 23 26. 4 27. 2 28. 18 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 2 729 81 9 3 (6 ) 3 (4 ) 2 = 3( ) 6!4=2 0 2 4 2 1 5 (!4)5 2 1 (or any power of 1) 64. ! 1 6 25 25 169 169 9.6091 9.6109 216 216 58. 59. 60. 61. 62. 63. 3•3 3 • 3• 3 • 3 6 3 3 • 3• 3 • 3 • 3• 3 6 3 = 729 9 • 81 = 729 ( ) ( ) ( ) 3 2 •3 4 = 36 2+4= 6 43 56 34 (!1)9 10 5 10 2 [ ] a b • a c = a b+c 65. 66. 67. 68. 69. ab ac = a[ b"c ] 64 4096 4096 2 2 2 2 3 •3 • 3 •3 5•5•5 ( ( ) )( ) = 4( ) 70. 4 2 3 6 71. 3 ! 2 = 6 [ ] b c 72. a = a bc ( ) 73. I. a b • a c = a b +c b II. a c = a b"c a ! III. (a ) b c 10 74. 3 , rule I 5 75. 2 , rule II 10 76. 2 , rule III 16 77. 3 , rule III 3 78. 3 , rule II 2 79. 5 , rule III 7 80. 10 , rule I 7 81. 9.3 ! 10 "5 82. 4.2 ! 10 83. 5.07 84. !3.2 " 10 85. 1403.0 86. 0.0911 87. .000004 42 88. 10 30 89. 10 2 90. 6.2 ! 10 2 91. 2.0 ! 10 2 92. 5.0 ! 10 7 93. 1.6 ! 10 5 94. 7.5 ! 10 95. 5 2 96. 25; 5 = 25 2 97. 16; 4 = 16 2 98. 6; 6 = 36 2 99. 13; 13 = 169 100. 9 101. 64 102. 8 103. 7 104. 10 105. 5 106. 13 107. 8 108. 5 109. 8 110.9 = a bc ! TOPIC 5: EQUATIONS and EXPRESSIONS ! A. Operations with literal symbols (letters): When letters are used to represent numbers, addition is shown with a “plus sign” (+), and subtraction with a “minus sign” (–). Multiplication is often show by writing letters together: example: ab means a times b So do a • b, a ! b, and (a )(b ) example: 7 • 8 = 7 ! 8 = ( 7)(8) 13 example: Compare 4x + 7x = (4 + 7)x = 11x with ac + bc = ( a + b )c ; example: a( b + c ) means a times (b + c) Problems 1-4: What is the meaning of: a = 4, c = x, b = 7 example: 4( 2 + 3) = 4 • 2 + 4 • 3 (The 2 1. abc 2. 2a 3. c 4. 3( a ! 4) distributive property says this has value 20, whether you do 4 • 5 or 8 + 12 .) example: 4a + 6x ! 2 = 2(2a + 3x ! 1) To show division, fractions are often used: example: 3 divided by 6 may be shown: 3 ÷ 6 or 63 , or 6 3 , and all have value 12 , or .5. ) 5. What does ! 4a 3b ! 25. 6( x ! 3) = 26. 4( b + 2 ) = mean? Problems 6-15: Write a fraction form and reduce 6. 36 ! ÷9= 7. 4 ÷ 36 = 8. 10 36 = 11. 12 ÷ 5y = 12. 6b ÷ 2a = 13. 8r ÷ 10s = 9. 14. a ÷ a = B. Evaluation of an expression by substitution: 15. 2x ÷ x = example: Find the value of 7 ! 4x , if x = 3 : ) 1.2) .06 = 7 ! 4x = 7 ! 4 • 3 = 7 ! 12 = !5 In the above exercises, notice that the fraction forms can be reduced if there is a common (shared) factor in the top and bottom: example: example: example: !example: !example: !example: 36 = 9•4 = 99 • 41 = 1• 4 = 4 9 9•1 4 •1 = 44 •9 = 44 • 19 = 1• 19 = 19 36 36 = 2•18 = 185 (or 3 53 ) 10 2•5 6b = 2•3•b = 3b 2a 2•a a a =1 a 2x != 21 • xx = 21 • 1 = 2 •1 = = 2•x x 1•x example: If a = !7 and b = !1, then 2 2 !Problems 16-24: Reduce and simplify: !16. !17. 18. ! 19. ! 20. ! ! ! 27. (3 ! x )2 = 28. 4b ! 8c = 29. 4x ! x = (Hint: think 4x ! 1x , or 4 cookies minus 1 cookie) 30. !5 (a ! 1) = 33. 3a ! a = 31. 5a + 7a = 34. 5x ! x + 3x = 32. 3a ! a = 35. x (x + 2) = ! ! to find the value (if possible): 10. 2x ÷ a = ! Problems 25-35: Rewrite, using the distributive property: 3x = 3 3x = 4x x = 2x 12x = 3x 12x = 3 (Hint: = 21. abc 3ac 22. 23. ! 24. 6x = 8xy 2 15x = 10x 6x •5= 5 ! 6x !• 5 5 = 6x 5 • 51 = 6x •5 ...) 5•1 ! The distributive property says a( b + c ) = ab + ac . Since equality (= ) goes both ways, the distributive property ! can also be written ab + ac = a( b + c ) . Another form it often takes is (a + b)c = ac + bc , or ac + bc = ( a + b )c . example: 3( x ! y ) = 3x ! 3y . Comparing this with a( b + c ) = ab + ac , we see a = 3 , b = x , and c = ! y 2 a b = ( !7) (!1) = 49(!1) = !49 2 example: If x = !2 , then 3x + x ! 5 2 = 3( !2 ) + (!2 ) ! 5 = 3 • 4 ! 2 ! 5 = 12 ! 2 ! 5 = 5 example: 8 ! c = 12 Add c to each, giving 8 ! c + c = 12 + c , or 8 = 12 + c . Then subtract 12, to get !4 = c, or c = !4 . example: a5 = 43 Multiply each by the lowest common denominator 15: a5 •15 = 43 • 15 , or a • 3 = 4 • 5 Then divide by 3: a3•3 = 20 , so a = 20 3 3 !Problems 36-46: Given x = !1, y = 3 and z = !3 , ! find the value: !42. 2x + 4y = 36. 2x =! 2 37. ! z = 43. 2x ! x ! 1 = 2 38. xz = 44. ( x + z ) = 2 2 39. y + z = 45. x + z = 2 40. z + 3 = 46. ! x z = 2 2 41. y + z = Problems 47-54: Find the value, given a = !1, b = 2, c = 0, x = !3, y = 1 , and z = 2 : 47. 48. 49. ! 50. 6 = b x = a x = 3 a"y b 51. 52. 53. = ! ! ! ! ! ! ! 54. 4 x"3y 3y"2x b = c " bz = c = z = 14 C. Solving a linear equation in one variable: Add or subtract the same thing on each side of the equation and/or multiply or divide each side by the same thing, with the goal of getting the variable alone on one side. If there are fractions, you can eliminate them by multiplying both sides of the equation by a common denominator. If the equation is a proportion, you may wish to cross-multiply. Problems 71-72: In d = rt , substitute, then solve for the variable: example: 3x = 10 Divide both sides by 3, to get 1x : 3x3 = 103 , or x = 103 example: 5 + a = 3 Subtract 5 from each side, to get 1a (which is a): 5 + a ! 5 = 3 ! 5 or 74. 75. 76. 77. a = !2 ! example: ! y 3 71. t = 5, r = 50; d = 72. d = 50, r = 4; t = 73. On a 40 mile hike, a strong walker goes 3 miles per hour. How much time will the person hike? Write an equation, then solve it. = 12 Multiplying both sides by 3, to y 3 get y : • 3 = 12 • 3, which gives y = 36 . example: b ! 4 = 7 Add 4, to get 1b : b ! 4 + 4 = 7 + 4 , or b = 11 ! ! ! Problems 55-65: Solve: 55. 56. 57. 58. 59. 60. 2x = 94 3 = 6x5 3x + 7 = 6 x = 45 3 5! x =9 ! x = 2x5 + 1 61. 62. 63. 64. 65. 4x ! 6 = x x " 4 = x2 + 1 6 ! 4x = x 7x ! 5 = 2x + 10 4x + 5 = 3 ! 2x Problems 66-70: Substitute the given value, then solve for the other variable: ! example: If n = r + 3 and r = 5 find the value of n: Replacing r with 5 gives n = 5 + 3 = 8 . ! 82. If a = 5 and b = 8 , find P. 83. If a = 7 and P = 40 , find b. Problems 84-85: The difference of two numbers x and 12 is 5. 69. 5x = y ! 3, x = 4 70. 5x = y ! 3, y = 3 84. If x is the larger, an equation, which says this same thing could be x ! 12 = 5 . Write an equation if x is the smaller of the two numbers x and 12. 85. Find the two possible values of x by solving each equation in problem 84. D. Word Problems: If an object moves at a constant rate of speed r, the distance d it travels in time t is given by the formula d = rt . Problems 86-87: Write an equation, which says: example: If t = 5 and d = 50 , find r: Substitute the given values in d = rt and solve: 50 = r • 5 , giving r = 10 . Answers: 1. a times b times c 2. 2 times a 3. c times c 4. 3 times (a ! 4) 78. The sum of two numbers is 43. One of the two numbers is 17. What is the other? 79. Write an equation which says that the sum of a number n and 17 is 43. 80. Write an equation which says the amount of simple interest A equals the product of the invested principle P, the rate of interest r, and the time t. 81. Use the equation of problem 80: P = $200, r = 7% and t = 5 years. Find the amount of interest A. Problems 82-83: In a rectangle which has two sides of length a and two sides of length b, the perimeter P is found by adding all the side lengths, or P = 2a + 2b . ! 66. n = r + 3, n = 5 67. n = r + 3, n = 1 68. a2 = b, b = 6 “Product” “Quotient” “Difference” “Sum” 86. n is 4 more than 3. 87. 4 less than x is 3. 88. Solve the two equations you wrote for problems 86 and 87. 5. 4a divided by 3b 6. 369 = 4 7. 8. 4 36 36 10 9. 10. = 19 = 185 11. ! ! ! ! ! ! .06 1.2 2x a 12 5y 6 = 120 = 1 20 15 12. 13. 14. ! ! ! ! ! ! ! ! ! 6b = 3b 2a a 8r 4r = 10s 5s a =1 a 2x =2 x 15. 16. x 17. 43 18. 12 19. 4 20. 4x 21. b3 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 3 4y 3x 2 6x 6x ! 18 4b + 8 6 ! 2x 4( b ! 2c ) (4 ! 1) x = 3x !5a + 5 (5 + 7)a = 12a (3 ! 2)a = 1a = a ! 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 3 3 0 0 18 10 2 16 10 3 3 3 –1 –1 " 53 no value (undefined) –1 0 x = 47 x= 5 2 57. x = " 13 58. x = 154 ! ! 2a 7x ! 2 x + 2x –2 ! 59. x = !4 60. x = 53 61. x = 2 62. x = 10 63. x = 65 ! ! ! ! x =3 64. 65. 66. 67. 68. 69. 70. 71. 72. x = " 13 5 = r + 3; r = 2 1 = r + 3; r = !2 a 2 = 6; a =12 5 • 4 = y ! 3; y = 23 5x = 3 ! 3; x = 0 d = 50 • 5 ; d = 250 50 = 4t; t = 25 2 40 3 73. 40 = 3t; t = hours 74. Multiply 75.!Divide 76. Subtract 77. Add 78. 26 79. n + 17 = 43 80. A = P r t 81. A = $70 82. P = 26 83. b = 13 84. 12 ! x = 5 85. x = 17 or 7 86. n = 4 + 3 87. x ! 4 = 3 88. n = 7; x = 7 TOPIC 6: GEOMETRY ! A of A. Formulas for perimeter P and area rectangles, squares, parallelograms, and triangles: Rectangle with base b and altitude (height) h: P = 2b + 2h A = bh example: A square with side s = 11cm has P = 4s = 4 ! 11 = 44 cm 2 2 A = s = 11 = 121 cm 2 (sq. cm) A parallelogram with base b and height h and other side a: h b If a wire is bent in this shape, the perimeter P is the length of the wire, and the area A is the number of square units enclosed by the wire. example: A rectangle with b = 7 and h = 8 : P = 2b + 2h = 2 • 7 + 2 • 8 = 14 + 16 = 30 units A = bh = 7• 8 = 56 square units A = bh P = 2a + 2b h a b example: A parallelogram has sides 4 and 6; 5 is the length of the altitude perpendicular to the side 4. P = 2a + 2b = 2 • 6 + 2 • 4 5 6 = 12 + 8 = 20 units A = bh = 4 • 5 = 20 square units 4 A square is a rectangle with all sides equal, so the rectangle formulas apply (and simplify). If the side length is s: In a triangle with side lengths a, b, and c, and altitude height h to side b: P = 4s A = s2 P = a+ b+c A = 12 bh = bh2 s s ! a c h b 16 example: P = a+ b+c = 6 + 8 + 10 = 24 units A = 12 bh = 12 (10)( 4.8) = 24 square units Problems 1-8: Find P and A: ! 1. 2. 3. 4. 5. Rectangle with sides 5 and 10. Rectangle with sides 1.5 and 4. Square with sides 3 miles. Square with sides 43 yards. Parallelogram with sides 36 and 24, and height 10 (on side 36). 6. Parallelogram, all sides 12, altitude 6. !with sides 5, 12, and 13. 7. Triangle Side 5 is the altitude 13 5 on side 12. 12 8. Triangle 5 3 shown: 4 B. Formulas for circumference C and area A of a circle: A circle with radius r r (and diameter d = 2r ) has a distance around (circumference) C = !d = 2!r (If a piece of wire is bent into a circular shape, the circumference is the length of the wire.) example: A circle with radius r = 70 has d = 2r = 140 and exact circumference C = 2!r = 2 • ! • 70 = 140! units If ! is approximated by 22 , 7 ! C = 140" # 140( 22 # 440 units (approx.) 7) If ! is approximated by 3.1, C ! 140(3.1) ! = 434 units 2 The area of a circle is A = !r example: If r = 8 , exact area is A = !r 2 = ! • 82 = 64! square units Problems 9-11: Find the exact C and A for a circle with: 9. radius r = 5 units 10. r = 10 feet 11. diameter d = 4 km Problems 12-14: A circle has area 49! : 12. What is its radius length? 13. What is the diameter? 14. Find its circumference. Problems 15-16: A parallelogram has area 48 and two sides each of length 12: 15. How long is the altitude to those sides? 16. How long are each of the other two sides? 17. How many times the P and A of a 3cm square are the P and A of a square with sides all 6 cm? 18. A rectangle has area 24 and one side 6. Find the perimeter. Problems 19-20: A square has perimeter 30: 19. How long is each side? 20. What is its area? 21. A triangle has base and height each 7. What is its area? C. Pythagorean theorem: In any triangle with a 90° (right) angle, the sum of the squares of the legs equals the square of hypotenuse. (The legs are the two shorter sides; the hypotenuse is the longest side.) If the legs have lengths a and b, and c is the hypotenuse 2 2 2 length, then a + b = c . In words: “ In a right triangle, leg squared plus leg squared equals hypotenuse squared.” example: A right triangle has hypotenuse 5 and one 2 2 2 leg 3. Find the other leg. Since leg + leg = hyp , 2 2 2 3 + x =5 2 9 + x = 25 x 2 = 25 ! 9 = 16 x = 16 = 4 Problems 22-24: Find the length of the third side of the right triangle: !22. one leg: 15, hypotenuse: 17 23. hypotenuse: 10, one leg: 8 24. legs: 5 and 12 Problems 25-26: Find x: 25. 27. 26. 17 28. In right !RST with right angle R, SR = 11 and TS = 61 . Find RT. (Draw and label a triangle to solve.) 29. Would a triangle with sides 7, 11, and 13 be a right triangle? Why or why not? F B Problems 30-32: Use this figure: 38. 0 39. 12 40. " D E A B 41. C ! 30. Find and name two similar triangles. ! 31. Draw the triangles separately and label them. ! 32. List the three pairs of corresponding sides. If two triangles are similar, any two corresponding sides have the same ratio (fraction value): example: the ratio a to x, or ax , is the same as by and cz . Thus a x = by , a x = cz , and b y = cz . Each of these equations is called a proportion. ! ! ! ! ! ! 33. Draw the similar triangles separately, label them, 10 and write 9 5 proportions for 3 the corresponding sides. 8 4 Problems 34-37: Solve for x: example: Find x by writing and solving a proportion: 4 2 5 3 ! x 2 5 ! ! = x3 , so cross multiply and get 2x = 15 or x = 7 12 ! E 34. AC = 10; EC = 7 BC = 4; DC = x D A 55 B 15 -2 E A 20 30 x 4 C D E A B 53 A 4 10 x Problems 38-45: Name the point with given coordinate: 53 36 10 4 3 D. Graphing on the number line: example: !ABC and !FED are similar: The pairs of sides which correspond are AB and D FE , BC and ED , AC and FD . 36 37. x 36. Similar triangles are triangles which are the same shape. If two angles of one triangle are equal respectively to two angles of another triangle, then the triangles are similar. C 7 35. -1 F 0 1 42. "1.5 G 2 43. 2.75 1 2 4 3 44. " 23 45. 1.3 Problems 46-51: On the number line above, what is the distance!between the listed points? (Remember that distance is always positive.) 46. D and G 47. A and D 48. A and F 49. B and C 50. B and E 51. F and G Problems 52-55: On the number line, find the distance from: 52. –7 to –4 53. –7 to 4 54. –4 to 7 55. 4 to 7 Problems 56-59: Draw a sketch to help find the coordinate of the point…: 56. Halfway between points with coordinates 4 and 14. 57. Midway between –5 and –1. 58. Which is the midpoint of the segment joining –8 and 4. 59. On the number line the same distance from –6 as it is from 10. E. Coordinate plane graphing: To locate a point on the plane, an ordered pair of numbers is used, written in the form ( x, y ) . Problems 60-63: Identify coordinates x and y in each ordered pair: 60. (3,0) 61. ("2,5) 62. (5,"2) 63. (0,3) To plot a point, start at the origin and make the moves, first in the x-direction (horizontal) and 55 C 3 18 then the y-direction (vertical) indicated by the ordered pair. 65. Two of the lines drawn in problem 64 cross each other. What are the coordinates of the crossing point? 66. In what quadrant is the point (a,b) if a > 0 and b < 0 ? example: (-3, 4) Start at the origin, move left 3 (since x = !3 ), Problems 67-69: ABCD is a square, with C (5,"2) and D ("1,"2) . Find: then (from there), up 4 (since y = 4 ), 67. the length of each side. 68. the coordinates of A. 69. the coordinates of the midpoint of DC . put a dot there to indicate the point (-3, 4) A y B x D C Problems 70-72: Given A(0,5) , B(12,0) : 70. Sketch a graph. Draw AB . Find its length. 71. Find the midpoint of AB and label it C. Find the coordinates of C. 72. What is the area of the triangle formed by A, B, and the origin? 64. On graph paper, join these points in order: (–3,–2), (1,–4), (3, 0), (2, 3), (–1, 2), (3, 0), (–3, –2), (–1, 2), (1,–4). Answers: 1. 30 units, 50 units 2 (units 2 means square units) 2 2. 11units, 6 units 3. 12!miles, 9 miles 2 4. 3 yards, 169 yards 2 ! ! ! ! ! ! ! ! ! ! ! ! ! 5. 120 un., 360 un.2 ! un., 72 un.2 6. 48 7.!30 un., 30 un.2 8.! 6 un.2 !12 un., ! 9. 10" un., 25" un.2 ! 10. 20" ft., 100" ft.2 ! 11. ! 4" km, 4" km 2 12. !7 13. 14 ! 14. 14" ! 15. 4 16. Cannot tell 17. P is 2 times, A is 4 times 18. 20 19. 7 12 ! 20. 225 4 21. 22. 23. 24. 24 12 8 6 13 25. 26. 27. 28. 29. 30. 31. 9 41 10 60 No, because 7 2 +112 "132 "ABE ~ "ACD D E A ! B A ! C 32. AB , AC ; AE , AD ; BE , CD 33. 39 = 155 = 124 14 ! !34. ! or!2 45 5 ! !35. 2 4 or 14 5 5 ! 36. 452 ! ! ! ! ! 37. 407 38. ! D 39. E 40. C 41. F 42. B 43. G 44. B 45. F 46. 2.75 47. 2 48. 3 13 65. (0,–1) 66. IV 67. 6 68. ("1,4 ) ! 69. (2,"2) 70. 13 71. (6, 52 ) ! 72. 30 ! ! 49. 1 50. 2 51. 17 12 52. 3 53. 11 54. 11 55. 3 56. 9 57. –3 58. –2 59. 2 60. x = 3,y = 0 61. x = "2,y = 5 62. x = 5,y = "2 63. x = 0,y = 3 64.