– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – 61. d. It is ironic that in a place

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– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – 61. d. It is ironic that in a place where there are so many ways to describe one food (indicating that this food is a central part of the culture), Thomas is hungry. The passage does not mention the language of the reservation, so choice a is incorrect. The sentence does not show any measure of how hungry Thomas is, so choice b is incorrect. The sentence does not describe fry bread or make it sound in any way appealing, so choice c is also incorrect. The passage tells us that...

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– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – 64. d. That his father would not realize that someone 61. d. It is ironic that in a place where there are so else was living in the house—that he would not many ways to describe one food (indicating that notice, for example, different furniture arranged this food is a central part of the culture), in a different way—suggests that his father did Thomas is hungry. The passage does not men- not pay any attention to things around him and tion the language of the reservation, so choice a just went through the motions of his life by is incorrect. The sentence does not show any habit. Being habitual is different from being measure of how hungry Thomas is, so choice b stubborn, so choice a is incorrect. The author is is incorrect. The sentence does not describe fry writing about his father and seems to know him bread or make it sound in any way appealing, so quite well, so choice b is incorrect. We do not choice c is also incorrect. The passage tells us know if the author’s father was inattentive to his that it was Thomas’s hunger, not the number of needs (choice c), though if he did not pay atten- ways to say fry bread, that provided his inspira- tion to things around him, he likely did not pay tion, so choice e is incorrect. much attention to his children. Still, there is not 62. c. The author tells us that the new house was in enough evidence in this passage to draw this conclusion. His father may have been very “the best neighborhood in town,” and the neigh- attached to the old house (choice e), but the borhood’s “prestige outweighed its deadliness” incident doesn’t just show attachment; it shows (lines 5–8). There is no indication that their old a lack of awareness of the world around him. house was falling apart (choice a) or that they needed more room (choice b). The neighbor- 65. b. The bulk of this excerpt is the story that the hood is clearly not great for children (“it was not author ﬁnds “pathetic,” so the most logical con- a pleasant place to live [especially for chil- clusion regarding his feelings for his father is dren]”), so choice d is incorrect. The author tells that he lived a sad life. We know that his busi- us that business was going well for his father— ness was going well, but the author does not dis- so well, in fact, that he could pay for the house cuss his father’s methods or approach to in cash—but that does not mean the house was business, so choice a is incorrect. Choice c is affordable (choice e). In fact, if it was in the likewise incorrect; there is no discussion of his most prestigious neighborhood, it was probably father’s handling of ﬁnancial affairs. Choice d is expensive. incorrect because there is no evidence that his 63. a. The author tells us that his father was “always a father was ever cruel. His father may have been impressive and strong (choice e), but the domi- man of habit”—so much so that he forgot he’d nant theme is his habitual nature and the sad moved and went to his old house, into his old fact that he did not notice things changing room, and lay down for a nap, not even noticing around him. that the furniture was different. This suggests that he has a difﬁcult time accepting and adjust- ing to change. There is no evidence that he is a calculating man (choice b). He may be unhappy with his life (choice c), which could be why he chose not to notice things around him, but there is little to support this in the passage, while there is much to support choice a. We do not know if he was proud of the house (choice d). We do know that he was a man of habit, but we do not know if any of those habits were bad (choice e). 375
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – G lossar y of Terms: Language the particular choice and use of words diction Arts, Reading drama literature that is meant to be performed dramatic irony when a character’s speech or actions have an unintended meaning known to the audience the repetition of sounds, especially at the alliteration but not to the character beginning of words elegy a poem that laments the loss of someone or antagonist the person, force, or idea working against something the protagonist exact rhyme the repetition of exactly identical antihero a character who is pathetic rather than stressed sounds at the end of words tragic, who does not take responsibility for his or her exposition in plot, the conveyance of background destructive actions information necessary to understand the complica- aside in drama, when a character speaks directly to tion of the plot the audience or another character concerning the eye rhyme words that look like they should rhyme action on stage, but only the audience or character because of spelling, but because of pronunciation, addressed in the aside is meant to hear they do not autobiography the true account of a person’s life falling action the events that take place immediately written by that person after the climax in which “loose ends” of the plot are ballad a poem that tells a story, usually rhyming abcb tied up blank verse poetry in which the structure is con- feet in poetry, a group of stressed and unstressed trolled only by a metrical scheme (also called metered syllables verse) ﬁction prose literature about people, places, and characters people created by an author to carry the events invented by the author action, language, and ideas of a story or play ﬁgurative language comparisons not meant to be climax the turning point or high point of action and taken literally but used for artistic effect, including tension in the plot similes, metaphors, and personiﬁcation closet drama a play that is meant only to be read, ﬂashback when an earlier event or scene is inserted not performed into the chronology of the plot comedy humorous literature that has a happy free verse poetry that is free from any restrictions of ending meter and rhyme commentary literature written to explain or illumi- functional texts literature that is valued mainly for nate other works of literature or art the information it conveys, not for its beauty of form, complication the series of events that “complicate” emotional impact, or message about human experience the plot and build up to the climax genre category or kind; in literature, the different conﬂict a struggle or clash between two people, kinds or categories of texts forces, or ideas haiku a short, imagistic poem of three unrhymed connotation implied or suggested meaning lines of ﬁve, seven, and ﬁve syllables, respectively context the words and sentences surrounding a half-rhyme the repetition of the ﬁnal consonant at word or phrase that help determine the meaning of the end of words that word or phrase hyperbole extreme exaggeration not meant to be couplet a pair of rhyming lines in poetry taken literally, but done for effect denotation exact or dictionary meaning iambic pentameter a metrical pattern in poetry in denouement the resolution or conclusion of the which each line has ten syllables (ﬁve feet) and the action stress falls on every second syllable dialect language that differs from the standard lan- imagery the representation of sensory experiences guage in grammar, pronunciation, and idioms (natu- through language ral speech versus standard English); language used by inference a conclusion based upon reason, fact, or a speciﬁc group within a culture evidence dialogue the verbal exchange between two or more people; conversation 376
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – see dramatic irony, situational irony, or verbal the overall sound or “musical” effect of the irony rhythm irony pattern of words and sentences any written or published text sarcasm sharp, biting language intended to ridicule literature literary texts literature valued for its beauty of its subject form, emotional impact, and message(s) about the satire a form of writing that exposes and ridicules its human experience subject with the hope of bringing about change main idea the overall fact, feeling, or thought a writer setting the time and place in which a story unfolds wants to convey about his or her subject simile a type of ﬁgurative language that compares two melodrama a play that starts off tragic but has a things using like or as happy ending situational irony the tone that results when there is memoir an autobiographical text that focuses on a incongruity between what is expected to happen and limited number of events and explores their impact what actually occurs metaphor a type of ﬁgurative language that com- soliloquy in drama, a speech made by a character pares two things by saying they are equal who reveals his or her thoughts to the audience as if meter the number and stress of syllables in a line of he or she is alone and thinking aloud poetry sonnet a poem composed of fourteen lines, usually monologue in drama, a play or part of a play in iambic pentameter, with a speciﬁc rhyme scheme performed by one character speaking directly to the speaker in poetry, the voice or narrator of the poem audience stage directions in drama, the instructions pro- narrator in ﬁction, the character or person who tells vided by the playwright that explain how the action the story should be staged, including directions for props, cos- nonﬁction prose literature about real people, places, tumes, lighting, tone, and character movements and events stanza a group of lines in a poem, a poetic paragraph ode a poem that celebrates a person, place, or thing structure the manner in which a work of literature is omniscient narrator a third-person narrator who organized; its order of arrangement and divisions knows and reveals the thoughts and feelings of the style the manner in which a text is written, composed characters of word choice, sentence structure, and level of for- onomatopoeia when the sound of a word echoes its mality and detail meaning subgenre a category within a larger category paragraph a group of sentences about the same idea suspense the state of anxiety caused by an unde- personiﬁcation ﬁgurative language that endows cided or unresolved situation nonhuman or nonanimal objects with human symbol a person, place, or object invested with spe- characteristics cial meaning to represent something else plot the ordering of events in a story theme the overall meaning or idea of a literary work poetry literature written in verse thesis the main idea of a nonﬁction text point of view the perspective from which something thesis statement the sentence(s) that express an is told or written author’s thesis prose literature that is not written in verse or dra- tone the mood or attitude conveyed by writing or matic form voice protagonist the “hero” or main character of a story, topic sentence the sentence in a paragraph that the one who faces the central conﬂict expresses the main idea of that paragraph pun a play on the meaning of a word tragedy a play that presents a character’s fall due to quatrain in poetry, a stanza of four lines a tragic ﬂaw readability techniques strategies writers use to tragic hero the character in a tragedy who falls from make information easier to process, including the use greatness and accepts responsibility for that fall of headings and lists tragic ﬂaw the characteristic of a hero in a tragedy rhyme the repetition of an identical or similar that causes his or her downfall stressed sound(s) at the end of words 377
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS – a tragic play that includes comic when the intended meaning of a word tragicomedy verbal irony scenes or phrase is the opposite of its expressed meaning a statement that is deliberately voice in nonﬁction, the sound of the author speaking understatement restrained directly to the reader 378
PART VI The GED Mathematics Exam T his section covers the material you need to know to prepare for the GED Math- ematics Exam. You will learn how the test is structured so you will know what to expect on test day. You will also review and practice the fundamental math- ematics skills you need to do well on the exam. Before you begin Chapter 40, take a few minutes to do the pretest that follows. The questions and problems are the same type you will ﬁnd on the GED. When you are ﬁn- ished, check the answer key carefully to assess your results. Your pretest score will help you determine how much preparation you need and in which areas you need the most care- ful review and practice. 3 79
– THE GED MATHEMATICS EXAM – P retest: GED Mathematics Question 3 is based on the following ﬁgure. 3b Directions: Read each of the questions below carefully a+ and determine the best answer. To practice the timing of the GED exam, please allow 18 minutes for this pretest. Record your answers on the answer sheet provided here and the answer grids for 3a + b 2a + b questions 9 and 10. Note: On the GED, you are not permitted to write in the test booklet. Make any notes or calculations on a sep- arate piece of paper. ANSWER SHEET 3a + 2b 3. What is the perimeter of the ﬁgure? 1. a b c d e a. 8a + 5b 2. a b c d e b. 9a + 7b 3. a b c d e c. 7a + 5b 4. a b c d e d. 6a + 6b 5. a b c d e 6. e. 8a + 6b a b c d e 7. a b c d e 8. a b c d e 4. Jossie has $5 more than Siobhan, and Siobhan has $3 less than Michael. If Michael has $30, how 1. On ﬁve successive days, a motorcyclist listed his much money does Jossie have? mileage as follows: 135, 162, 98, 117, 216. a. $30 If his motorcycle averages 14 miles for each b. $27 gallon of gas used, how many gallons of gas did c. $32 he use during these ﬁve days? d. $36 a. 42 e. Not enough information is given. b. 52 c. 115 d. 147 e. 153 2. Bugsy has a piece of wood 9 feet 8 inches long. He wishes to cut it into 4 equal lengths. How far from the edge should he make the ﬁrst cut? a. 2.5 ft. b. 2 ft 5 in. c. 2.9 ft. d. 29 ft. e. 116 in. 380
– THE GED MATHEMATICS EXAM – Questions 5 and 6 are based on the following graph. 8. Mr. DeLandro earns $12 per hour. One week, Mr. DeLandro worked 42 hours; the following week, he worked 37 hours. Which of the Personal Service following indicates the number of dollars Mr. 12% DeLandro earned for 2 weeks? Manufacturing a. 12 × 2 + 37 All Others There are 180,000 33% 17% employees total. b. 12 × 42 + 42 × 37 c. 12 × 37 + 42 Trade and d. 12 + 42 × 37 Finance Food 25% Service e. 12(42 + 37) 5% Professional 9. What is the slope of the line that passes through 8% points A and B on the coordinate graph below? 5. The number of persons engaged in Food Service Mark your answer in the circles in the grid in the city during this period was below. a. 900. b. 9,000. y c. 14,400. 5 B (3,5) 4 d. 36,000. 3 e. 90,000. A (1,3) 2 1 6. If the number of persons in trade and ﬁnance is x −5 −4 −3 −2 −1 represented by M, then the approximate number 12345 −1 in manufacturing is represented as −2 −3 a. M5 −4 b. M + 3 −5 c. 30M d. 43 M e. Not enough information is given. / / / Question 7 is based on the following ﬁgure. • • • • • A B E 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 C D 4 4 4 4 4 In the ﬁgure AB | | CD, CE bisects ∠BCD, and 7. 5 5 5 5 5 m∠ABC = 112°. Find m∠ECD. 6 6 6 6 6 a. 45° b. 50° 7 7 7 7 7 c. 56° 8 8 8 8 8 d. 60° 9 9 9 9 9 e. Not enough information is given. 381
– THE GED MATHEMATICS EXAM – What is the value of the expression 3(2x − y) + 10. 3b (3 + x)2, when x = 4 and y = 5? Mark your a+ answer in the circles on the grid below. 3a + b / / / 2a + b • • • • • 0 0 0 0 1 1 1 1 1 3a + 2b 2 2 2 2 2 3. b. To ﬁnd the perimeter of the ﬁgure, ﬁnd the sum 3 3 3 3 3 of the lengths of the four sides: 2a + b + a + 3b 4 4 4 4 4 + 3a + b + 3a + 2b = 9b + 7b. 5 5 5 5 5 4. c. Michael has $30. Siobhan has $30 − $3 = $27. 6 6 6 6 6 Jossie has $27 + $5 = $32. 7 7 7 7 7 8 8 8 8 8 Personal 9 9 9 9 9 Service 12% Manufacturing All Others There are 180,000 Pretest Answers and Explanations 33% 17% employees total. 1. b. First, ﬁnd the total mileage; 135 + 162 + 98 + Trade and Finance 117 + 216 = 728 miles. Divide the total mileage Food 25% Service (728) by the number of miles covered for each 5% gallon of gas used (14) to ﬁnd the number of Professional gallons of gas needed; 728 ÷ 14 = 52 gallons. 8% 5. b. To ﬁnd 5% of a number, multiply the number 2. b. 1 ft. = 12 in. 9 ft. 8 in. = 9 × 12 + 8 = 116 in.; by .05: 180,000 × .05 = 9,000. There are 9,000 116 ÷ 4 = 29 in. = 2 ft. 5 in. food service workers in the city. 6. d. M = number of persons in trade and ﬁnance. Since M = 25% of the total, 4M = total number of city workers. Number of persons in manufac- turing = total number of workers = 43 . M 3 382
– THE GED MATHEMATICS EXAM – 9.1. A B E y 5 B (3,5) 4 C D 3 A (1,3) 2 7. c. Since pairs of alternate interior angles of parallel 1 lines have equal measures, m∠BCD = m∠ABC. x Thus, m∠BCD = 112°. −5 −4 −3 −2 −1 12345 −1 −2 m∠ECD = 1 m∠BCD = 1 (112) = 56° −3 2 2 −4 −5 8. e. In two weeks, Mr. Delandro worked a total of (42 + 37) hours and earned $12 for each hour. Therefore, the total number of dollars he earned was 12(42 + 37). 1 / / / • • • • • 0 0 0 0 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 The coordinates of point A are (1,3). The coordinates of point B are (3,5). Use the slope formula: y2 − y1 x2 − x1 Substitute and solve: 5−3 = 2 , or 1 =1 3−1 2 1 383
– THE GED MATHEMATICS EXAM – 10. 58. Pretest Assessment How did you do on the math pretest? If you answered 58 seven or more questions correctly, you have earned the equivalent of a passing score on the GED Mathematics Test. But remember that this pretest covers only a frac- / / / tion of the material you might face on the GED exam. It • • • • • is not designed to give you an accurate measure of how you would do on the actual test. Rather, it is designed to 0 0 0 0 help you determine where to focus your study efforts. 1 1 1 1 1 For success on the GED, review all of the chapters in this 2 2 2 2 2 section thoroughly. Focus on the sections that corre- 3 3 3 3 3 spond to the pretest questions you answered incorrectly. 4 4 4 4 4 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 9 9 9 9 9 3(2x − y) + (3 + x)2, x = 4 and y = 5. 3(2 × 4 − 5) + (3 + 4)2 = 3(8 − 5) + (7)2 = 3(3) + 49 = 9 + 49 = 58. 384
CHAPTER About the GED 40 Mathematics Exam IN THIS chapter, you will learn all about the GED Mathematics Exam, including the number and type of questions, the topics and skills that will be tested, guidelines for the use of calculators, and recent changes in the test. W hat to Expect on the GED Mathematics Exam The GED Mathematics Exam measures your understanding of the mathematical knowledge needed in everyday life. The questions are based on information presented in words, diagrams, charts, graphs, and pictures. In addi- tion to testing your math skills, you will also be asked to demonstrate your problem-solving skills. Examples of some of the skills needed for the mathematical portion of the GED are: understanding the question ■ organizing data and identifying important information ■ selecting problem-solving strategies ■ knowing when to use appropriate mathematical operations ■ setting up problems and estimating ■ computing the exact, correct answer ■ reﬂecting on the problem to ensure the answer you choose is reasonable ■ This section will give you lots of practice in the basic math skills that you use every day as well as crucial problem-solving strategies. 385
– ABOUT THE GED MATHEMATICS EXAM – The GED Mathematics Test is given in two separate Formula Page A page with a list of common formulas is provided with sections. The ﬁrst section permits the use of a calculator; all test forms. You are allowed to use this page when you the second does not. The time limit for the GED is 90 are taking the test. It is necessary for you to become minutes, meaning that you have 45 minutes to complete familiar with the formula page and to understand when each section. The sections are timed separately but and how to use each formula. An example of the formula weighted equally. This means that you must complete page is on page 388 of this book. both sections in one testing session to receive a passing grade. If only one section is completed, the entire test must be retaken. Gridded-Response and Set-Up The test contains 40 multiple-choice questions and Questions There are ten non-multiple-choice questions in the math ten gridded-response questions for a total of 50 ques- portion of the GED. These questions require you to ﬁnd tions overall. Multiple-choice questions give you several an answer and to ﬁll in circles on a grid or on a coordi- answers to choose from and gridded-response questions nate axis. ask you to come up with the answer yourself. Each multiple-choice question has ﬁve answer choices, a through e. Gridded response questions use a standard S TANDARD G RID - IN Q UESTIONS When you are given a question with a grid like the one grid or a coordinate plane grid. (The guidelines for below, keep these guidelines in mind: entering a gridded-response question will be covered later in this section.) First, write your answer in the blank boxes at the ■ top of the grid. This will help keep you organized Test Topics as you “grid in” the bubbles and ensure that you The math section of the GED tests you on the following ﬁll them out correctly. subjects: You can start in any column, but leave enough ■ columns for your whole answer. measurement and geometry ■ You do not have to use all of the columns. If your algebra, functions, and patterns ■ ■ answer only takes up two or three columns, leave number operations and number sense ■ the others blank. data analysis, statistics, and probability ■ You can write your answer by using either frac- ■ tions or decimals. For example, if your answer Each of these subjects is detailed in this section along is 1 , you can enter it either as a fraction or as a with tips and strategies for solving them. In addition, 100 4 decimal, .25. practice problems and their solutions are given at the end of the subject lessons. The slash “/” is used to signify the fraction bar of the fraction. The numerator should be bubbled to the left of Using Calculators the fraction bar and the denominator should be bubbled The GED Mathematics Test is given in two separate in to the right. See the example on the next page. booklets, Part I and Part II. The use of calculators is per- mitted on Part I only. You will not be allowed to use your own. The testing facility will provide a calculator for you. The calculator that will be used is the Casio fx-260. It is important for you to become familiar with this calcula- tor as well as how to use it. Use a calculator only when it will save you time or improve your accuracy. 386
– ABOUT THE GED MATHEMATICS EXAM – S ET-U P Q UESTIONS .25 These questions measure your ability to recognize the 4 /1 correct procedure for solving a problem. They ask you to / / / / / choose an expression that represents how to “set up” the problem rather than asking you to choose the correct • • • • • • • • • solution. About 25 percent of the questions on the GED Mathematics Test are set-up questions. 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 Example: Samantha makes $24,000 per year at a new 2 2 2 2 2 2 2 2 2 job. Which expression below shows how much 3 3 3 3 3 3 3 3 3 3 she earns per month? 4 4 4 4 4 4 4 4 4 a. $24,000 + 12 b. $24,000 − 12 5 5 5 5 5 5 5 5 5 c. $24,000 × 12 6 6 6 6 6 6 6 6 6 6 d. $24,000 ÷ 12 7 7 7 7 7 7 7 7 7 7 e. 12 ÷ $24,000 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 Answer: d. You know that there are 12 months in a year. To ﬁnd Samantha’s monthly income, you would divide the total ($24,000) by the number When your answer is a mixed number, it must be ■ of months (12). Option e is incorrect because it represented on the standard grid in the form of means 12 is divided by $24,000. an improper fraction. For example, for the answer 1 1 , grid in 5 . Graphics 4 4 When you are asked to plot a point on a coordi- ■ Many questions on the GED Mathematics Test use nate grid like the one below, simply ﬁll in the diagrams, pie charts, graphs, tables, and other visual bubble where the point should appear. stimuli as references. Sometimes, more than one of these questions will be grouped under a single graphic. Do not 6 let this confuse you. Learn to recognize question sets by 5 reading both the questions and the directions carefully. 4 What’s New for the GED? 3 The structure of the GED Mathematics Test, revised in 2 2002, ensures that no more than two questions should include “not enough information is given” as a correct 1 answer choice. Given this fact, it is important for you to −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 pay attention to how many times you select this answer −1 choice. If you ﬁnd yourself selecting the “not enough information is given” for the third time, be sure to check −2 the other questions for which you have selected this −3 choice because one of them must be incorrect. −4 The current GED has an increased focus on “math in everyday life.” This is emphasized by allowing the use of −5 a calculator on Part I as well as by an increased empha- −6 sis on data analysis and statistics. As a result, gridded- response questions and item sets are more common. The number of item sets varies. 387
Formulas Area of a: Area = side2 square rectangle Area = length width parallelogram Area = base height 1 triangle Area = base height 2 1 trapezoid Area = (base1 + base2) height 2 Area = π radius2; π is approximately equal to 3.14 circle Perimeter of a: square Perimeter = 4 side rectangle Perimeter = 2 length + 2 width triangle Perimeter = side1 + side2 + side3 Circumference = π diameter; π is approximately equal to 3.14 Circumference of a circle Volume of a: Volume = edge3 cube rectangular solid Volume = length width height 1 (base edge)2 square pyramid Volume = height 3 π height π is approximately equal to 3.14 radius2 cylinder 1 π height; π is approximately equal to 3.14 radius2 cone Volume = 3 (x2 – x1)2 + (y2 – y1)2; (x1,y1) and (x2,y2) are two points Coordinate Geometry distance between points = in a plane slope of a line = y2 – y1 ; (x1,y1) and (x2,y2) are two points on the line x2 – x1 Pythagorean Relationship a2 + b2 = c2; a and b are legs and c is the hypotenuse of a right triangle mean = x1 + x2 + . . . + xn , where the x's are the values for which a mean is desired, Measures of n Central Tendency and n is the total number of values for x. median = the middle value of an odd number of ordered scores, and halfway between the two middle values of an even number of ordered scores. Simple Interest interest = principal rate time Distance distance = rate time Total Cost total cost = (number of units) (price per unit) Adapted from ofﬁcial GED materials. 388
CHAPTER 41 Measurement and Geometry THE GED Mathematics Test emphasizes real-life applications of math concepts, and this is especially true of questions about meas- urement and geometry. This chapter will review the basics of meas- urement systems used in the United States and other countries, performing mathematical operations with units of measurement, and the process of converting between different units. It will also review geometry concepts you’ll need to know for the exam, such as prop- erties of angles, lines, polygons, triangles, and circles, as well as the formulas for area, volume, and perimeter. T measurement enables you to form a connection between mathematics and the real world. HE USE OF To measure any object, assign a unit of measure. For instance, when a ﬁsh is caught, it is often weighed in ounces and its length measured in inches. This lesson will help you become more familiar with the types, conversions, and units of measurement. Also required for the GED Mathematics Test is knowledge of fundamental, practical geometry. Geometry is the study of shapes and the relationships among them. A comprehensive review of geometry vocabulary and con- cepts, after this measurement lesson, will strengthen your grasp on geometry. 389
– MEASUREMENT AND GEOMETRY – T ypes of Measurements 5 feet = how many inches? 5 feet × 12 inches (the number of inches in a single foot) = 60 inches The types of measurements used most frequently in the Therefore, there are 60 inches in 5 feet. United States are listed below: Try another: Units of Length Change 3.5 tons to pounds. 12 inches (in.) = 1 foot (ft.) 3.5 tons = how many pounds? 3 feet = 36 inches = 1 yard (yd.) 3.5 tons × 2,000 pounds (the number of pounds in 5,280 feet = 1,760 yards = 1 mile (mi.) a single ton) = 6,500 pounds Therefore, there are 6,500 pounds in 3.5 tons. Units of Volume 8 ounces* (oz.) = 1 cup (c.) To change a smaller unit to a larger unit, simply ■ 2 cups = 16 ounces = 1 pint (pt.) divide the speciﬁc number of smaller units by the 2 pints = 4 cups = 32 ounces = 1 quart (qt.) number of smaller units in only one of the larger 4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon units. (gal.) For example, to ﬁnd the number of pints in 64 ounces, simply divide 64, the smaller unit, by 16, Units of Weight the number of ounces in one pint. 16 ounces* (oz.) = 1 pound (lb.) 2,000 pounds = 1 ton (T.) speciﬁc number of the smaller unit the number of smaller units in one larger unit Units of Time 64 ounces 60 seconds (sec.) = 1 minute (min.) 16 ounces = 4 pints 60 minutes = 1 hour (hr.) 24 hours = 1 day Therefore, 64 ounces are equal to four pints. 7 days = 1 week 52 weeks = 1 year (yr.) Here is one more: 12 months = 1 year 365 days = 1 year Change 24 ounces to pounds. 32 ounces *Notice that ounces are used to measure both the volume and 16 ounces = 2 pounds weight. Therefore, 32 ounces are equal to two pounds. C onverting Units B asic Operations with When performing mathematical operations, it is neces- Measurement sary to convert units of measure to simplify a problem. Units of measure are converted by using either multipli- It will be necessary for you to review how to add, sub- cation or division: tract, multiply, and divide with measurement. The mathematical rules needed for each of these operations To change a larger unit to a smaller unit, simply ■ with measurement follow. multiply the speciﬁc number of larger units by the number of smaller units that makes up one of Addition with Measurements the larger units. To add measurements, follow these two steps: For example, to ﬁnd the number of inches in 5 feet, simply multiply 5, the number of larger units, 1. Add like units. by 12, the number of inches in one foot: 2. Simplify the answer. 390
– MEASUREMENT AND GEOMETRY – Example: Add 4 pounds 5 ounces to 20 ounces. Multiplication with Measurements 1. Multiply like units. 4 lb. 5 oz. Be sure to add ounces to ounces. 2. Simplify the answer. + 20 oz. 4 lb. 25 oz. Because 25 ounces is more than 16 Example: Multiply 5 feet 7 inches by 3. ounces (1 pound), simplify by 5 ft. 7 in. Multiply 7 inches by 3, then multiply 5 dividing by 16. Then add the 1 ×3 feet by 3. Keep the units separate. pound to the 4 pounds. 15 ft. 21 in. Since 12 inches = 1 foot, simplify 21 inches. 4 lb. + 25 oz. 15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches = 16 feet 9 inches 1 lb. 4 lb. + 16 25 Example: Multiply 9 feet by 4 yards. −16 First, change yards to feet by multiplying the 9 oz. number of feet in a yard (3) by the number of 4 pounds 25 ounces = yards in this problem (4). 4 pounds + 1 pound 9 ounces = 3 feet in a yard × 4 yards = 12 feet 5 pounds 9 ounces Then, multiply 9 feet by 12 feet = Subtraction with Measurements 108 square feet. 1. Subtract like units. (Note: feet × feet = square feet) 2. Regroup units when necessary. 3. Write the answer in simplest form. Division with Measurements 1. Divide into the larger units ﬁrst. For example, to subtract 6 pounds 2 ounces 2. Convert the remainder to the smaller unit. from 9 pounds 10 ounces, 3. Add the converted remainder to the existing 9 lb. 10 oz. Subtract ounces from ounces. smaller unit if any. − 6 lb. 2 oz. Then, subtract pounds from pounds. 4. Then, divide into smaller units. 3 lb. 8 oz. 5. Write the answer in simplest form. Sometimes, it is necessary to regroup units when Example: subtracting. Divide 5 quarts 4 ounces by 4. Example: Subtract 3 yards 2 feet from 5 yards 1 1 qt. R1 First, divide 5 ounces foot. 1. 4 5 by 4, for a result of 1 −4 quart and a reminder 4 4 5 yd. 1 ft. 1 of one. − 3 yd. 2 ft. 2. R1 = 32 oz. Convert the remainder 1 yd. 2 ft. to the smaller unit (ounces). From 5 yards, regroup 1 yard to 3 feet. Add 3 3. 32 oz. + 4 oz. = 36 oz. Add the converted feet to 1 foot. Then subtract feet from feet and remainder to the yards from yards. existing smaller unit. 4. 9 oz. Now divide the smaller 4 36 units by 4. 5. 1 qt. 9 oz. 391
– MEASUREMENT AND GEOMETRY – M etric Measurements The chart shown here illustrates some common rela- tionships used in the metric system: The metric system is an international system of meas- Length Weight Volume urement also called the decimal system. Converting units in the metric system is much easier than converting 1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L units in the English system of measurement. However, 1 m = .001 km 1 g = .001 kg 1 L = .001 kL making conversions between the two systems is much 1 m = 100 cm 1 g = 100 cg 1 L = 100 cL more difﬁcult. Luckily, the GED test will provide you 1 cm = .01 m 1 cg = .01 g 1 cL = .01 L with the appropriate conversion factor when needed. 1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL The basic units of the metric system are the meter, 1mm = .001 m 1 mg = .001 g 1 mL = .001 L gram, and liter. Here is a general idea of how the two sys- tems compare: Conversions within the Metric System M ETRIC S YSTEM E NGLISH S YSTEM An easy way to do conversions with the metric system is 1 meter A meter is a little more than a to move the decimal point to either the right or the left yard; it is equal to about 39 inches. because the conversion factor is always ten or a power of 1 gram A gram is a very small unit of ten. As you learned previously, when you change from a weight; there are about 30 grams large unit to a smaller unit, you multiply, and when you in one ounce. change from a small unit to a larger unit, you divide. 1 liter A liter is a little more than a quart. Making Easy Conversions within Preﬁxes are attached to the basic metric units listed the Metric System above to indicate the amount of each unit. When you multiply by a power of ten, you move the dec- For example, the preﬁx deci means one-tenth ( 110 ); imal point to the right. When you divide by a power of therefore, one decigram is one-tenth of a gram, and one ten, you move the decimal point to the left. decimeter is one-tenth of a meter. The following six pre- To change from a large unit to a smaller unit, move ﬁxes can be used with every metric unit: the decimal point to the right. kilo hecto deka UNIT deci centi milli Kilo Hecto Deka Deci Centi Milli (k) (h) (dk) (d) (c) (m) To change from a small unit to a larger unit, move the 1 1 1 1,000 100 10 10 100 1,000 decimal point to the left. Examples: Example: ■ 1 hectometer = 1 hm = 100 meters Change 520 grams to kilograms. 1 ■ 1 millimeter = 1 mm = 1,000 meter = .001 meter Step 1: Be aware that changing meters to kilome- ■ 1 dekagram = 1 dkg = 10 grams ters is going from small units to larger units, and 1 ■ 1 centiliter = 1 cL* = 100 liter = .01 liter thus, you will move the decimal point three places to the left. ■ 1 kilogram = 1 kg = 1,000 grams 1 ■ 1 deciliter = 1 dL* = 10 liter = .1 liter Step 2: Beginning at the UNIT (for grams), you *Notice that liter is abbreviated with a capital letter—“L.” need to move three preﬁxes to the left. k h dk unit dc m 392
– MEASUREMENT AND GEOMETRY – Step 3: Move the decimal point from the G eometr y end of 520 to the left three places. 520. As previously deﬁned, geometry is the study of shapes and the relationships among them. Basic concepts in Place the decimal point before the 5. .520 geometry will be detailed and applied in this section. The Your answer is 520 grams = .520 kilograms. study of geometry always begins with a look at basic vocabulary and concepts. Therefore, here is a list of def- Example: initions of important terms: You are packing your bicycle for a trip from New York City to Detroit. The rack on the back area—the space inside a two-dimensional ﬁgure of your bike can hold 20 kilograms. If you bisect—cut in two equal parts exceed that limit, you must buy stabilizers for circumference—the distance around a circle the rack that cost $2.80 each. Each stabilizer can diameter—a line segment that goes directly through hold an additional kilogram. If you want to pack the center of a circle—the longest line you can 23,000 grams of supplies, how much money will draw in a circle you have to spend on the stabilizers? equidistant—exactly in the middle of hypotenuse—the longest leg of a right triangle, Step 1: First, change 23,000 grams to kilograms. always opposite the right angle line—an inﬁnite collection of points in a straight kg hg dkg g dg cg mg path Step 2: Move the decimal point three places to the point—a location in space left. parallel—lines in the same plane that will never 23,000 g = 23.000 kg = 23 kg intersect perimeter—the distance around a ﬁgure Step 3: Subtract to ﬁnd the amount over the limit. 23 kg − 20 kg = 3 kg perpendicular—two lines that intersect to form 90- degree angles Step 4: Because each stabilizer holds one kilogram quadrilateral—any four-sided closed ﬁgure and your supplies exceed the weight limit of the radius—a line from the center of a circle to a point rack by three kilograms, you must purchase three on the circle (half of the diameter) stabilizers from the bike store. volume—the space inside a three-dimensional ﬁgure Step 5: Each stabilizer costs $2.80, so multiply $2.80 by 3: $2.80 × 3 = $8.40. 393
– MEASUREMENT AND GEOMETRY – A ngles An acute angle is an angle that measures less than ■ 90 degrees. An angle is formed by an endpoint, or vertex, and two rays. Acute Angle y ra A right angle is an angle that measures exactly 90 ■ ray degrees. A right angle is represented by a square at the vertex. Endpoint (or Vertex) Naming Angles There are three ways to name an angle. Right Angle B An obtuse angle is an angle that measures more ■ D than 90 degrees, but less than 180 degrees. 1 2 A C Obtuse Angle 1. An angle can be named by the vertex when no other angles share the same vertex: ∠A. 2. An angle can be represented by a number written across from the vertex: ∠1. 3. When more than one angle has the same vertex, A straight angle is an angle that measures 180 ■ three letters are used, with the vertex always degrees. Thus, its sides form a straight line. being the middle letter: –1 can be written as ∠BAD or as ∠DAB; –2 can be written as ∠DAC Straight Angle or as ∠CAD. 180° Classifying Angles Angles can be classiﬁed into the following categories: acute, right, obtuse, and straight. 394

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