 What 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. CHAPTER About the GED 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. 40 385 The GED Mathematics Test is given in two separate sections. The first section permits the use of a calculator; the second does not. The time limit for the GED is 90 minutes, meaning that you have 45 minutes to complete each section. The sections are timed separately but weighted equally. This means that you must complete both sections in one testing session to receive a passing grade. If only one section is completed, the entire test must be retaken. The test contains 40 multiple-choice questions and ten gridded-response questions for a total of 50 ques- tions overall. Multiple-choice questions give you several answers to choose from and gridded-response questions 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 grid or a coordinate plane grid. (The guidelines for entering a gridded-response question will be covered later in this section.) Test Topics The math section of the GED tests you on the following subjects: ■ measurement and geometry ■ algebra, functions, and patterns ■ number operations and number sense ■ data analysis, statistics, and probability Each of these subjects is detailed in this section along with tips and strategies for solving them. In addition, 100 practice problems and their solutions are given at the end of the subject lessons. Using Calculators The GED Mathematics Test is given in two separate 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. Formula Page A page with a list of common formulas is provided with all test forms. You are allowed to use this page when you are taking the test. It is necessary for you to become familiar with the formula page and to understand when and how to use each formula. An example of the formula page is on page 388 of this book. Gridded-Response and Set-Up Questions There are ten non-multiple-choice questions in the math portion of the GED. These questions require you to find an answer and to fill in circles on a grid or on a coordi- nate axis. S TANDARD G RID - IN Q UESTIONS When you are given a question with a grid like the one below, keep these guidelines in mind: ■ First, write your answer in the blank boxes at the top of the grid. This will help keep you organized as you “grid in” the bubbles and ensure that you fill them out correctly. ■ You can start in any column, but leave enough columns for your whole answer. ■ You do not have to use all of the columns. If your answer only takes up two or three columns, leave the others blank. ■ You can write your answer by using either frac- tions or decimals. For example, if your answer is ᎏ 1 4 ᎏ , you can enter it either as a fraction or as a decimal, .25. The slash “/” is used to signify the fraction bar of the fraction. The numerator should be bubbled to the left of the fraction bar and the denominator should be bubbled in to the right. See the example on the next page. – ABOUT THE GED MATHEMATICS EXAM – 386 ■ When your answer is a mixed number, it must be represented on the standard grid in the form of an improper fraction. For example, for the answer 1 ᎏ 1 4 ᎏ , grid in ᎏ 5 4 ᎏ . ■ When you are asked to plot a point on a coordi- nate grid like the one below, simply fill in the bubble where the point should appear. S ET -U P Q UESTIONS These questions measure your ability to recognize the 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. Example: Samantha makes $24,000 per year at a new job. Which expression below shows how much she earns per month? a. $24,000 + 12 b. $24,000 − 12 c. $24,000 × 12 d. $24,000 ÷ 12 e. 12 ÷ $24,000 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 of months (12). Option e is incorrect because it means 12 is divided by $24,000. Graphics Many questions on the GED Mathematics Test use diagrams, pie charts, graphs, tables, and other visual stimuli as references. Sometimes, more than one of these questions will be grouped under a single graphic. Do not let this confuse you. Learn to recognize question sets by reading both the questions and the directions carefully. What’s New for the GED? The structure of the GED Mathematics Test, revised in 2002, ensures that no more than two questions should include “not enough information is given” as a correct answer choice. Given this fact, it is important for you to pay attention to how many times you select this answer choice. If you find yourself selecting the “not enough information is given” for the third time, be sure to check the other questions for which you have selected this choice because one of them must be incorrect. The current GED has an increased focus on “math in everyday life.” This is emphasized by allowing the use of a calculator on Part I as well as by an increased empha- sis on data analysis and statistics. As a result, gridded- response questions and item sets are more common. The number of item sets varies. 1 2 −3 4 −5 −6 0 1 −2 3 4 5 6 −1 2 3 −4 5 −6 −1 −2−3−4 −5 6 1 2 3 4 5 6 7 8 9 • 1 2 3 4 5 6 7 8 9 0 • / 1 2 3 4 55 6 7 8 9 0 / 1 3 4 6 7 88 9 0 • / 1 2 3 4 6 7 9 0 • 2 . 5 1 2 3 4 5 6 7 8 9 • 1 2 3 4 5 6 7 8 9 0 • / 2 3 4 5 6 7 8 9 0 • / 1 2 3 4 5 6 7 8 9 0 • 1 2 3 5 6 7 8 9 0 • 1 / 4 – ABOUT THE GED MATHEMATICS EXAM – 387 Area of a: square Area = side 2 rectangle Area = length ϫ width parallelogram Area = base ϫ height triangle Area = ᎏ 1 2 ᎏ ϫ base ϫ height trapezoid Area = ᎏ 1 2 ᎏ ϫ (base 1 + base 2 ) ϫ height circle Area = π ϫ radius 2 ; π is approximately equal to 3.14 Perimeter of a: square Perimeter = 4 ϫ side rectangle Perimeter = 2 ϫ length + 2 ϫ width triangle Perimeter = side 1 + side 2 + side 3 Circumference of a circle Circumference = π ϫ diameter; π is approximately equal to 3.14 Volume of a: cube Volume = edge 3 rectangular solid Volume = length ϫ width ϫ height square pyramid Volume = ᎏ 1 3 ᎏ ϫ (base edge) 2 ϫ height cylinder π ϫ radius 2 ϫ height π is approximately equal to 3.14 cone Volume = ᎏ 1 3 ᎏ ϫ π ϫ radius 2 ϫ height; π is approximately equal to 3.14 Coordinate Geometry distance between points = ͙(x 2 – x ෆ 1 ) 2 + (y ෆ 2 – y 1 ) ෆ 2 ෆ ; (x 1 ,y 1 ) and (x 2 ,y 2 ) are two points in a plane slope of a line = ᎏ y x 2 2 – – y x 1 1 ᎏ ; (x 1 ,y 1 ) and (x 2 ,y 2 ) are two points on the line Pythagorean Relationship a 2 + b 2 = c 2 ; a and b are legs and c is the hypotenuse of a right triangle Measures of mean = ᎏ x 1 + x 2 + n .+x n ᎏ , where the x's are the values for which a mean is desired, 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 ϫ rate ϫ time Distance distance = rate ϫ time Total Cost total cost = (number of units) ϫ (price per unit) Adapted from official GED materials. 388 Formulas T HE USE OF measurement enables you to form a connection between mathematics and the real world. 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. CHAPTER 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. 41 389  Types of Measurements The types of measurements used most frequently in the United States are listed below: Units of Length 12 inches (in.) = 1 foot (ft.) 3 feet = 36 inches = 1 yard (yd.) 5,280 feet = 1,760 yards = 1 mile (mi.) Units of Volume 8 ounces* (oz.) = 1 cup (c.) 2 cups = 16 ounces = 1 pint (pt.) 2 pints = 4 cups = 32 ounces = 1 quart (qt.) 4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon (gal.) Units of Weight 16 ounces* (oz.) = 1 pound (lb.) 2,000 pounds = 1 ton (T.) Units of Time 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day 7 days = 1 week 52 weeks = 1 year (yr.) 12 months = 1 year 365 days = 1 year *Notice that ounces are used to measure both the volume and weight.  Converting Units When performing mathematical operations, it is neces- sary to convert units of measure to simplify a problem. Units of measure are converted by using either multipli- cation or division: ■ To change a larger unit to a smaller unit, simply multiply the specific number of larger units by the number of smaller units that makes up one of the larger units. For example, to find the number of inches in 5 feet, simply multiply 5, the number of larger units, by 12, the number of inches in one foot: 5 feet = how many inches? 5 feet × 12 inches (the number of inches in a single foot) = 60 inches Therefore, there are 60 inches in 5 feet. Try another: Change 3.5 tons to pounds. 3.5 tons = how many pounds? 3.5 tons × 2,000 pounds (the number of pounds in a single ton) = 6,500 pounds Therefore, there are 6,500 pounds in 3.5 tons. ■ To change a smaller unit to a larger unit, simply divide the specific number of smaller units by the number of smaller units in only one of the larger units. For example, to find the number of pints in 64 ounces, simply divide 64, the smaller unit, by 16, the number of ounces in one pint. = 4 pints Therefore, 64 ounces are equal to four pints. Here is one more: Change 24 ounces to pounds. = 2 pounds Therefore, 32 ounces are equal to two pounds.  Basic Operations with Measurement It will be necessary for you to review how to add, sub- tract, multiply, and divide with measurement. The mathematical rules needed for each of these operations with measurement follow. Addition with Measurements To add measurements, follow these two steps: 1. Add like units. 2. Simplify the answer. 32 ounces ᎏᎏ 16 ounces 64 ounces ᎏᎏ 16 ounces specific number of the smaller unit ᎏᎏᎏᎏᎏ the number of smaller units in one larger unit – MEASUREMENT AND GEOMETRY – 390 Example: Add 4 pounds 5 ounces to 20 ounces. 4 lb. 5 oz. Be sure to add ounces to ounces. + 20 oz. 4 lb. 25 oz. Because 25 ounces is more than 16 ounces (1 pound), simplify by dividing by 16. Then add the 1 pound to the 4 pounds.  4 lb. + 25 oz.  1 lb. 4 lb. + 16ͤ25 ෆ −16 9 oz. 4 pounds 25 ounces = 4 pounds + 1 pound 9 ounces = 5 pounds 9 ounces Subtraction with Measurements 1. Subtract like units. 2. Regroup units when necessary. 3. Write the answer in simplest form. For example, to subtract 6 pounds 2 ounces from 9 pounds 10 ounces, 9 lb. 10 oz. Subtract ounces from ounces. − 6 lb. 2 oz. Then, subtract pounds from pounds. 3 lb. 8 oz. Sometimes, it is necessary to regroup units when subtracting. Example: Subtract 3 yards 2 feet from 5 yards 1 foot. 5 4 ΋ yd. 1 4 ΋ ft. − 3 yd. 2 ft. 1 yd. 2 ft. From 5 yards, regroup 1 yard to 3 feet. Add 3 feet to 1 foot. Then subtract feet from feet and yards from yards. Multiplication with Measurements 1. Multiply like units. 2. Simplify the answer. Example: Multiply 5 feet 7 inches by 3. 5 ft. 7 in. Multiply 7 inches by 3, then multiply 5 × 3 feet by 3. Keep the units separate. 15 ft. 21 in. Since 12 inches = 1 foot, simplify 21 inches. 15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches = 16 feet 9 inches Example: Multiply 9 feet by 4 yards. First, change yards to feet by multiplying the number of feet in a yard (3) by the number of yards in this problem (4). 3 feet in a yard × 4 yards = 12 feet Then, multiply 9 feet by 12 feet = 108 square feet. (Note: feet × feet = square feet) Division with Measurements 1. Divide into the larger units first. 2. Convert the remainder to the smaller unit. 3. Add the converted remainder to the existing smaller unit if any. 4. Then, divide into smaller units. 5. Write the answer in simplest form. Example: Divide 5 quarts 4 ounces by 4. 1 qt. R1 First, divide 5 ounces 1. 4ͤ5 ෆ ෆ by 4, for a result of 1 −4 quart and a reminder 1 of one. 2. R1 = 32 oz. Convert the remainder to the smaller unit (ounces). 3. 32 oz. + 4 oz. = 36 oz. Add the converted remainder to the existing smaller unit. 4. 9 oz. Now divide the smaller 4ͤ36 ෆ units by 4. 5. 1 qt. 9 oz. – MEASUREMENT AND GEOMETRY – 391  Metric Measurements The metric system is an international system of meas- urement also called the decimal system. Converting units in the metric system is much easier than converting units in the English system of measurement. However, making conversions between the two systems is much more difficult. Luckily, the GED test will provide you with the appropriate conversion factor when needed. The basic units of the metric system are the meter, gram, and liter. Here is a general idea of how the two sys- tems compare: M ETRIC S YSTEM E NGLISH S YSTEM 1 meter A meter is a little more than a yard; it is equal to about 39 inches. 1 gram A gram is a very small unit of weight; there are about 30 grams in one ounce. 1 liter A liter is a little more than a quart. Prefixes are attached to the basic metric units listed above to indicate the amount of each unit. For example, the prefix deci means one-tenth ( ᎏ 1 1 0 ᎏ ); therefore, one decigram is one-tenth of a gram, and one decimeter is one-tenth of a meter. The following six pre- fixes can be used with every metric unit: Kilo Hecto Deka Deci Centi Milli (k) (h) (dk) (d) (c) (m) 1,000 100 10 ᎏ 1 1 0 ᎏ ᎏ 1 1 00 ᎏ ᎏ 1,0 1 00 ᎏ Examples: ■ 1 hectometer = 1 hm = 100 meters ■ 1 millimeter = 1 mm = ᎏ 1,0 1 00 ᎏ meter = .001 meter ■ 1 dekagram = 1 dkg = 10 grams ■ 1 centiliter = 1 cL* = ᎏ 1 1 00 ᎏ liter = .01 liter ■ 1 kilogram = 1 kg = 1,000 grams ■ 1 deciliter = 1 dL* = ᎏ 1 1 0 ᎏ liter = .1 liter *Notice that liter is abbreviated with a capital letter—“L.” The chart shown here illustrates some common rela- tionships used in the metric system: Length Weight Volume 1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L 1 m = .001 km 1 g = .001 kg 1 L = .001 kL 1 m = 100 cm 1 g = 100 cg 1 L = 100 cL 1 cm = .01 m 1 cg = .01 g 1 cL = .01 L 1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL 1mm = .001 m 1 mg = .001 g 1 mL = .001 L Conversions within the Metric System An easy way to do conversions with the metric system is to move the decimal point to either the right or the left because the conversion factor is always ten or a power of ten. As you learned previously, when you change from a large unit to a smaller unit, you multiply, and when you change from a small unit to a larger unit, you divide. Making Easy Conversions within the Metric System When you multiply by a power of ten, you move the dec- imal point to the right. When you divide by a power of ten, you move the decimal point to the left. To change from a large unit to a smaller unit, move the decimal point to the right. kilo hecto deka UNIT deci centi milli To change from a small unit to a larger unit, move the decimal point to the left. Example: Change 520 grams to kilograms. Step 1: Be aware that changing meters to kilome- ters is going from small units to larger units, and thus, you will move the decimal point three places to the left. Step 2: Beginning at the UNIT (for grams), you need to move three prefixes to the left. یی ی k h dk unit d c m – MEASUREMENT AND GEOMETRY – 392 Step 3: Move the decimal point from the end of 520 to the left three places. 520.  Place the decimal point before the 5. .520 Your answer is 520 grams = .520 kilograms. Example: You are packing your bicycle for a trip from New York City to Detroit. The rack on the back of your bike can hold 20 kilograms. If you exceed that limit, you must buy stabilizers for the rack that cost $2.80 each. Each stabilizer can hold an additional kilogram. If you want to pack 23,000 grams of supplies, how much money will you have to spend on the stabilizers? Step 1: First, change 23,000 grams to kilograms. یی ی kg hg dkg g dg cg mg Step 2: Move the decimal point three places to the left. 23,000 g = 23.000 kg = 23 kg Step 3: Subtract to find the amount over the limit. 23 kg − 20 kg = 3 kg Step 4: Because each stabilizer holds one kilogram and your supplies exceed the weight limit of the rack by three kilograms, you must purchase three stabilizers from the bike store. Step 5: Each stabilizer costs $2.80, so multiply $2.80 by 3: $2.80 × 3 = $8.40.  Geometry As previously defined, geometry is the study of shapes and the relationships among them. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and concepts. Therefore, here is a list of def- initions of important terms: area—the space inside a two-dimensional figure bisect—cut in two equal parts circumference—the distance around a circle diameter—a line segment that goes directly through the center of a circle—the longest line you can draw in a circle equidistant—exactly in the middle of hypotenuse—the longest leg of a right triangle, always opposite the right angle line—an infinite collection of points in a straight path point—a location in space parallel—lines in the same plane that will never intersect perimeter—the distance around a figure perpendicular—two lines that intersect to form 90- degree angles quadrilateral—any four-sided closed figure radius—a line from the center of a circle to a point on the circle (half of the diameter) volume—the space inside a three-dimensional figure – MEASUREMENT AND GEOMETRY – 393  Angles An angle is formed by an endpoint, or vertex, and two rays. Naming Angles There are three ways to name an 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, three letters are used, with the vertex always being the middle letter: –1 can be written as ∠BAD or as ∠DAB; –2 can be written as ∠DAC or as ∠CAD. Classifying Angles Angles can be classified into the following categories: acute, right, obtuse, and straight. ■ An acute angle is an angle that measures less than 90 degrees. ■ A right angle is an angle that measures exactly 90 degrees. A right angle is represented by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees. ■ A straight angle is an angle that measures 180 degrees. Thus, its sides form a straight line. Straight Angle 180° Obtuse Angle Right Angle Acute Angle 1 2 A C D B Endpoint (or Vertex) ray ray – MEASUREMENT AND GEOMETRY – 394 [...]... In a right triangle with the other angles measuring 30 and 60 degrees: ■ ■ The leg opposite the 30-degree angle is half the length of the hypotenuse (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.) The leg opposite the 60-degree angle is ͙3 times ෆ the length of the other leg 60° s 2s 30° s√¯¯¯ 3 398 – MEASUREMENT AND GEOMETRY – Example D A 60° x 7 B 30°... states: a2 + b2 = c2, where a and b represent the legs and c represents the hypotenuse This theorem allows you to find the length of any side as along as you know the measure of the other two ■ The length of the hypotenuse is ͙2 multiplied by ෆ the length of one of the legs of the triangle ͙ෆ ᎏ The length of each leg is ᎏ2 multiplied by the 2 length of the hypotenuse 10 x c 2 y 1 a2 b2 x=y= c2 + = 12... the surface area of one of its sides by six Graphing Ordered Pairs The x-coordinate ■ The x-coordinate is listed first in the ordered pair and it tells you how many units to move to either the left or to the right If the x-coordinate is positive, move to the right If the x-coordinate is negative, move to the left The y-coordinate 4 ■ 4 Surface area of front side = 16 Therefore, the surface area of the. .. Find the slope of a line containing the points (3,2) and (8,9) (8,9) (3,2) Solution: 9−2 7 ᎏ ᎏ = ᎏᎏ 8−3 5 Therefore, the slope of the line is ᎏ7ᎏ 5 Note: If you know the slope and at least one point on a line, you can find the coordinates of other points on the line Simply move the required units determined by the slope In the last example, from (8,9), given the slope ᎏ7ᎏ, 5 move up seven units and to the. .. ᎏ2ᎏ = 6 Therefore the midpoint of ៮៮៮ is (3,6) AB – MEASUREMENT AND GEOMETRY – Slope I MPORTANT I NFORMATION The slope of a line measures its steepness It is found by writing the change in the y-coordinates of any two points on the line, over the change of the corresponding x-coordinates (This is also known as the rise over the run.) The last step is to simplify the fraction that results ■ ■ ■ ■ Example... point on the coordinate plane with the first number, or coordinate, representing the horizontal placement and the second number, or coordinate, representing the vertical placement Coordinate points are given in the form of (x,y) 4 Surface Area The surface area of an object measures the area of each of its faces The total surface area of a rectangular solid is double the sum of the areas of the three... length of a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points or by counting the spaces on the graph between them Example Find the lengths of ៮៮៮ and line ៮៮៮ AB BC (7,5) C (2,1) A B Solution: | 2 − 7 | = 5 = ៮៮៮ AB ៮៮៮ | 1 − 5 | = 4 = BC Midpoint To find the midpoint of a segment, use the following formula: x +x 1 2 Midpoint x = ᎏ2ᎏ y +y 1 2 Midpoint... whether this is enough information to prove the triangles are congruent Yes, two angles and the side between them are equal Using the ASA rule, you can determine that triangle ABD is congruent to triangle CBD Comparing Triangles Triangles are said to be congruent (indicated by the symbol Х) when they have exactly the same size and shape Two triangles are congruent if their corresponding parts (their... that the triangles are congruent If two triangles are congruent, one of the three criteria listed below must be satisfied Polygons and Parallelograms A polygon is a closed figure with three or more sides B Side-Side-Side (SSS) The side measures for both triangles are the same Side-Angle-Side (SAS) The sides and the angle between them are the same Angle-Side-Angle (ASA) Two angles and the side between them... 9 These two polygons are similar because their angles are equal and the ratios of the corresponding sides are in proportion m∠a + m∠b + m∠c + m∠d = 360° Interior Angles To find the sum of the interior angles of any polygon, use this formula: Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides S = 180(x − 2)°, with x being the number of polygon sides B C Example Find the . Expect on the GED Mathematics Exam The GED Mathematics Exam measures your understanding of the mathematical knowledge needed in everyday life. The questions. 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