About the GED Mathematics Exam

C H A P T E R 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 wi

 W h a t t o E x p e c t o n t h e G E D M a t h e m a t i c s E x a m

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

C H A P T E R

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

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3 8 5

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 quesques-tions 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 calculacalcula-tor 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 14, 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

■ 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 114, grid in 54

■ 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 analyempha-sis and statistics As a result, gridded-response questions and item sets are more common The number of item sets varies

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1 2 3 4 5 6 7 8 9

0

• /

1 2 3 4

6 7 8 9

0 /

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3 4

6 7

9

0

• /

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9

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1 / 4

– A B O U T T H E G E D M AT H E M AT I C S E X A M –

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Area of a:

square Area = side2

rectangle Area = length  width

parallelogram Area = base  height

triangle Area = 12 base  height

trapezoid Area = 12 (base1+ base2)  height

circle Area = π  radius2; π is approximately equal to 3.14

Perimeter of a:

square Perimeter = 4  side

rectangle Perimeter = 2  length + 2  width

triangle Perimeter = side1+ side2+ side3

Circumference of a circle Circumference = π  diameter; π is approximately equal to 3.14

Volume of a:

rectangular solid Volume = length  width  height

square pyramid Volume = 13 (base edge)2 height

cylinder π  radius2 height π is approximately equal to 3.14

cone Volume = 13 π  radius2 height; π is approximately equal to 3.14

Coordinate Geometry distance between points = (x2– x1)2+ (y2– y1)2; (x1,y1) and (x2,y2) are two points

in a plane slope of a line = y x2

2

– –

y x

1 1

; (x1,y1) and (x2,y2) are two points on the line

Pythagorean Relationship a2+ b2= c2; a and b are legs and c is the hypotenuse of a right triangle

Measures of mean = x1+ x2 +

n + x n

, 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  rate  time

Total Cost total cost = (number of units)  (price per unit)

Adapted from official GED materials

TH E U S E O F 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

C H A P T E R

Measurement and Geometry

THE GED Mathematics Test emphasizes real-life applications of

math concepts, and this is especially true of questions about 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

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 Ty p e s o f M e a s u r e m e n t s

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.

 C o n v e r t i n g U n i t s

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

 B a s i c O p e r a t i o n s w i t h

M e a s u r e m e n t

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

16 ounces

64 ounces

16 ounces

specific number of the smaller unit

 the number of smaller units in one larger unit

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 + 1625

−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

54

 yd 1 ft.4

− 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 45 by 4, for a result of 1

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

5 1 qt 9 oz

– M E A S U R E M E N T A N D G E O M E T R Y –

3 9 1

 M e t r i c M e a s u r e m e n t s

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 (110);

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

1,000 100 10 110 1100 1,0100

Examples:

■ 1 hectometer = 1 hm = 100 meters

■ 1 millimeter = 1 mm = 1,0100meter =

.001 meter

■ 1 dekagram = 1 dkg = 10 grams

■ 1 centiliter = 1 cL* = 1100liter = 01 liter

■ 1 kilogram = 1 kg = 1,000 grams

■ 1 deciliter = 1 dL* = 110liter = 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:

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

  

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.

  

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

 G e o m e t r y

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

– M E A S U R E M E N T A N D G E O M E T R Y –

3 9 3

 A n g l e s

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

D B

Endpoint (or Vertex)

ray ray