Ged Basics in Mathematics

GED

BASICS IN MATHEMATICS

Henry R. Varela

© Copyright 2004 Henry R. Varela. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author.

Printed in Victoria, Canada.

Note for Librarians: a cataloguing record for this book that includes Dewey Classif cation and US Library of Congress numbers is avail-able from the National Library of Canada. The complete cataloguing record can be obtained from the National Library˜s online database at: www.nlc-bnc.ca/amicus/index-e.html

ISBN: 1-4120-1681-9

___________________________________________________________

This book was published on-demand in cooperation with Trafford Publishing. On-demand publishing is a unique process and service of making a book available for retail sale to the public taking advantage of on-demand manufacturing and Internet marketing. On-demand publishing includes promotions, retail sales, manufacturing, order fulf lment, accounting and collect-ing royalties on behalf of the author.

Suite 6E, 2333 Government St., Victoria, B.C.  V8T 4P4, CANADA

Phone     250-383-6864     Toll-free    1-888-232-4444 (Canada & US)

Fax         250-383-6804     E-mail         [email protected]

Web site  www.trafford.com                   trafford publishing is a division of trafford holdings, ltd.

Trafford Catalogue # 03-2058                 www.trafford.com/robots/03-2058.html

Contents

ACKNOWLEDGEMENTS

INTRODUCTION

1.0… Decimals

Calculators

Reading Decimals

Exercise 1:  Reading and Comparing Decimals

Rounding  Decimals

Adding and Subtracting Decimals

Adding and Subtracting Decimals

Exercise 3:  Problems of Decimal Adding and Subtracting.

Multiplying and Dividing Decimals

Multiplying  and  Dividing  Decimals

Multiplying and Dividing Decimals

Multiplying and Dividing Decimals

Multiplying and Dividing Decimals

Use of a Standard Grid

Exercise 4: Problems of Multiplication and Division

Exercise 4: Problems of Multiplication and Division

Answers to Chapter 1.0 …Decimals

Answers to Chapter 1.0

2.0 ... Fractions

Equivalent Fractions

Mixed Numbers and Improper Fractions

Representing Fractions and Decimals

Relating Fractions and Decimals

Relating Fractions to Fractions

The Least Common Denominator

Exercise 3: Converting Fractions and Decimals

Adding and Subtracting Fractions

Exercise 6: Adding and Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Rounding Fractions

Fractions in a Standard Grid

Exercise 7: Multiplying  and  Dividing  Fractions

Answers to Chapter 2.0 …Fractions

Answers to Chapter 2.0

Answers to Chapter 2.0

3.0 … Powers, Roots, and Scientific Notation

Exercise 1: Powers (Exponents)

Order of Operations

Exercise 2: Order of Operations

Exercise 3: Evaluating Formulas

Scientific Notation

Scientific Notation

Scientific Notation

Answers to Chapter  3.0 … Powers, Roots, and Scientific Notation

Answers to Chapter 3.0

4.0 ... Ratio and Proportion

Ratio Word Problems

Proportion

Exercise 3: Solving Proportions

Answers to Chapter  4.0 Ratio and Proportion

5.0 ... Percentage

Exercise 1:  Changing Percents to fractions or Decimals

Percent Problems

Exercise 3: Identifying the Rate, Base, and Amount.

Percent Problems

Estimated Solutions

Words Versus Mathematical Operations

Exercise 5: Percent Word Problems  

Answers to Chapter  5.0 Percentage

Exercise 5: Percent Word Problems.  Page 65.

6.0 ... Measurements  

Exercise 1: Converting Units

Exercise 2: Measurement Operations

Common Metric Measurement Units  

Temperature Scales

Scale Drawings

Answers to Chapter  6.0 Measurements

7.0 ... Statistics

Bar and Line Graphs

Probability

Dependent and Independent Probability

Answers to Chapter  7.0 Statistics

8.0 ... Geometry

Angles and Lines

Exercise 1: Angle Problems

Triangles

Pythagorean Theorem

Perimeters

Circles

Exercise 5 Perimeter Problems

Area

Volume

Exercise 7: Volume Problems

Exercise 8: Review of Geometry

Answers to Chapter 8.0 Geometry

9.0 ... Algebra

Exercise 1: Equations

Exercise 2: Solving Equations  

Exercise 3: Word Phrases Translated To Algebra

Exercise 4: Sentences / Equations / Solutions  

Eliminating Brackets or Parenthesis

Exercise 7: Setting Up and Solving Word Problems

Signed Numbers

Operations with Signed Numbers

Exercise 8: Signed Number Operations

Multiplication and Division of Signed Numbers

Quadratic Equations

Graphing

The Distance Between Two Points

Inequalities

Answers to Chapter 9.0 Algebra

The GED Mathematics Practice Test

Formulas

Calculator

GED Mathematics Practice Test

Answer Sheet  Part I

GED Mathematics Practice Test

Answer Sheet Part II

GED Mathematics Practice Test

Answers

ACKNOWLEDGEMENTS

Some of the definitions of the word acknowledgement are: 1. To show appreciation of or admit obligation for; express thanks for.  2.  Recognition of the existence or truth of something. When I think of these definitions, I immediately want to express my sincere thanks to the Lord and recognize that he has been with me from the beginning of this work helping me all the way to its completion. Then, I want to thank my patient wife, Ingrid, who understood why I had to spend so much time on my computer neglecting the little projects she wanted done around the house. I will always be thankful for her understanding, encouragement, and moral support.

I also would like to express my sincere appreciation to the GED Testing Service for providing the material containing the directions for properly recording the answers to any Practice Test questions. These recording directions are included in this document to familiarize the student with the method used in the latest GED test.      

INTRODUCTION

GED

GED stands for the Tests of General Educational Development. The GED Test is a national examination developed by the GED Testing Service of the American Council on Education. The GED Test consists of five subjects, which are: Language Arts, Writing; Language Arts, Reading; Social Studies; Science; and Mathematics. Upon passing the test, a certificate is received that is recognized by employers in private industry and government, as well as admissions officers in colleges and universities. The GED certificate is accepted as equivalent to a high school diploma.

Each state has its own requirement for how many tests you take in a day or testing period, therefore, you must check with your local adult education center for the requirements in your state, province, or territory.

Problem Solving

Most of the problems on the Mathematics Test are word problems. In preparation for these or any type of math problem, the basics of addition, subtraction, multiplication, and division of numbers are essential. These basics will be presented in the chapters that follow. The format and method of presenting the material is to include many examples and exercises to encourage the student to work with the material rather than to just read the written explanations. The student is encouraged to go through every chapter working the examples and exercises, and checking each exercise with the answer to ensure that the problem is being solved correctly. At times, the student may want to draw a picture to help understand a problem, or estimate an answer, which can then be used as a guide to solve the problem.

Practice Test

In order to offer further help in the areas of geometry and algebra, a review of geometry and algebra is included at the end of these chapters. Finally, a practice test is presented that consists of multiple-choice and alternate format questions of the type that are on the newest GED test. The practice test Part I consists of 29 questions with optional use of a calculator and Part II consists of 28 questions without the use of a calculator. The actual test will consist of 25 questions in each part to be completed in 90 minutes. However, for added practice we have included a few more questions to improve your test-taking skills. Formulas you may need for the test and some calculator directions are also included.

1.0… Decimals

The decimal, or base 10, system is a universally adopted system and consists of ten digits, which are: 0,1,2,3,4,5,6,7,8, and 9. The position of a digit within a given number denotes the value of the digit. These values are shown in the chart below.

The whole numbers are to the left of the decimal point, and the fractions, or parts of a whole are to the right of the decimal point. For example, the number 1,324.53 is read as follows: one thousand three hundred twenty four and fifty three hundredths. Every digit in this number has its own positional value and is represented as follows :(1 x 1000, or 1000)+(3 x 100, or 300)+(2 x 10, or 20)+(4 x 1, or 4) and (5 x 1/10, or 5/10)+(3 x 1/100, or 3/100). The .53 is read as 53-hundredths because the last positional value is in the hundredths position.

Note that the whole part is read first, but the word and is only used for the decimal point, then, in the fractional part, the last positional value is named.

Figure 1

Now that we understand the positional values, let’s read out the number in the above Chart.. Forty five trillion eight hundred thirty nine billion four hundred ninety million three hundred sixty one thousand three hundred twenty four and fifty three hundredths. Adding zeros to the right of the last decimal digit 3 does not change the value of the number, but they are used here to show the value of their position.

Note that the commas in the whole number part occur every third digit, starting from the decimal point and moving to the left, so as to group the values. To express numbers greater than a trillion, as for example quadrillions, quintillions, or greater, we can use exponents, or powers of ten. This unique system is explained in chapter 3.

Calculators

The GED Mathematics Test will allow you to use a calculator on Part I of the test. The scientific calculator, the CASIO fx-260solar, will be provided by the testing center for this purpose. Although you should be able to solve every problem in this book without the use of a calculator, they can save time and provide a check on your calculations. Therefore, the keys that will be the most helpful to you are labeled below and it is recommended that you obtain your own calculator so that you can obtain the practice of using one before the test. To become familiar with it’s operation, we can perform a few basic operations: Press the ON key, which will clear the memory and set the display to 0. The letters DEG will appear on the top of  the display window.

Press the AC key to clear all numbers and operations from the display. The C key is used to erase only the last number or operation that you entered and is used when you know that you have entered a wrong number.

Examples:

Add 12 to 14:  Enter 12, press the + key, enter 14, press the = key to obtain 26.

Subtract 20 from 54: Enter 54, press the – key, enter 20, press the = key to obtain 34.

Multiply 22 times 6:  Enter 22, press the x key, enter 6, press the = key to obtain 132.

Divide 84 by 4:  Enter 84, press the ÷ key, enter 4, press the = to obtain 21

Reading Decimals

Example 1: Write the whole number, 248, in words.

Solution: Recognize that the 2 is in the hundreds position, (or place), the 4 is in the tens place, and the 8 is in the ones place. Therefore, the number, 248, is written as two hundred forty eight.

Example 2: Write the decimal number, 34.573, in words.

Solution:

step 1: Write out the whole number part: thirty four.

step 2: Write and in place of the decimal point.  

step 3: Write the decimal part as though it is a whole number: "five hundred

           seventy three."

step 4: Finally, name the place of the last digit: thousandths.

          Thus, 34.573, is written as:  "thirty four and five hundred seventy three

           thousandths."

Example 3:  a). Write ninety four thousandths in decimal form.

Solution: Insert a zero into the number to place the number 4 in the thousandths

position: .094

                    b). Write eighty three and forty four ten thousandths in decimal form.

Solution: Write down 83, then insert the decimal point for the word and plus a

couple of zeros to place the number 4 in the ten-thousandths position.

resulting in: 83.0044

Note: Many occasions arise where it will be necessary to compare numbers.

The symbols used to indicate the relationship between two numbers are:

Symbol                          Definition                         Example

>                           is greater than                         8 > 7

<                                is less than                         3 < 5    

                          =                                 is equal to                          6 = 6

Example 4:  Compare .16 to .156, and determine which number is greater.

Solution: Add a zero to .16 (adding a zero to the last decimal digit of a decimal

fraction does not change the value of the number) in order that the two numbers

will have the same number of decimal places.

Thus:             .16 =.160   and

                   .156 =.156

It is now easier to see that:   .160 > .156    or    .156 < .160

Exercise 1:                      Reading and Comparing Decimals

Select the matching decimal number for the written value by underlining A, B, C, or D.

1. thirty six hundredths                       A)  .36       B)  .036      C)  3600      D)  30.06

2. ninety five thousandths             A)  95,000         B)  .950    C)  9.005      D)  .095        

3. five and seven thousandths        A)  5,007     B)  5.070    C)  5.007      D)  5.700

4. two hundred sixteen and two hundred sixteen thousandths  A)  216,216

  B)  216.216     C)  2.16      D)  .216

5. nine and fourteen ten thousandths  A)  9.0014     B)  914,000    C)  .914    D)  9.14

6. Select the matching expression for the two numbers given. Underline A, B, or C.

 a)  .004 and .04                  A)  .004 = .04        B)   .04 < .004     C)  .04 > .004

 b)  8.1 and 8.12                  A)  8.1 > 8.12        B)   8.1 < 8.12      C)  8.1 = 8.12  

 c) .110 and .1011               A) .1011 > .110      B) .110 > .1011    C) .110 =.1011

 d)  4.02 and 4.020             A)  4.020 > 4.02      B)  4.02 = 4.020     C) 4.02 < 4.020

7. Write the decimal number equivalent of each written value given:

 a)  two thousand and eighty hundredths

 b)  four tenths

 c)  one hundred ten and one thousandth

 d)  three hundred one and twenty thousandths

 e)  ten - ten thousandths

8. From the following numbers, which is the nearest in value to 6?

 a) 5.9       b) 6.001        c) 6.1        d) 5.9999        e) 6.0100

Rounding  Decimals

Rounding numbers is best explained by using an example. To round the number 376 to the nearest tens place, as an example, it is necessary to look at the number one place to the right of the 7. Since this number 6 is greater than or equal to five, the seven is rounded up to an eight. Therefore, 376 rounded to the nearest tens place is 380.

Example 2: Round off $ 43.345 to the nearest cent.    

Solution: When dealing with money, we round off to the nearest hundredths. So we examine the number to the right of the hundredths place to determine if it is greater than or equal to five. In this case, it is a five, so we round up the four in the hundredths place to a five. Thus the value of $ 43.345 to the nearest cent is,  $ 43.35

Example 3: Round off 238.731 to the nearest whole number

Solution: The number just to the right of the decimal point is a seven, which is greater than or equal to five, so rounded off, the nearest whole number to 238.731 is to 239.

Exercise 2:                                  Rounding Decimals

1. A temperature is 98.63 degrees. What is that temperature to the nearest degree?

2. The thickness of a piece of sheet metal measures .034 inches. This measurement to

   the nearest hundredth of an inch is?

3. An interest rate is 8.75 percent, but to the nearest tenth of a percent, it is?

4. If milk costs $ 2.567