Fundamentals of technical mathematics

Solutiona × + We write the numbers in a vertical column and then we multiplyeach number in the multiplicand from right to left by the rightmostmultiplier.. Therefore, to reduce a fractio

Fundamentals of Technical

Mathematics

Sarhan M Musa

Prairie View A&M University

AMSTERDAM  BOSTON  HEIDELBERG  LONDON  NEW YORK  OXFORD

PARIS  SAN DIEGO  SAN FRANCISCO  SINGAPORE  SYDNEY  TOKYO

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ISBN: 978-0-12-801987-0

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To my late father, Mahmoud; my mother, Fatmeh; my wife, Lama; and my children, Mahmoud, Ibrahim, and Khalid

Fundamentals of Technical Mathematics introduces applied mathematics for engineering

technologists and technicians Through a simple, engaging approach, the book reviews basic

mathematics including whole numbers, fractions, mixed numbers, decimals, percentages,

ratios, and proportions The text covers conversions to different units of measure (standard

and/or metric) and other topics as required by specific businesses and industries Building on

these foundations, it then explores concepts in arithmetic; introductory algebra; equations,

inequalities, and modeling; graphs and functions; measurement; geometry; trigonometry; and

matrices, determinants, and vectors It supports these concepts with practical applications in

a variety of technical and career vocations, including automotive, allied health, welding,

plumbing, machine tool, carpentry, auto mechanics, HVAC, and many other fields In

addition, the book provides practical examples from a vast number of technologies and uses

two common software programs, Maple and Matlab

This book has eight chapters Chapter 1 provides basic concepts in arithmetic Chapter 2

introduces algebra Chapter 3 presents equations, inequalities, and modeling Chapter 4

presents graphs and functions Chapter 5 presents measurement Chapter 6 introduces

geometry Chapter 7 reviews trigonometry Finally, Chapter 8 introduces matrices,

determinants, and vectors

Sarhan M Musa

xi

It is my pleasure to acknowledge the outstanding help and support of the team at Elsevier

in preparing this book, especially from Cathleen Sether, Steven Mathews, Katey Birtcher,

Sarah J Watson, Amy Clark, and Anitha Sivaraj

Thanks for professors John Burghduff and Mary Jane Ferguson for their support,

understanding, and being great friends

I would also like to thank Dr Kendall T Harris, his college dean, for his constant

support Finally, this book would never have seen the light of day if not for the constant

support, love, and patience of our family

xiii

Chapter 1

Basic Concepts in Arithmetic

You must do the thing you think you cannot do

Eleanor Roosevelt

nAlessandro Volta (1745e1827), an Italian physicist, chemist and a pioneer of electrical science He is

most famous for his invention of the electric battery Alessandro has a special talent for languages

Before he left school, he had learned Latin, French, English and German His language talents helped

him in later life, when he traveled around Europe, discussing his work with scientists in Europe’s

centers of science

INTRODUCTION

In this chapter, we will discuss the most useful and important basic topics in

arithmetic (science of numbers), which are essential refreshment for people

who have forgotten or do not like math Mastering the fundamental

con-cepts in mathematics is vital in strengthening the foundation for learning

advanced material and forming a like for mathematics

1.1 BASIC ARITHMETIC

In arithmetic numbers used are known The purpose of this section is to

introduce the basic principles/laws of arithmetic such as properties for the

Fundamentals of Technical Mathematics http://dx.doi.org/10.1016/B978-0-12-801987-0.00001-0

operations of addition (þ), subtraction (), multiplication (), and division(O) Next, we introduce fractions, decimals, and percents.

Numbers are categorized into six categories First is real numbers, whichincludes all numbers Whole numbers are numbers 0, 1, 2, 3,., while nat-ural numbers do not include zero, i.e., 1, 2, 3, Integer numbers are wholenumbers including their negative counterparts, i.e.,., 3, 2, 1, 0, 1, 2,

3, Rational numbers, i.e.,1

(b)How much is 3þ 5 þ 13?

(c) How much is 12þ 15 þ 28 þ 13?

Solution(a)3þ 5 ¼ 8(b)3þ 5 þ 13 ¼ 21(c) We can use columns of numbers to get the total or sum manynumbers:

We write the numbers in a vertical column and then subtract the

bottom numbers from the top numbers We start from right to left

When the bottom number is larger than the top number, we

borrow from the number in the top of the next column and add ten

to the top number before subtracting, at the same time we reduce

the top number in the next column by 1

Since 2 is smaller than 3, 2 borrows 1 from 8 to become 12, and

8 becomes 7

Multiplication is a quick way to add many similar numbers The symbol ()

represents the multiplication operation

For example, 7 5 is the same as adding 7 for 5 times, 7 þ 7 þ

7þ 7 þ 7 ¼ 35 This is the same as 7  5 ¼ 35 by multiplication table

1.1 Basic arithmetic 3

Example 3(a)Multiply 23 with 12.

(b)Multiply 57 with 18

Solution(a)

×

+

We write the numbers in a vertical column and then we multiplyeach number in the multiplicand from right to left by the rightmostmultiplier If the product is greater than 9, add the tens number tothe product of numbers in the next column Next, we multiplyeach number in the multiplicand by the next number in themultiplier and we place the second partial product under thefirstpartial product, by moving one space to the left We continue theprocess for each number in the multiplier, and then we add thepartial products to get thefinal answer

Properties of signs for multiplication

Multiplication sign Result

We see from the table that, like signs give (þ) and unlike signs give ().

Any number written without a sign except 0 is considered to be positive, for

example, 3¼ þ3.0 is a neural number, neither positive nor negative

Division is used for separating a number to several equal groups of

numbers The symbol (O) represents the division operation

By dividing 22 by 2 we get 11 without remainder

When we have very large numbers to divide, long division is

performed with the method as shown below:

(b)

63428

22

5674

18

quotient dividend

Long division allows us to use two numbers of the divided at a

time, making the division process easier We divide 63 by 28, we

get 2 of 28 in the 63, we put the 2 as the quotient We multiply 2

by 28 to give us 56 Then we subtract 56 from 63 to give us 7

The 4 is brought down so we have 2 numbers to divide So,

634O 28 ¼ 22 þ (18/28) This means that 634 can be divided into

22 groups of 28 and there will be 18 left over, which is not enough to

make a 23rd group of 28

1.1 Basic arithmetic 5

Signs are important in division as in multiplication The tablebelow summarizes the rules of signs in division.

Properties of signs for division

þb

a ¼ b a

b

a ¼ b a

a

Example 6(a)8

4 ¼ 2(b)21

(c) 32

2 ¼ 16(d)15

NumeratorDenominatorFor example, 3/16 is a fraction Also, the denominator of any fractioncannot be zero, that is, bs 0 But, when the numerator of a fraction iszero, then the value of the fraction is zero For example, 0/5¼ 0 Any wholenumber has a denominator equal to 1, that is, a/1¼ a

6 CHAPTER 1 Basic Concepts in Arithmetic

There are two types of fractions: proper and improper.

Definition of proper fraction

A proper fraction is a fraction with a numerator less than its denominator

9are proper fractions.

Definition of improper fraction

An improper fraction is a fraction with a numerator greater than or equal

7 are improper fractions.

If the numerator a¼ denominator b, then the fraction is equal to 1, such as

a/a¼ 1, where a is a whole number

Fractions can be at lowest term when both the numerator and denominator can

only be divided by 1 Therefore, to reduce a fraction to lowest terms, we need

to divide the numerator and the denominator by the largest number in which

both can be divided equally This process is reducing or simplifying

1.1 Basic arithmetic 7

Example 8(a)Reduce the fraction15

35to lowest terms.

(b)Simplify the fraction12

30to lowest terms.

Solution(a)We divide both the numerator and the denominator by 5, as15

30 ¼ 12=630=6 ¼ 2

5.

Definition of mixed number

A mixed number is a number that consists of a whole number and afraction

For example, 31

2and 5

3

4are mixed numbers.

To convert an improper fraction into an equivalent mixed number, we need

to divide the denominator into the numerator and write the answer as awhole number, and any remainder (which is smaller than the numerator) be-comes the numerator of the fraction part of the mixed number

Example 9(a)Convert5

3into a mixed number.

(b)Convert9

2into a mixed number.

8 CHAPTER 1 Basic Concepts in Arithmetic

9 8

To convert mixed numbers into improper fractions, you need to

multiply the whole number by the denominator of the fraction and add

the answer to the numerator of the fraction, then use the answer as the

new numerator and keep the same denominator as it is from before

(a)Multiply the whole number by the denominator of the fraction and add

the answer of it to the numerator of the fraction, then use it on the

numerator and keep denominator as it is from before:

5 ¼ 3 5 þ 2

3

(b)Multiply the whole number by the denominator of the fraction and add

the answer of it to the numerator of the fraction, then use it on the

numerator and keep the denominator as it is from before:

7 ¼ 5 7 þ 1

5

1.1 Basic arithmetic 9

1.1.1.6 Adding or subtracting of fractions

When we want to add or subtract fractions, we need to have the same nominator, which is called a common denominator When we add and sub-tract fractions, we use the lowest common denominator (LCD)

de-When adding or subtracting fractions with the same denominator, we need

to add or subtract just their numerators and keep the denominator as it is.Example 11

Solve the following:

(a)3

8þ28(b) 6

13þ 5

13 113Solution(a)3

8þ2

8 ¼ 3 þ 2

8(b) 6

we need tofind common denominator by multiplying thedenominators together and multiply each numerator with the otherdenominator, then add or subtract them

Example 12Solve the following:

(a)2

3þ54

(b)5

2þ7

314Solution(a)2

1.1.1.7 Multiplying fractions

In multiplying fractions operation, all we have to do is to multiply

numer-ators together and denominnumer-ators together

In fractions multiplications, we can simplify before the multiplications, if

there is any, to save the calculation time

You need to change the division operator (e, / or O) between the fractions

into multiplication operation () and flip the divided by fraction (replace

the numerator with its denominator and replace the denominator with its

numerator) to apply the fraction multiplications method, also called the

1.1 Basic arithmetic 11

It is essential to know the correct order of operations in which any sion can be simplified.

expres-Rules for order of operations

1 First, parentheses

2 Second, multiplications and divisions, starting from left to right

3 Third, addition and subtraction, starting from left to right

Example 16Perform the indicated operations

(a)3þ 8 O 2(b)(4þ 8) O 4(c) 3 (4 þ 1)(d)5 2 þ 3  3(e) 7 3 þ 12 O 2  8  5(f) 4 (3 þ 2) O 6  9  2Solution

(a)3þ (8 O 2) ¼ 3 þ 4 ¼ 7(b)(4þ 8) O 4 ¼ 12 O 4 ¼ 3(c) 3 (4 þ 1) ¼ 3  5 ¼ 15(d)(5 2) þ (3  3) ¼ 10 þ 9 ¼ 19(e) ð7  3Þ þ ð12 O 2Þ  ð8  5Þ ¼ 21 þ 6  40 ¼ 27  40 ¼ 13(f) 4 ð3 þ 2Þ O 10  9  2 ¼ 4  5 O 10  9  2

> (9/5) þ (7/3)

6215

> (11/3)  (8/5)

3115

> (2/5)(4/3)

25

> ð6=5Þð4=3Þ

910

1.1 EXERCISES

Write each of the improper fractions in problems 1e4 as a mixed number

1 53

2 72

3 215

4 399Write each of the mixed numbers in problems 5e8 as an improper fraction

5 112

6 24

7 325

8 345

14 CHAPTER 1 Basic Concepts in Arithmetic

Write the fractions in simplest form as in problems 9e12.

34 23

2 715

35 53

418

36 31

254

37 21

5O3 110

38 51

3O217

39 1

35

8 420

40 3

714

112215

41 1

3þ133

42 11

5 þ35

43 2

3þ17

16 CHAPTER 1 Basic Concepts in Arithmetic

47 Adam paid the following bills: house payment $825, energy bill

$125, water bill $50, car payment $272, and insurance premium $65

What is the total amount of the bills?

48 John drives 45 miles per h (mph) for 2 h and 65 mph for 4 h

Deter-mine the average speed for the total driving time

Hint: Average rate ¼ Total distance

Total time .

49 A welded support base is cut into three pieces: A, B, C Find the

fractional part of the total length that each of the three pieces

represents when piece A¼ 18 in, piece B ¼ 6 in, and piece C ¼ 32 in

50 The operation time sheet for machining aluminum housing identifies 2 h

for facing, 3 4/5 h for milling, 4/10 h for drilling, 6/10 h for tapping, and

3/5 h for setting up Determine the total time allotted for this operation

51 Find the length of the grip of the bolt shown where all dimensions

are in inches

3 7/8

2 4/32 Grip

52 Calculate the distance (D) between the center lines of thefirst and

fifth rivets connecting the two metal plates shown in the figure

Rivet Spacing = 4 ½ Diameters

⅝ inch = Diameter

D = ?

53 A steel block has the following parameters:

634= in long, 3712= in wide, and 5/6 in thick What is the volume

of the block?

Hint: volume¼ length  width  thickness

1.1 Exercises 17

54 Assume that a light bulb draws 0.4 A current (I) at an input voltage(V) of 240 V.

Find the resistance (R) of thefilament and the power (P)dissipated

Hint: P ¼ VI and R ¼ V

I

55 An electric heater is rated 1100 watts at 110 volts Find:

1e the current (I)

58 How many joules (J) of work (W) are done when a force of 200 tons (N) is applied for a distance of 100 meters?

new-59 If a rider and a bicycle weighing 180 lb travel 600 ft in 30 s, howmuch power (P) is developed?

60 If a rider and a bicycle with a mass of 80 kg move 400 ft in 25 s,how much power (P) is developed?

61 If a force of 40 lb is applied to move a load 15 ft, how much torque

66 Calculate RTin the circuit of the following Figure.

67 Consider the series circuit in the Figure given below Find:

(a) the total resistance RT

1.1 Exercises 19

69 Find the total current Itfor the circuit in the following Figure.

For example, we can write the fraction 5

10in decimal as 0.5 We use a period

to indicate that the number is a decimal Both 5

10 and 0.5 are rationalnumbers

It is necessary to know the value of each digit in a decimal

Place Values names

Now we can read the decimal 0.5 asfive tenths or zero point five When a

decimal is less than 1, we place a zero before the decimal point, for

example, we write 0.5 not 5

In naming a decimal, we read the number from left to right as we read a

whole number, and then we use the place value name for last digit of the

(a)First we write in words of 3521, and then we write the place

value name for the last number, which is 1 after it: Three

thou-sand five hundred twenty one thousandths or zero point three five

two one

(b)Sixty-four ten thousandths or zero point zero zero six four

(c) Seventy-two thousand three hundred forty-five hundred thousandths

(d)First we write in words of 9, and then we write the place value name

for the last number, which is 9 after it: Nine millionths or zero point

zero zero zero zero zero nine

If we have a whole number and a decimal, the decimal point is written using

the word“and.”

(a)Eleven and seven hundred twenty-five thousandths

(b)Forty-one and ninety-three hundredths

(c) Eighty-one and thirteen hundredths

(d)Two hundred seventy-six andfive tenths

1.2 Decimals 21

1.2.1 Rules to rounding off decimals

The following rules are used in rounding off numbers:

1 When the number following the last digit to be retained is greater than 5,increase the last number by 1 For example, 0.016 becomes 0.02 and0.047 becomes 0.05

2 When the number following the last digit to be retained is less than 5,retain the last number For example, 0.071 becomes 0.07 and 0.093becomes 0.09

3 When the number following the last digit to be retained is exactly 5,and the number to be retained is odd, increase the last number by 1.For example, 0.375 becomes 0.38 and 0.835 becomes 0.84

4 When the number following the last digit to be retained is exactly 5,and the number to be retained is even, retain the last number Forexample, 0.125 becomes 0.12 and 0.645 becomes 0.64

So, we can conclude that to round a decimal to a specific place value, weneed to do the following steps:

1 If the digit to the right of the number needs to be rounded is 0, 1, 2, 3,

or 4, the number remains the same

2 If the digit to the right of the number needs to be rounded is 5, 6, 7, 8,

or 9, we add one to the number

3 After the rounding, all digits to the right of the rounded number aredropped

Example 3Round the following decimals to the given specific place value:

(a)3.731 to the nearest one(b)0.25 to the nearest tenth(c) 0.3412 to the nearest hundredth(d)48.6723 to the nearest hundredth(e) 32.5468 to the nearest thousandthSolution

(a)4(b)0.3(c) 0.34(d)48.67(e) 32.547Zeros at the end of a decimal can be affixed on the right side ofthe decimal point Also, zeros can be dropped if they are located atthe end of a decimal on the right side of the decimal point

22 CHAPTER 1 Basic Concepts in Arithmetic

Example 4

0.75 can be written as 0.750 or 0.7500

0.560 can be written as 0.56

Now, we will go over the basic operations additions, subtraction,

multiplica-tion, and division

We can use MATLAB to do rounding as below:

In order to add two or more decimals, we put the decimals in a column,

placing the decimal points of each number in a vertical line, then adding

the numbers at the same place value starting from the rightmost If there

are fewer digits in the added decimals prefix as many zeros as needed

1.2.3 Subtraction of decimals

In order to subtract two decimals, we put the decimals in a column, placingthe decimal points of each number in a vertical line, then subtracting thenumbers at the same place value starting from the farthest right

Example 6Subtract the following decimals:

(a)56.963 17.35(b)16.4 8.368Solution

Example 7Multiply the following decimals:

(a)3.04 0.2(b)42.8 0.007(c) 0.002 0.02(d)23.4 0.57(e) 13.32 4.3Solution(a)

1 digit from right

0.608

3.04

0.2

×

2 digits from right

3 digits from right

(b)

0.007

4 digits from right

42.8

3 digits from right

24 CHAPTER 1 Basic Concepts in Arithmetic

0.002

0.02

5 digits from right

3 digits from right

(d)

1638

23.4

0.57

3 digits from right

1 digit from right

3 digits from right

2 digits from right

1

1.2.5 Division of decimals

The division of decimals based on two situations:

First, when a decimal is divided by a whole number, in this case, we place

the decimal point in the quotient directly above the decimal point in the

dividend

Example 8

Find the value of the divisions up to the tenths digits value135:1

48Solution

135.1

48 2.8

390 384

We can use MATLAB to do decimal operations as the following:

>> 1.2 þ 2.7ans¼3.9000

>> 2.2  1.3ans¼0.9000

>> 0.23 * 0.15ans¼0.0345

>> 0.75/0.35ans¼2.1429

We can use Maple to do decimal operations as the following:

> 2.6 þ 5.37.9

> 5.8  3.42.4

> (6.3) (4.7)29.61

> 8:6

2:33.739130435

>

26 CHAPTER 1 Basic Concepts in Arithmetic

13 John paid $2.88 for six pounds of bananas How much do the

ba-nanas cost per pound?

14 If Sarah burns about 328.15 calories while walking fast on a

tread-mill for 40.5 min, about how many calories does she burn per

minute?

15 If a car travels at a speed of 65 miles per hour for 15 min, how far

will the car travel?

Hint: Distance¼ rate  time

16 If one gallon of water weighs 8.25 pounds, approximately how much

does a 30-gallon container of water weigh?

17 Find the decimal fraction of distance 1 inch of distance 4 inches

1 inch

4 inches

1.2 Exercises 27

18 Find the perimeter of the following Figure.

19 A charge of 6.5 Cflows through an element for 0.3 s; find the amount

of current through the element

20 An electric motor delivers 50,000 J of energy (W) in 2 min mine the power

To convert a percent to a decimal, we remove the percent sign and move the

decimal point two places to the left For example, 35% becomes 0.35, and

3.7% becomes 0.037

In order to convert a decimal to percent, we move the decimal places to

the right then add the percent sign % For example, the decimal number

0.56 becomes a percent number as 56%

To convert a percent to a fraction, we divide the percent number by 100,

and then we reduce it to the lowest term

Tofind the percentage of a number, the percent is converted to decimal and

multiplied by the number

1.3 Percents (%) 29

Example 13What is 50% of 100?

Increase 100 by 50%

Solution50%¼ 0.50.5 100 ¼ 50

100þ 50 ¼ 150

So, 100 increased by 50% is 150

Example 15What is 8% of 75?

Solution8%¼ 8/100 ¼ 0.080.08 75 ¼ 6

Determine the answers for problems 9e12.

9 16% of 38

10 15% of 2.5

11 250 is 35% of what amount?

12 120 is 15% of what amount?

13 An electric motor consumes 878 watts (W) and has an output of

1.14 horsepower (hp) Find the percent efficiency of the motor

Hint:1 hp¼ 746 W, percent efficiency ¼ output

input  100

14 An electric motor consumes 975 watts (W) and has an

output 1.25 horsepower (hp) Find the percent efficiency of the

motor

15 In a technical mathematics class of 30 students, 75% are boys If

65% of them are working part time, how many boys in the class are

working part time?

16 If a motorcycle requires 25 units of actual work and 65 units of

theo-retical work, compute the mechanical efficiency

17 Richard and Matt arrive late for a seminar and miss 5%

of it The seminar is 115 min long How many minutes did they

miss?

18 The price of gasoline drops from $3.00 per gallon to $2.35 per gallon

What is the percent of decrease?

19 Thirty percent of the people at a restaurant selected the lunch special

If 60 people did not select the special, how many people are eating at

the restaurant?

20 A machine on a production line produces parts that are not acceptable

by company standards 2% of the time If the machine produces 900

parts, how many will be defective?

21 Jeff’s monthly utility bill is equal to 40% of his monthly rent, which

is $700 per month How much is Jeff’s utility bill each month?

22 Sam has completed 80% of his 110-page report How many pages

has he written?

CHAPTER 1 REVIEW EXERCISES

Reduce each of the following fractions in problems 1e4 to lowest

3 2442

4 65135Change each of the following mixed numbers in problems 5e8 to improperfractions:

5 723

6 534

7 812

8 625Find the fraction of each portion of the following figures in problems 9and 10

9

10

11 A batch of concrete is made by mixing 20 kg of water, 65 kg ofcement, 100 kg of sand, and 225 kg of aggregate Determine the totalweight of the mixture

12 Texas contains 268,820 square miles (mi2), and Louisiana 52,

271 mi2 How much larger is Texas than Louisiana?

13 The current to a lamp is 4.3 A when the line voltage is 110 V.Calculate the power dissipated in the lamp

Hint: Power¼ voltage  current

32 CHAPTER 1 Basic Concepts in Arithmetic

14 A gear in a machine rotates at the speed of 1603 revolutions/min.

How many revolutions will it make in 6 min?

15 If 800 shares of a stock are valued at $72,000, what is the value of

each share?

16 If 600 shares of a stock are valued at $42,000, what is the value of

each share?

17 A charge (Q) of 90 coulombs (C) passes a given point of an electric

circuit in 20 seconds (s) Find the current (I) in the circuit

Hint: I ¼ Q

t.

18 The power (P) dissipated in an electric circuit element is 120 W and

a current (I) of 20 A isflowing through it Determine the voltage (V)

across and the resistance (R) of the element

Hint: P ¼ VI and R ¼ V

I.

19 Calculate the total resistance (RT) in each of the circuits shown in the

Figure given below, where

R1¼ 1 ohm, R2¼ 2 ohms, R3¼ 3 ohms

Hint: The total resistance in a series circuit is RT¼ R1þ R2þ R3

120 volts

20 Calculate the total resistance (RT) in each of the circuits shown in the

Figure given below, where

R1¼ 2 ohm, R2¼ 4 ohms, R3¼ 6 ohms

Hint: The total resistance in a parallel circuit is RT ¼ 1 1

21 Determine V1and V2in the following Figure, using voltage division.

23 Determine V1, V2, and V3in the circuit of the following Figure

5 Ω

+ V1 - +

25 Find the voltage Vabin the circuit of the following Figure.

+ 50V +

-+

-

10 V

- +

Vab

Hint: When voltage sources are connected in series, the total voltage

is the sum of the voltage of the individual sources

28 John drove 87 miles and Jane drove 48 miles What fraction of the

entire trip did Jane drive?

29 A water tank, whenfilled, holds 825,000 L If there are 386,000 L of

liquid in the tank, what fraction represents the unfilled portion of the

32 A mechanic has 21 bolts, each 35

7in long If he placed the bolts end

to end, how long a string of bolts would be formed?

33 If a piece of metal trim215

7in long is cut into 6 pieces of equallength, what will be the length of each piece?

34 The maximum continuous load on a circuit breaker device is limited

to 85% of the device rating If the circuit breaker device is rated

40 A, what is the maximum continuous load permitted on the device?

35 A fuse must be sized no less than 120% of the continuous load If the

load is 75 A, what will be the smallest-sized fuse?

36 Increase 55 by 30%

37 Increase 35 by 25%

Chapter 1 Review exercises 35