PART ONE MATHEMATICAL PRINCIPLES Chapter 1 …

[Pages:30]PART ONE

1 C h a p t e r

MATHEMATICAL PRINCIPLES Basic Math Skills

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Introduction

Numbers used to express counted, measured, or calculated values can be found in various forms. This chapter is designed to review the various ways numbers can be expressed and manipulated. A review of proper order of operations, the isolation of x , and a review of the use of ratios and proportions in problem solving also are included.

Whole Numbers, Fractions, and Mixed Numbers

n Definition m Whole Numbers

Whole numbers are sometimes referred to as natural numbers or counting numbers. This set of numbers includes all of the positive integers.

1, 2, 3, 4, 5 . . . These numbers are exact and have a fractional part of zero. m

Counted numbers in horticulture can be used to express the number of plants to be planted in a landscaping project, the number of potted plants or cut flowers to be grown in a greenhouse, or the number of paving units required for a paving project. Paving units are purchased as a whole unit and plants are not useful unless they are whole.

n Definition m Fraction or Rational Number

A fraction or rational number is a number expressed in a ratio format.

a , where b = 0 b

m

A fraction represents a part of a whole. The word rational number comes from the fact that a rational number or fraction is expressed as a ratio. The number a is called the numerator, and the number b is called the denominator. The number b may not be equal to zero.

1

2 Part One { Chapter 1 Basic Math Skills

?

Figure 1-1

1 4

is

a

fraction.

The

one

is

the

numerator

and

four

is

the

denominator,

which

is

greater than zero.

1 4

also

can

be

described

as

one

of

four

parts,

as

illustrated

in

Figure

1-1.

Fractions can be used and defined in many ways. The preferred way to write

a fraction is in the form of a proper fraction, which is defined below. Improper

fractions often are the result of performing mathematical operations with fractions.

Mixed numbers combine whole numbers and fractions to express a number. Horticul-

turists are called on to manipulate fractions especially when performing mathematical

operations on measures of length.

n Definition m Proper Fraction

A proper fraction is one in which:

a

< 1, and a, b > 0 b

m

1 8

is

a

proper

fraction

because

it

is

less

than

1

and

8,

the

denominator,

is

greater

than zero.

24 32

also

is

a

proper

fraction,

but

it

could

be

written

in

the

more

efficient

reduced

or simplified form. Reducing a fraction requires dividing both the numerator and

denominator by the greatest common factor. This results in a fraction that is expressed

in lowest terms.

n Definition m Greatest Common Factor (GCF)

The greatest common factor is the largest number that divides two numbers evenly. The greatest common factor is used to reduce fractions to lowest terms. m

EXAMPLE 1-1 Reducing Fractions Using the GCF

24 Reduce to lowest terms by using the GCF.

32 Step 1

Factor the numerator and the denominator.

24 32

=

8 4

? ?

3 8

Whole Numbers, Fractions, and Mixed Numbers 3

Step 2

Find the fraction that equals 1 and isolate it.

8 4

? ?

3 8

=

3 4

?

8 8

Step 3 Reduce the fraction.

or

Step 4 Check using division.

3 ? 8 = 3 ?1= 3

48 4

4

24 ? 8 = 3 32 8 4

24 = 3 32 4

24 = 24 ? 32 = 0.75 32

3 = 3 ? 4 = 0.75 4

n Definition m Improper Fraction

An improper fraction is one in which:

a 1, where a b and a, b > 0 b

m

9 8

is

an

improper

fraction

because

it

is

greater

than

1

and

the

numerator

is

greater

than or equal to the denominator.

An improper fraction can be rewritten in the form of a mixed number. For

example,

the

improper

fraction

9 8

may

be

written

as

the

mixed

number

1

1 8

.

n Definition m Mixed Numbers

Mixed numbers are a combination of a whole number and a fractional part. It

may be written as the sum of a whole number and a proper fraction.

n+ a b

m

Mixed numbers are more accurate than whole numbers when estimating a value.

For

example,

a

garden

bed

may

measure

61

3 4

inches

(in.)

deep.

This

is

a

more

accurate

estimate than 61 or 62 in. Mixed numbers are commonly used when estimating length

using the U.S. Customary units of measurement.

4 Part One { Chapter 1 Basic Math Skills

EXAMPLE 1-2 Converting Improper Fractions to Mixed Numbers and vice versa

Convert

134 25

to

a

mixed

number.

Step 1

Divide the numerator by the denominator.

134 ? 25 = 5.36

Step 2

Write down the whole number. 5

Step 3 Calculate the remainder by subtracting the product of the whole number times the denominator from the numerator.

134 - (5 ? 25) = 134 - 125 = 9

Step 4

Combine the whole number with the remainder over the denominator. 9

5 25

TO REVERSE THE PROCESS:

Convert

5

9 25

to

an

improper

fraction.

Step 1

Multiply the whole number times the denominator.

5 ? 25 = 125

Step 2 Add the numerator to the solution in step one.

9 + 125 = 134

Step 3 Place the solution in step two in the fraction as the numerator.

134 25

L Practice Problem Set 1-1 L Whole Numbers, Fractions, and Mixed Numbers

Identify each of the following numbers as a whole number, proper fraction, improper fraction, or mixed number:

9 1.

8

Adding, Subtracting, Multiplying, and Dividing Fractions 5

3 2.

4 3. 7

5 4. 1

8 25 5. 25 Convert each of the following improper fractions to a mixed number: 11 6. 8 67 7. 32 128 8. 5 13 9. 4 117 10. 16

Adding, Subtracting, Multiplying, and Dividing Fractions

Algebraic manipulation of fractions can be challenging. Here is a review of the proper

ways to combine fractions when adding, subtracting, multiplying, and dividing.

a + c = ad + bc

bd

bd

a - c = ad - bc

bd

bd

a ? c = ac b d bd

a ? c = ad b d bc

EXAMPLE 1-3 Adding Fractions

a + c = ad + bc

bd

bd

3

+

1

=

(3)(2)

+

(4)(1)

=

10

=

5

=

1 1

42

(4)(2)

84 4

6 Part One { Chapter 1 Basic Math Skills

Note that the fraction is reduced at the end of the computation.

10 ? 2 = 5 8 24

Then the improper fraction is changed to a mixed number.

5

=

1 1

44

EXAMPLE 1-4 Subtracting Fractions

a - c = ad - bc

bd

bd

15 - 1 = (15)(4) - (16)(1)

16 4

(16)(4)

60 - 16 = 44 = 11

64

64 16

EXAMPLE 1-5 Multiplying Fractions

a ? c = ac b d bd 5 ? 2 = (5)(2) = 10 = 5 8 3 (8)(3) 24 12

EXAMPLE 1-6 Dividing Fractions

a ? c = ad b d bc

5

?

2

=

(5)(4)

=

20

=

5

=

2 1

6 4 (6)(2) 12 3 3

L Practice Problem Set 1-2 L Adding, Subtracting, Multiplying, and Dividing Fractions

Solve the following problems involving fractions: 1. 3 + 15

32 16 2. 3 + 1

82

3. 10 - 1 32 4

4. 7 - 1 83

5. 3 ? 1 84

6. 2 ? 1 16 10

7. 10 ? 3 16 8

8. 1 ? 1 23

Decimal Numbers, Place Value, and Decimal Fractions 7

Decimal Numbers, Place Value, and Decimal Fractions

n Definition m Decimal Number

A decimal number is the base-10 system used for expressing a mixed number. In other words, it is a way of naming the values that lie between whole numbers. The whole number is separated from the fractional portion of the number with a decimal point. m

An example of a decimal number is five and four-tenths, and it is written as 5.4. Five is the whole number, and four-tenths is the fractional part of the number. The number five and four-tenths lies between the whole numbers five and six. A decimal point, written as a dot or a period, separates the whole number from the fractional part of the number.

n Definition m Decimal Point

A decimal point is a period or dot in a decimal number that serves to separate the whole number portion of a number from the fractional part of the number. m

When writing decimal numbers, it is important to understand the concept of place value. The benchmark for place value is the decimal point because it separates the whole number portion of the decimal number from the fractional portion of the decimal number (see Table 1-1). Notice that place value changes by a magnitude of ten as digit placement moves to the left or the right of the decimal point.

8 Part One { Chapter 1 Basic Math Skills

TABLE 1-1 ? PLACE VALUE TABLE

1,000

100

10 1 .

1

1

10

100

1 1,000

1 10,000

Thousands Hundreds Tens Ones Decimal Point One Tenth One Hundredth One Thousandth One Ten-Thousandth

n Definition m Place Value

Place value is the value of a digit in a number. A digit's value depends on its position in relation to the decimal point. m

Place value aids in understanding and comparing the value of numbers. It also explains the relationship between digits in a number. For example, place value can be used to compare the whole numbers 302 and 320. Although the numbers are similar as written, they are very different in value. By identifying that the 2 in 302 is 2 ones, you can see that it is a smaller number than 320, since the 2 in 320 is 2 tens. Notice that, as the placement of a digit moves to the left, its value increases by a magnitude of ten, and as the placement of the digit moves to the right, its value decreases by a magnitude of ten. Comparing decimal numbers can be more difficult because a number of digits can be located to the left and to the right of the decimal point. Writing a decimal number in an expanded form using a place value table can aid in comparing one number to another and in comparing digits within a number.

Table 1-2 demonstrates how the number 3,924.1256 is written in expanded form. Here is the expanded form of 3,924.1256 without the use of a table:

(3 ? 1,000) + (9 ? 100) + (2 ? 10) + (4 ? 1) + 1 ? 1 + 2 ? 1

10

100

+ 5? 1 + 6? 1

1,000

10,000

TABLE 1-2 ? PLACE VALUE TABLE FOR THE NUMBER 3,924.1256

1,000

100

10 1 .

1

1

10

100

1 1,000

1 10,000

Thousands Hundreds Tens Ones Decimal Point One Tenth One Hundredth One Thousandth One Ten-Thousandth

3

9

24 .

1

2

5

6

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