`MATH 20053 CHAPTER 6 EXAMPLES & DEFNINITIONS Section 6.1 Ex. A) Last year Karen taught part time, and had 79 students. This year, teaching full time, she has 175 students. How does last year’s student load compare with this year’s?

Definitions: A fraction is a number of the form

!

ab

or a/b where a and b are any

number except b ≠ 0. A rational number is a number of the form

!

ab

where a and

b are integers and b ≠ 0. Fractions that represent the same point on the number line are called equivalent fractions. Ex. B) Represent with a fraction the part of the collection of dots: (a) outside the rectangle, (b) both inside the triangle AND outside the rectangle. • • • • • • • • • • • • • • • • • • • • • • • •

Ex. C) Use fraction bars to represent the fraction

!

12

in as many ways as possible.

Ex. D) Using standard pattern blocks, let 2 hexagons equal one whole. Make a key to show the value of (a) an equilateral triangle, (b) a trapezoid, and (c) a parallelogram.

FUNDAMENTAL LAW OF FRACTIONS:

Let b ≠ 0. For any fraction a/b and any natural number c,

!

ab

=acbc

and cacb

=ab

.

MATH 20053 CHAPTER 6 EXAMPLES 2

Definition: A fraction

!

ab

is in its simplest form when b is positive and a and b

are relatively prime. This is also called a reduced fraction. Ex. E) Is 30/78 a reduced fraction? If not, reduce it. Ex. F) Write –416/512 as a simplified fraction. Ex. G) Write the expressions below in simplest form:

(a)

!

xy + xzx2y + x2z

(b)

!

x2 + 36x

(c)

!

210 "28

213 "211

Ex. H) Are 9/48 and 8/40 equivalent fractions? To determine whether fractions are equivalent:

1. Simplify both and compare, or 2. Write both with the same denominator, or

3. Cross-multiply: The fractions

!

ab

and

!

cd

, where b ≠ 0 and d ≠ 0,

are equivalent if and only if ad = bc.

NOTE: If

!

pq

and rs

are rational numbers with positive denominators, then

!

pq

<rs

if

and only if ps < qr.

Ex. I) Is

!

2479

<47

159 ?

Ex. J) Estimate the values of: (a) 25/48 (b) 19/60 DENSITY PROPERTY OF RATIONAL NUMBERS: For any two unequal rational numbers, it is always possible to find another rational number between them.

Ex. K) (a) Find a rational number between

!

14

and

!

15

.

(b) How would one find two more rational numbers between

!

14

and

!

15

.

MATH 20053 CHAPTER 6 EXAMPLES 3

Section 6.2

Definitions: A proper fraction

!

pq

is one in which | p | < | q |. An improper

fraction

!

pq

is one in which | p | ≥ | q |.

A mixed fraction is made up of an integer and a rational number. In general, we write mixed fractions in simplest form, so that the rational part of the fraction is a

proper fraction.

Ex. L) Are the following proper, improper, or mixed fractions? Write any improper fraction as a proper or mixed fraction. Write any mixed fractions as improper fractions.

(a)

!

"1413

(b)

!

5 23

(c)

!

1212

(d)

!

89

(e) 235

ADDITION/SUBTRACTION OF RATIONAL NUMBERS:

(a) If

!

ab

and

!

cb

are rational numbers with equal denominators, then

!

ab

±cb

=a ± c

b.

(b) If

!

ab

and

!

cd

are any two rational numbers, then

!

ab

±cd

=ad ± bc

bd.

Ex. M) Perform the operation: (a)

!

78"

34

(b)

!

23

+45

Ex. N) Solve the pattern blocks addition problem shown on the next page.

Ex. O) Perform the operations: (a)

!

739

+291

"421

(b)

!

2x5y2

+2

xy4

MATH 20053 CHAPTER 6 EXAMPLES 4

Ex. P) Michelle wants to jog a total of 10 miles this week. On Monday she jogged

!

2 14

miles, on Tuesday,

!

2 18

miles, on Wednesday

!

2 12

miles, and on Thursday,

!

134

miles. If she will not be able to jog during the weekend, how far must she jog on

Friday to reach her goal of 10 miles?

Ex. Q) Perform the operations: (a)

!

8 23

+ 1 712

(b)

!

9 16" 5 2

3

Ex. R) Estimate

!

8 23

+ 1 712

in three ways.

Ex. S) Common errors in the addition and subtraction of rational numbers--look over each of the following and identify the errors.

1.

!

38 +

15 =

313

2.

!

23 +

12 =

13

3.

!

23 +

12 =

24

4.

!

25+ 7 =

25 +

27

5.

!

2 = 63 =3+ 3

3 =33 + 3 =1+ 3 = 4

6.

!

xy + zx

= y + z

7.

!

2+ 53+ 5 =

23

8.

!

4 58 "323 =1

35

MATH 20053 CHAPTER 6 EXAMPLES 5

Section 6.3 Ex. T) While rearranging her baby’s bedroom, which is square, a mother notices that the crib’s length is about 2/5 of the wall, and its width takes up about 1/3 of the wall. How much of the room’s area is occupied by the crib?

MULTIPLICATION OF RATIONAL NUMBERS:

If

!

ab

and

!

cd

are any two rational numbers, then

!

ab"

cd

=acbd.

Ex. U) Perform the operations: (a) 516

!43

(b)

!

34" 7

MULTIPLICATIVE INDENTITY FOR FRACTIONS:

!

aa

, where a ≠ 0.

Definitions: If the product of two numbers equals 1, the two numbers are

multiplicative inverses. Each rational number

!

xy

has a multiplicative inverse

!

yx

if x ≠ 0, y ≠ 0. The multiplicative inverse of a rational number is also called a reciprocal.

Ex. V) Find the reciprocal of

!

"2 45

.

Ex. W) Multiply the mixed numbers:

!

3 12" 6 3

8

DIVISION OF RATIONAL NUMBERS:

If

!

ab

and

!

cd

are any two rational numbers and

!

cd

≠ 0, then

!

ab

÷cd

=ab

. dc

MATH 20053 CHAPTER 6 EXAMPLES 6

Algorithm for the Division of Fractions or

WHY do we “invert the divisor and multiply?” •Consider what division of rational numbers might mean – we know that the

definition of division is that a ÷ b = c means that c × b = a.

Thus,

!

34

÷58

= x means that

!

x " 58

=34

.

Multiplying both sides of the equation by

!

85

in order to isolate the variable, we get:

!

x " 58"85

=34"85

or

!

x = 34"85

=2420

=65

.

We can see that we found

!

34

÷58

by multiplying

!

34"85

.

•In the middle grades, the Fundamental Law of Fractions

!

ab

=acbc

is used to

justify the division algorithm for fractions:

!

34

÷58

=

3458

=

3458

"

8585

=

34"85

58"85

=

34"85

1=

34"85

Thus,

!

34

÷58

=

!

34"85

.

>> continued

MATH 20053 CHAPTER 6 EXAMPLES 7

•For an alternative approach for developing an algorithm for the division of

fractions, let’s consider dividing fractions with equal denominators:

!

67

÷27

= 6 ÷ 2 = 3 So if the denominators are equal, we simply divide the numerators. Then if we

want to divide fractions with different denominators, we simply rename the

fractions so that their denominators are equal! In general:

!

ab

÷cd

=ab"dd

#

$ %

&

' ( ÷

cd"bb

#

$ %

&

' ( =

adbd

÷bcbd

= ad ÷ bc or adbc

Rewriting this last expression, we realize that we have

!

ab"dc

.

NOW: you know WHY we “multiply by the reciprocal” in order to divide by a fraction. [Above material adapted from A Problem Solving Approach to Mathematics for Elementary Teachers by Billstein et al. 7th edition.]

Ex. X) Mary bought

!

2 13

yards of ribbon to make bows. Each bow uses

!

38

of a

foot of ribbon. How many bows can she make?

MATH 20053 CHAPTER 6 EXAMPLES 8

Ex. Y) Common errors in the multiplication and division of rational numbers--look over each of the following and identify the errors.

1.

!

14 "5 =

520

3.

!

1443 =

13

5.

!

7+ 36 =

72

7.

!

4 + 316 "9 =

212 =

16

2.

!

416 "12 =

112

4.

!

2 12 "325 = ( 2 "3 )+ (

12 "

25 )

6.

!

34

= 68

so 32

= 38

8.

!

x2

x + xy= 1

y

9. 627÷214= 447÷ 94= 117÷ 91= 117! 19= 1163

NOTE ABOUT THE RATIONAL NUMBERS: The rational numbers possess all of the properties of the integers. We have now added two new properties: Closure for division, and a multiplicative inverse. Thus, the rational numbers with the operations of addition and multiplication form a field.

Practice Problems for Section 6.3

1. Multiply the mixed numbers using the Distributive Property: 7 25! 2 18

[Hint: write the fractions as (7+ 25) ! (2+ 1

8) and then multiply using the Distributive

Property.]

>> continued

MATH 20053 CHAPTER 6 EXAMPLES 9

Practice Problems section 6.3, continued: 2. The product of 20 and 12½ is how much greater than the product of 20½ and 12? 3. The Chess Club has adopted ten miles of highway to keep free from litter. The

section assigned to the club starts at mile-marker 12 and extends to mile-marker 22.

When the chess club members have cleaned

!

58

of their section, between which two

mile-markers will they be?

4. Paul owns

!

38

of the stock of the company of which he is President. His wife owns

!

13

of the stock. (a) What part of the stock will his daughter need to purchase for the

family to own 3/4 of the stock? (b) There are 30,000 shares of this company being

traded. How many shares should the daughter purchase?

5. Perform the operation:

!

5 37

÷ 2 19

6. Meredith just purchased a new cat food designed for elderly cats. The

instructions say to give a cat

!

34

of a cup of cat food per day. Meredith would like to

continue using the scoop that she has in the cat food bin. She determines that the

scoop will hold

!

23

of a cup of cat food. Since Meredith has four cats, how many

scoops of cat food will she be giving them per day?

7. Find the sum of 12÷12

and 13÷ 3 . Remember to show all work.