Download Section 6.1 – The Set of Fractions

Chapter 6. Section 1
Page 1
Section 6.1 – The Set of Fractions
Homework (page 226) problems 1-12
A Historical Perspective:
• To understand fractions we need to have a concept of discrete versus continuous
• As soon as we want to divide a single whole into parts we are faced with the inadequacy of whole
numbers
• 'Parts of a whole' is one way to explain fractions. But there are other ways. In cutting a cake, what
is passed to each person at a party is a single piece of cake, not a tenth of a cake. If we have another
piece, we think of two pieces, not two tenths. This is the way most ancient people treated fractions,
they avoided them by resorting to smaller units. So 3/4 was not regarded as 3 parts of a whole, but
as three of a new and smaller entity
• For example, we use ideas of part of a whole without using fractions every day. We split the unit up
into smaller 'units'. Can you think of two?
Definitions:
•
•
•
•
•
•
a
, where a and b are whole numbers
b
( b ≠ 0 ). Here we have a equal parts (or portions) of all parts (or the whole) b
We can represent fractions visually with set, area or number line models
Example, page 228 number 2. Illustrate 4/7
The part to whole model of a fraction is represented by
Example, page 228 number 1a. (see figure)
5
8
Example, page 228 number 1d. (see figure)
This is a bit misleading, because you don't want to assume that the 'whole' is 12
9
Doing so you would arrive at the solution
(which is wrong)
12
In reality, when you use a ruler, you measure parts of the whole out of one unit
1
1 9
So you would therefore have 2 + = 2 =
4
4 4
a
In the fraction , a is called the numerator and b is called the denominator
b
Chapter 6. Section 1
Page 2
•
•
Two fractions are equivalent if they represent the same relative amount
For example, 1/2 and 2/4 are equivalent because they represent the same amount relative to the
denominator
•
We can also test for equivalency using the relation that
•
•
•
•
•
a c
= if and only if ad = cb
b d
Example. Show 1/2 and 2/4 are equivalent using the above relation
1 2
= ⇔ (1)(4) = (2)(2), 4 = 4
2 4
A fraction is written in its simplest form or lowest terms when its numerator and denominator have
no common prime factors
294
Example, page 228 number 8b. Rewrite
in simplest form
63
294 7 ⋅ 42 42 3 ⋅14 14
=
=
=
=
63
7⋅9
9
3⋅ 3
3
A fraction where the numerator is greater than the denominator is called an improper fraction (this
means the numeric value is greater than 1)
4
i.e.
3
Another form of writing a fraction greater than 1 is called mixed number where you have the whole
number portion sitting right next to its fractional portion
4 3 1
1
i.e. = + = 1
3 3 3
3
(this will be studied further in section 6.2)
Inequalities with Fractions:
•
•
•
a
b
a b
and , then < if and only if a < b
c
c
c c
a
c
a c
If they do not have the same denominator, that is
and , then < if and only if ad < bc
b
d
b d
NOTE: There is a 'problem' with the above theorem!
1 1
i.e. we know that − < . In cross multiplying we find 3 < – 4? What went wrong?
4 3
Between any two fractions there is another fraction
For any two fractions with the same denominator, that is
Chapter 6. Section 1
Page 3
•
•
•
a
c
a c
a +c
and
be any two fractions where < . Then
is between them
b
d
b d
b+d
43 50
Example, page 226 number 11b. Order the fractions
,
and find one in between
567 687
50
43
43(687) = 29541 and 50(567) = 28350. So 50(567) < 43(687) ⇒
<
687 567
50 + 43
93
31
A fraction in between would be
=
=
687 + 567 1254 418
Let
Example, page 229 number 12. (see book)
72,000,000 under 18
50,000,000 under 18 with 2 parents
19,000,000 under 18 with 1 parent
16,000,000 under 18 with mom
3,000,000 under 18 with dad
2,000,000 under 18 with other
Let's make a Venn Diagram out of this data
Lived with both parents?
50,000,000 50
69
=
≈ 69.4% ≈
72,000,000 72
100
Lived with 1 parent?
19,000,000 19
26
=
≈ 26.4% ≈
72,000,000 72
100
Of those with 1, those that lived with mom?
16,000,000 16
84
=
≈ 84.2% ≈
19,000,000 19
100
Lived with other?
2,000,000
2
3
=
≈ 2.8% ≈
72,000,000 72
100