jovan algebra | Page 4

these
“mini-tests.” If you are patient and take your time to work through all of the material, you
will become comfortable with algebra and maybe even find it fun!
Rhonda Huettenmueller

1
chapter 1
Fractions
Being able to perform arithmetic with fractions is one of the most basic skills
that we use in algebra. Though you might feel the fraction arithmetic that we
learn in this chapter is not necessary (most calculators can do these computa-
tions for us), the methods that we develop in this chapter will help us when
working with the kinds of fractions that frequently occur in algebra.
CHAPTER OBJECTIVES
In this chapter, you willMultiply and divide fractions
•
Simplify fractions•
Add and subtract fractions having like denominators•
Add and subtract fractions having unlike denominators•
Convert between mixed numbers and improper fractions•
Simplify compound fractions•
Translate English sentences into mathematical symbols•

2 alGebra D e myst i fieD DeMYstiFieD / algebra DeMYstiFieD / Gibilisco / 000-0 / Chapter 1
Fraction Multiplication
To illustrate concepts in fraction arithmetic, we will use pie charts. For example,
we represent the fraction
1
3 with the shaded region in Figure 1-1. That is, 13 is
one part out of three equal parts. Let us now develop the rule for multiplying fractions,
ab c
d ac
bd
⋅=. For example,
using this rule we can compute
2 3 1
4
⋅ by multiplying the numerators, 2 and 1, and
the denominators, 3 and 4. Doing so, we obtain
2 3 1
4 21
34 2
12
⋅=
⋅
⋅ = (we will concern
ourselves later with simplifying fractions). Let us see how to represent the
product
2 3 1
4
⋅ on the pie chart. We can think of this fraction as “two-thirds of
one-fourth.” We begin with one-fourth represented by a pie chart in Figure 1-2. Let us see what happens to the representation of one-fourth if we divide the
pie into twelve equal parts as in Figure 1-3.
FIGURE 1-1 FIGURE 1-2 FIGURE 1-3
Now we see that the fraction 1 4 is the
same as
3
12. We can also see that when 1 4 is
itself divided into three equal pieces, each
piece represents one-twelfth, so two-thirds
of
1 4 is two-twelfths. (See Figure 1-4.) This
is why
2 3 1
4
⋅ is 2
12.
EXAMPLE
Perform the multiplication with the rule a
b c
d ac
bd
⋅
 = .

2 3 4
5
⋅
According to the rule, we multiply the numerators, 2 and 4, and the
denominators, 3 and 5 to obtain
2
3 4
5 24
35 8
15
⋅
⋅
⋅
  
=  = .
One-third
of one-fourth
FIGURE 1-4
EXAMPLE
Perform the multiplication with the rule

Chapter 1 FraCtions 3
DeMYstiFieD / algebra DeMYstiFieD / Gibilisco / 000-0 / Chapter 1
PRACTICE
Perform the multiplication with the rule a
b c
d ac
bd
⋅
 = .
1.
7 6 1
4
⋅=
2. 8
15 6
5
⋅=
3. 5
3 9
10
⋅=
4. 40
9 2
3
⋅=
5. 3
7 30
4
⋅=
SOLUTIONS
1. 7
6 1
4 71
64 7
24
⋅=
⋅
⋅ =
2. 8
15 6
5 86
15 5 48
75
⋅=
⋅
⋅ =
3. 5
3 9
10 59
31 0 45
30
⋅=
⋅
⋅ =
4. 40
9 2
3 40
2
93 80
27
⋅=
⋅
⋅ =
5. 3
7 30
4 33
0
74 90
28
⋅=
⋅
⋅ =
Multiplying Fractions and Whole Numbers
We now develop a rule for multiplying a whole number and a fraction. To see
how we can multiply a fraction by a whole number, we use a pie chart to find
the product
4
2
9
⋅.
The shaded region in Figure 1-5 represents
2
9.
We want a total of four of these shaded regions. See Figure 1-6.
As we can see, four of the
2
9 regions give us a total of eight 1
9’s. This is
why
4 2
9
⋅ is 4
2
9 42
9 8
9
⋅=
⋅
=.
In general, when multiplying
W a
b
⋅ (where
W is a whole number), we have
Wa⋅ of the 1
b fractions. This fact gives us the multiplication rule W a
b Wa
b
⋅=.
SOLUTIONS✔
PRACTICE
Perform the multiplication with the rule

4 alGebra D e myst i fieD DeMYstiFieD / algebra DeMYstiFieD / Gibilisco / 000-0 / Chapter 1
That is, the numerator of the product is the whole number times the fraction’s
numerator, and the denominator is the fraction’s denominator.
An alternate method for finding the product of a whole number and a frac-
tion is to treat the whole number as a fraction—the whole number over one—
and then multiply as we would any two fractions. This method gives us the
same rule:

Wa
b Wa
bWa
b Wa
b
⋅= ⋅=
⋅
⋅ =
11
EXAMPLE
Find the product 5 2
3
⋅ with each