Fractions Of Whole Numbers Worksheet Page 5
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1
2
3
4
5
6
7
8
9Solution.
6
3 2
3
9
3 3
3
6
9
Since
=
=
and
=
=
, we find
=
.
14
7 2
7
21
7 3
7
14
21
The following theorem shows that two fractions are equivalent if and only if
their cross-products are equal.
Theorem 18.1
a
c
=
if and only if ad = bc.
b
d
Proof.
a
c
a
ad
c
bc
ad
bc
Suppose first that
=
. Since
=
and
=
, we must have
=
.
b
d
b
bd
d
bd
bd
bd
But this is true only when ad = bc.
ad
bc
Conversely, if ad = bc then
=
. By the Fundamental Law of Fractions
bd
bd
a
ad
c
bc
a
c
we have
=
and
=
. Thus,
=
.
b
bd
d
bd
b
d
Example 18.3
12
x
Find a value for x so that
=
.
42
210
Solution.
By the above theorem we must have 42 x = 210 12. But 210 12 = 60 42
42 60
so that x =
= 60.
42
Simplifying Fractions
a c
a
a c
When a fraction
is replaced with
, we say that
has been simplified.
b c
b
b c
a
We say that a fraction
is in simplest form (or lowest terms) if a and b
b
3
have no common divisor greater than 1. For example, the fraction
.
7
a
We write a fraction
in simplest form by dividing both a and b by the
b
GCF(a,b).
Example 18.4
Find the simplest form of each of the following fractions.
240
399
(a)
(b)
.
72
483
Solution.
4
3
2
(a) First, we find GCF(240,72). Since 240 = 2
3 5 and 72 = 2
3
we find
3
GCF (240, 72) = 2
3 = 24. Thus,
240
240
24
10
=
=
.
72
72
24
3
5
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