Page 82 - ArithBook5thEd ~ BCC

The Fundamental Property of Fractions:

If we multiply both the numerator and denominator of a fraction by any nonzero
c,then the new fraction is equivalent to the original one, and represents the same
number:
a a · c
= .
b b · c

2
Example 89. Write two fractions equivalent to .
3
Solution. We use the fact that
2 2 · c
=
3 3 · c
for any nonzero c. Picking two values for c, say, 6 and 7, we get two fractions equivalent to 2/3:

2 2 · 6 12 2 2 · 7 14
= = and = = .
3 3 · 6 18 3 3 · 7 21
Of course, other choices of c would have produced other fractions equivalent to 2/3.

3.5.1 Cancellation and Lowest Terms

The boxed rule above produces equivalent fractions with higher (larger) terms. It is sometimes possible
to go the other way, producing lower (smaller) terms. If both numerator and denominator have a
common factor – a whole number greater than 1 which divides them both with zero remainder – we
can “cancel” it by division. The two quotients become the terms of an equivalent fraction with lower
terms. For example, the numerator and denominator of 18 have the common factor 6. Thus
24
18 18 ÷ 6 3
= = .
24 24 ÷ 6 4
In general,

For any non-zero c,
a a ÷ c
= .
b b ÷ c

This method of obtaining lower terms is called cancellation or cancelling out.It is often indicated as
follows:
✚❃
18 ✚ 18 3 3
= = .
24 ✚❃ 4 4
✚ 24
This is useful short-hand, but it has one disadvantage: the common factor (6, in this case) is not made
visible. To ensure accuracy, you can show the common factor explicitly, before cancelling it. Thus,
explicitly,
18 3 · 6 3 · ✁ 6 ✁✕ 3
= = = .
24 4 · 6 4 · ✁ 6 ✁✕ 4

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