18. Efficient ways to Multiply Fractions
[Play
Video] 2-3 minutes. Multiplying Fractions with
cancellation of common factors done first (recommended) or not,
with more simplification to be done later.
After reading this page,
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calculate a few products of fractions with and with
the cancellation methods described below for "efficient"
multiplication, or more precisely efficient or easier simplification
after (cross) cancellation of common factors.
-
Find examples in which non-cancellation of
factors makes the numerators and denominator multiples of ten.
For such examples, is it easier to cancel the common factors (powers
of 2 and 5) before calculating the product?
-
If one of the fractions in a product can be
simplified (reduced), should you do so to make the simplification of
the product easier?
In general, we may multiply fractions as
follows:.
In the resulting fraction, the the numerator (top) is a product of the
numerators of the factors and the denominator (bottom) is a product
of the denominator of the factors. The foregoing describes the
first method for multiplying fractions. After that, we would simplify the
resulting fraction by canceling common factors in the products numerator
and denominator. The rule here is multiply first and cancel second.
But this order can be changed. Cancellation first leads
to smaller numbers and a quicker way (usually) to get the simplified form
of the product.
Example:
25
33
|
×
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44
75
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=
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25×44
33×75
|
Now instead of compute the products of the numerators and denominators
(and then factoring the products to cancel common factors), we take
advantage of the situation that the original numerators and denominators
provide factors of the product numerators, and factor further to locate
common factors that will cancel. Cancelled factors are
crossed-out.
25
33
|
×
|
44
75
|
=
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25×44
33×75
|
=
|
25×4×11
3×11×3×25
|
=
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4
3×3
|
=
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4
9
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Here we kept the original numerators and denominators and then factored
them in a way that would help simplification (lowering terms) in the
product fraction. So we cancelled the 25 and 11 after factorization. Then
after no further factors could be cancelled, computed the decimal
representation of the product numerator and denominator in reduced form.
Here is the above product computation revisited with in place
cancellation - the same calculation with a cosmetic change.
25
33
|
×
|
44
75
|
=
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25
3×11
|
×
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4×11
3×25
|
=
|
1
3
|
×
|
4
3
|
=
|
4
9
|
The first way we did the cancellation (that is, multiplying the
fractions together and then factoring to reduce) provides justification
for the cancellation of common factors in the original fractions before
multiplication is done.
Triple Product Example
22
21
|
×
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48
33
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×
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147
64
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=
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22×48 ×147
21 × 33 ×64
|
|
|
|
|
|
=
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2 ×11×3 ×16×3 × 49
3 × 7 ×3×11×4×16
|
|
|
|
|
|
=
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2 ×11×3
×16×3 × 49
3 × 7 ×3×11×4×16
|
|
|
|
|
|
=
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3×49
7 ×4
|
|
|
|
|
|
=
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3×7×7
7 ×4
|
|
|
|
|
|
=
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3×7
4
|
|
|
|
|
|
=
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21
4
|
|
|
|
|
|
=
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5
|
1
4
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Observe the product is a fraction given by the product of the numerators
of the fraction fractions over the product of the denominators of the
denominators of the factors. But instead of calculating those products,
we factor them. The factorization may be complete as in the case of
prime factorization or it may partial. Above we or I saw that 16 was a
common factor or divisor for both 48 and 64. Such common factors can be
cancelled immediately and do not need to be factored further.
Algebraic View
for reading as part of algebra skill development - optional
reading for now
Algebraic Shorthand Description
A×B
C×D
|
×
|
D×E
B×F
|
=
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A×B
C×D
|
×
|
D×E
B×F
|
=
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A
C
|
×
|
E
F
|
This description is rather complicated, it can be ignored. None the less,
the challenge is to understand what is says or suggests, good luck.
Understanding is a sign (not a guarantee) of algebra mastery.
Euclid's Algorithm, Prime Decomposition factorization, and Rules for
recognizing multiples of whole numbers 2, 3, 5, 9, 10, 25, all
provide methods to identify and cancel common factors. These
methods were presented briefly or not in the previous lesson.
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Secondary
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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Appetite.
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