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Taking the Fear
out of Math
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#6
Dividing
1 ÷1
3
3
Common
Fractions
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Division of Fractions
Using the Adjective/Noun Theme
Wherever there is multiplication,
division cannot be far behind.
Our adjective/noun theme gives us an
easy way to convert division problems
that involve fractions into equivalent
division problems that involve only
whole numbers.
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Let’s begin by computing the
quotient of 6 apples ÷ 2 apples.
By the definition of “unmultiplying”,
6 apples ÷ 2 apples means the number
we must multiply 2 apples by to obtain
6 apples as the product.
Clearly, 6 apples ÷ 2 apples = 3.
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That is, we must multiply 2 by 3 apples
to obtain 6 apples as the product.
In other words, 2 × 3 apples = 6 apples.
Notes
The answer is not 3 apples. 3 apples
would be the answer to the division
problem 6 apples ÷ 2.
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Notes
One way to remember this is that the nouns
behave just the same way as numbers do
when we divide.
In the same way that we can cancel a
common factor from the numerator and the
denominator, we can also cancel a noun if it
appears as a factor in both the numerator
and the denominator. In other words…
6 apples
2 apples = 3
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Notes
If we divide two amounts that have the
same noun, we are finding the relative
size of one of the amounts compared
to the other.
Thus, for example, when we say that
6 apples ÷ 2 apples = 3,
we’re saying that compared to 2 apples,
6 apples are 3 times as much.
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Notes
Applying this to fractions, if we are
dividing two fractions that have the
same denominator we can “cancel”
the denominators.
For example, 6/7 ÷ 2/7 = 31
That is, compared to 2/7, 6/7 is 3 times as much.
note
1 This result might be easier to see if we rewrite the problem as
6 sevenths ÷ 2 sevenths = 3.
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Let’s apply this idea to the problem
of dividing two fractions that have
different denominators.
For example, suppose we want to find
the answer to the problem, 3/7 ÷ 2/5. In
light of our above discussion, we should
rewrite both fractions so that they have
the same denominator (and therefore we
can then cancel them).
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More specifically…
3
7
2
5
=
3×5
7×5
=
2×7
5×7
3
2
Hence…
÷
7
5
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=
15
35
=
14
35
15
14
=
÷
35
35
=
15
14
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Rather than seeing this abstract
approach, elementary school students
might relate better to the more visual corn
bread model.
In this model, 3/7 ÷ 2/5 becomes
3/ of a corn bread ÷ 2/ of a corn bread, and
7
5
since a common multiple of 5 and 7 is 35,
let’s assume that our corn bread is
presliced into 35 pieces of equal size.2
note
2
Notice how the corn bread model replaces the abstract symbol 1/35
by the more visual “1 piece of the corn bread”
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3/
7
Then…
of the corn bread ÷ 2/5 of the corn bread
= 3/7 of 35 pieces ÷ 2/5 of 35 pieces
= 15 pieces ÷ 14 pieces
= 15 ÷ 14
=
15/
14
Thus, 3/7 ÷ 2/5 = 15/14
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From a visual point of view, our corn bread
has been pre-sliced into 35 equally sized
pieces (a common multiple of 5 and 7).
corn bread
1 2 1/7
3 4 5 6 7 1/7
8 9 10 11 121/7
13 14 15 16 171/7
18 19 20 21 221/7
23 24 25 26 271/7
28 29 30 31 321/7
33 34 35
3/
3/ of 35 pieces = 15 pieces
of
the
corn
bread
=
7
7
corn bread
1 2 3 1/5
4 5 6 7 8 9 101/5
11 12 13 14 15 16 171/5
18 19 20 21 22 23 241/5
25 26 27 28 29 30 311/5
32 33 34 35
2/
2/ of 35 pieces = 14 pieces
of
the
corn
bread
=
5
5
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3/
7
Therefore…
of the corn bread ÷ 2/5 of the corn bread
corn bread
1 2 1/7
3 4 5 6 7 1/7
8 9 10 11 121/7
13 14 15 16 171/7
18 19 20 21 221/7
23 24 25 26 271/7
28 29 30 31 321/7
33 34 35
=
15
14
corn bread
1 2 3 1/5
4 5 6 7 8 9 101/5
11 12 13 14 15 16 171/5
18 19 20 21 22 23 241/5
25 26 27 28 29 30 311/5
32 33 34 35
15 pieces ÷ 14 pieces =
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Demystifying the
“Invert and Multiply ” Rule
The underlying theme of this course is to
show how the adjective/noun theme can be
used as an extra tool for you to use, no
matter what other “delivery systems”
you like to use.
It is not the goal of this course to
de-emphasize other ways of looking at
topics. With this in mind, we will look at
the more traditional way of explaining
the algorithm for dividing fractions.
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You may have heard the little ditty…
“Ours is not to reason why.
Just invert and multiply.”
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To begin to understand the “mystic
formula”, let’s discover what happens
when we compute the product of
4/ and 7/ .
7
4
Recall when we multiply two fractions, we
“multiply the numerators and we
multiply the denominators”.
4
×
7
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7
4
=
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4×7
7×4
28
=
28
= 1
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Notes
There was nothing special about our choice
of 4/7 and 7/4. Specifically, if we let n denote
the numerator of a fraction and let d denote
the denominator of the fraction, then d/n is
called the reciprocal of n/d .
What the above problem illustrates is
that the product of any fraction
and its reciprocal is 1.
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Notes
Namely…
n
×
d
d
n
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n×d
=
d×n
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n×d
=
n×d
=1
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Notes
The only rational number that doesn’t
have a reciprocal is 0. 3
1/
0
would be the answer to 1÷ 0 which is
the number which when multiplied by 0
would equal 1. However, since any
number multiplied by 0 is 0, there is
no such number.
note
3 In the language of fractions we talk about reciprocals. However, a fraction is
just one way to represent a rational number. Hence, when we talk about rational
numbers, rather than use the term “reciprocal”, we refer to it as the
multiplicative inverse.
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Notes
Knowing the fact that multiplying a number
by 1 doesn’t change the number gives us a
two step process to find the quotient of two
fractions.
For example, the quotient 5/11 ÷ 4/7 means
the number by which we must multiply
4/ to obtain 5/ as the product.
7
11
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Notes
In words, our approach might be something
like this…
Step 1: Start with…
Step 2: Multiply by…
4
7
7
4
(thus obtaining
1 as the product)
Step 3: Next, multiply by 5/11 to obtain
the correct answer (namely 5/11).
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Notes
Combining Steps 2 and 3, we multiplied
4/ by 7/ × 5/ to obtain 35/ ; and
7
4
11
44
as a check we see that…
4
35
=
×
7
44
4
35
×
7
44
=
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5
11
1
5
=
×
1
11
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Notes
To see how this is related to the “invert and
multiply rule”, we have now shown that
5 / ÷ 4/ = 5 / × 7/ .
11
7
11
4
We can then see that we obtained the
right side of the equation from the left
side by leaving the first fraction alone,
changing the division sign to a times sign,
and inverting the second fraction.
5/
7
4
÷ //47
11 ×
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Pedagogical Note
Even when the numbers are not whole
numbers, mathematicians still like to use
the fraction bar instead of the division
symbol. In other words, rather than write
the division of two fractions in the form…
3
7
2
÷
5
they prefer to write
(where the heavier fraction bar is used to
separate the dividend from the divisor).
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3
7
2
5
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The above expression is referred to as a
complex fraction. Complex fractions have
the same properties as common fractions;
at least in the sense that…
(1) If the denominator of the complex
function is 1, the value of the complex
fraction is equal to the numerator,
and (2) if we multiply numerator and
denominator of the complex fraction by the
same non-zero number, we obtain an
equivalent complex fraction.
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This gives us yet another way
to visualize why the “invert and
multiply” rule works.
Namely, given the
problem 3/7 ÷ 2/5 , we
rewrite it in the form…
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3
7
2
5
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3
7
2
5
×
×
5
2
=
5
2
=
3
7
×
1
5
2
3
5
=
×
7
2
If we multiply the denominator of the
complex fraction by 5/2 , it equals 1.
However, to make sure that the value of the
complex fraction doesn’t change, we must
make sure that if we multiply the
denominator by 5/2 we must also multiply
the numerator by 5/2. Thus, we obtain…
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An Important Caveat
Because of what happens in whole number
arithmetic, students identify multiplication
with “bigger” and division with “smaller”.
However, we’ve just seen that division
can tell us the size of one number
relative to the size of another number.
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Thus, the fact that 12 ÷ 1/2 = 24 simply
means that with respect to 1/2, 12 is 24
times as great.
In short, when we divide a given number
by a number less than 1 the quotient is
greater than the given number, but if we
divide it by a number that is greater than 1,
the quotient is less than the given number.
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In a similar way, if we multiply a given
number by a number that’s less than 1 then
the product is less than the given number.
For example, 2/3 × 5/7 (10/21) is less than
both 2/3 ( i.e., 14/21 ) and 5/7 (i.e., 15/21).
The reason is that 2/3 × 5/7 means 2/3 of 5/7,
and 2/3 of a number is less than the number.
In a similar way, 2/3 × 5/7 = 5/7 × 2/3 = 5/7 of 2/3
and 5/7 of a number is less than the number.
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From a more visual
point of view, the
rectangle whose
dimensions are
2/ inches by 5/ inches
3
7
is contained within the
2
rectangle whose
3
dimensions are
2/ inches by 1 inch.
3
Hence, 2/3 × 5/7 is
less than 2/3 × 1.
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5
7
1
1
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In a similar way it
is also contained within
the rectangle whose
dimensions are 5/7
inches by 1 inch.
5
7
2
3
1
Hence, 2/3 × 5/7 is
less than 5/7 × 1.
1
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3
5
÷13
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In our next presentation,
we will present a few
“real world” examples
that require us to divide
one fraction by another
fraction.
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