Transcript 5. Dividing Mixed Numbers

Taking the Fear
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Dividing
Mixed
Numbers
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As in our previous discussions
with mixed numbers, let’s begin
with a “real world” example…
What is the cost per pound
if 33/4 pounds of pears cost
$4.50?
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33/4 pounds is the same amount as 15
fourths of a pound.
If 15 fourths of a pound cost $4.50 (that is,
450 cents), then each fourth
of a pound costs…
450 cents ÷ 15 (or 30 cents).
And since there are 4 fourths of a pound in
a pound, each pound of pears costs
4 × 30 cents or $1.20.
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The cost per pound in dollars is given
by the formula…
dollars per pound =
total cost in dollars ÷ number of pounds.
This formula is the same whether we are
dealing with whole numbers,
common fractions or mixed numbers.
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Rewriting $4.50 as $41/2, our answer
takes the form…
cost per pound =
41/2 dollars ÷ 33/4 pounds
which can be rewritten as…
cost per pound =
(41/2 ÷ 33/4) dollars per pound
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If you are uncomfortable using mixed
numbers to perform the division, you
may use improper fractions and rewrite the
above computation in the form…
cost per pound =
(9/2 ÷ 15/4) dollars per pound
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From our study of fractions we know
that…
41/2 ÷ 33/4 dollars per pound
= 9/2 ÷ 15/4 dollars per pound
= 9/2 × 4/15 dollars per pound
= 36/30 dollars per pound
= 6/5 dollars per pound
= 11/5 dollars per pound
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And since there are 100 cents per
dollar…
1/ of
5
a dollar = 100 cents ÷ 5 = 20 cents
and therefore…
11/5 dollars per pound = $1.20 per pound
which agrees with our previous answer.
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As we shall show later in this
presentation, it is possible to divide
mixed numbers without having to first
convert them to improper fractions.
However, this can be quite tedious.
So while the idea of converting mixed
numbers to improper fractions is quite
common, it is especially useful when we
deal with division.
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However, converting the division of
mixed numbers to division of improper
fractions tends to obscure what is
actually happening.
For example, suppose we want to divide
41/2 by 21/2 , We could begin by rewriting
41/2 ÷ 21/2 as 9/2 ÷ 5/2. We would then use
the “invert and multiply” rule to obtain
9/ × 2/ or 9/ . Finally, we divide 9 by 5
2
5
5
and obtain 14/5 as the answer.
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Notes
Since the definition of division hasn’t
changed 41/2 ÷ 21/2 means the number
we must multiply by 2 1/2 to obtain 41/2
as the product.
A quick check shows that this is the case…
21/2 × 14/5 =
5/
2
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× 9/5 =
9/ =
2
41/2
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Notes
As a “reality check”, it is helpful to
estimate an answer before proceeding with
the actual computation. Noticing that 41/2 is
“around” 4 and 21/2 is “around” 2, we can
estimate that the answer should be
“around” 4 ÷ 2 or 2.
Actually, since 5 ÷ 21/2 is exactly 2,
41/2 ÷ 21/2 must be a “little less” than 2.
Thus, 14/5 is a plausible answer.
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Notes
In terms of relative size, what the quotient
tells us is that not only is 41/2 almost twice
as much as 21/2 but it’s exactly 14/5 times
as much.
As a practical application, suppose we can
buy 21/2 pounds of cheese for $ 41/2. Then the
price per pound (that is, “dollars per pound”)
is obtained by dividing $41/2 by 2 1/2 pounds.
The quotient tells us that the cost of the
cheese is $14/5 or $1.80 per pound.
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For additional practice, let’s use the
above method to express
6 1/3 ÷ 13/4 as a mixed number.
61/3 ÷ 13/4
= 19/3 ÷ 7/4
= 19/3 × 4/7
= 76/21
= 76 ÷ 21
= 313/21
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Notes
To check that our answer is at least
reasonable, we observe that rounded off to
the nearest whole number 61/3 = 6 and
13/4 = 2. Hence, our answer should be
“reasonably close to” 6 ÷ 2 or 3.
However, once we obtain our answer, we
can check to see if it’s correct by
remembering that 61/3 ÷ 13/4 = 313/21
means that 313/21 × 13/4 = 61/3.
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Thus, to check our answer we
compute the value of 313/21 × 13/4
to verify that it is equal to 61/3.
313/21 × 13/4
= 76/21 × 7/4
= (76 × 7)/(21 ×4)
= (19 × 2 × 2 × 7)/(3 × 7 × 2 × 2)
= 19/3
= 61/3
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As we mentioned earlier, while
converting mixed numbers to improper
fractions gives us the desired result when
we divide two mixed numbers, the fact is
that it doesn’t give us much insight as to
what is actually happening.
From a mathematical perspective, it would
be nice to know that the arithmetic of
mixed numbers is self-contained (at least
in the sense that we aren’t forced to
rewrite mixed numbers as improper
fraction in order to do the arithmetic).
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From a teaching point of view, we might
want to illustrate how we can divide mixed
numbers in ways that may be more intuitive
to students.
Using Mixed Numbers as Adjectives
Modifying the Same Noun.
To divide 41/2 by 21/2, think of both mixed
numbers as modifying, say, “a carton of
books” where each carton contains 2 books1
note
1 More generally, we would use the (least) common multiple of both denominators.
For example, had the problem been 41/3 ÷ 25/7 , we would have assumed that
each carton contained 21 books and then multiplied both the dividend and the
divisor by 21.
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In this case, 41/2 cartons is another
name for 9 books and 21/2 cartons is
another name for 5 books. Therefore, we
may visualize the problem in terms of the
following steps…
41/2 ÷ 21/2 = 41/2 cartons ÷ 21/2 cartons
= 9 books ÷ 5 books
= 9 ÷ 5 = 14/5
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Multiplying Both Numbers in a Ratio
by the Same Number
Students might find it interesting to see
that we can divide mixed numbers by
“translating” the problem into an
equivalent whole number problem without
having to refer to such things as books
and cartons.
The key is that it is still a fact that we do
not change a quotient when we multiply
both the dividend and the divisor by the
same (non zero) whole number.
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To this end, if we are given the
division problem 41/2 ÷ 21/2, we simply
multiply both numbers by 2…2
and obtain…
41/2 ÷ 21/2
= (41/2 × 2) ÷ (21/2 × 2)
=9÷5
= 14/5
note
2 When we multiply a mixed number by the denominator of its fractional part we always
obtain a whole number. For example, 5 × 42/5 = 5 × (4 + 2 2/5) = (5 × 4) + (5 × 2/5) = 20 + 2 = 22.
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We can now use the above method to
write 61/3 ÷ 23/5 as a mixed number.
Since the denominators are 3 and 5, we
can eliminate them by multiplying
both numbers by 15.
15 × 61/3
= (15 × 6) + (15 ×
1/ )
3
= 90 + 5
= 95
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15 × 23/5
= (15 × 2) + (15 × 3/5)
= 30 + 9
= 39
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Since…
15 × 61/3 = 95
and
15 × 23/5 = 39
…then, 61/3 ÷ 23/5
= 95 ÷ 39
= 217/39
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As a check we see that…
217/39 × 23/5
= 95/39 ×
13/
5
= (95 ×13)/(39 × 5)
= (19 × 5 × 13)/(3 × 13 × 5)
= 19/3
= 61/3
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Notes
In terms of relative size, the above result
tells us that 61/3 is approximately 21/2 times
as great as 23/5.
(It is actually 217/39 times as great.)
We don’t usually think about using
common denominators when we want to
divide two fractions, but we can. The
denominator of a fraction is the noun and
when we divide two numbers that
modify the same noun, the nouns “cancel”.
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Thus, if we decide to rewrite the mixed
numbers as improper fractions, we obtain…
61/3 ÷ 23/5
= 19/3 ÷
13/
5
= (19 × 5) /(3 × 5) ÷ (13 × 3)/(5 × 3)
= 95/15 ÷ 39/15
= 95 fifteenths ÷ 39 fifteenths
= 95 ÷ 39
= 217/39
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While the topic might be beyond the
scope of the elementary school student, it
is interesting to note that the long division
algorithm for whole numbers, as a form of
rapid subtractions, also applies to the
division of mixed number.
For example, 61/3 ÷ 23/5 = ( ) means the
same thing 23/5 × ( ) = 61/3.
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Using trial-and error to solve this
equation we see that…
23/5 × 1 = 23/5
23/5 × 2 = 46/5 = 51/5
less than 61/3
In other words, 61/3 is
more than twice as big
as 23/5 but less than
three times as big.
23/5 × 3 = 69/5 = 74/5
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greater than 61/3
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The difference between 61/3 and 51/5 is
61/3 – 51/5 = 65/15 – 53/15 = 12/15. Hence, if we use
the long division algorithm we see that…
2 R
23/5
61/3
51/5
122/15
15
…and if we now write
the remainder over the
divisor, we see that…
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61/3 ÷ 23/5 =
2/
1
15
2 +
23/5
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12/15
23/5
We can simplify the complex fraction
above by multiplying numerator and
denominator by 15.
15 × 12/15
= (15 × 1) + (15 × 2/15)
= 15 × 2
= 17
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15 × 23/5
= (15 × 2) + (15 ×
3/ )
5
= 30 + 9
= 39
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Therefore…
23/5
61/3
12/15
=
3
2 /5
17/
39
61/3 ÷ 23/5
2/
1
15
=2 +
23/5
= 2 + 17/39
= 217/39
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4/
5
× 100 = 80%
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This completes our
discussion of the four
basic operations of
arithmetic
using mixed numbers,
and in our next
presentation we shall
introduce the
notion of percents.
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