Download Fractions Are Numbers Too: Part 6

Mathematics
as a
Second Language
An Innovative Way
to
Better Understand Arithmetic
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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Part 6
Fractions are
numbers, too
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© 2007 Herbert I. Gross
Dividing
÷
Common Fractions
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© 2007 Herbert I. Gross
Our is not to
reason why.
Just invert and
multiply!
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© 2007 Herbert I. Gross
The mystery behind the “invert and multiply”
rule is not difficult to unravel once we
understand the following…
(1) To invert a fraction means to interchange
its numerator and denominator. Thus for
example, if we invert 2/3, we obtain 3/2.
(2) When we invert a fraction, it is called the
reciprocal of the given fraction. Every
rational number except zero (0) has a
reciprocal.
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© 2007 Herbert I. Gross
(3) Based on our definition of multiplication it
follows that the product of any common
fraction and its reciprocal is one.
Example
2/3 × 3/2 = 6/6 = 1
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© 2007 Herbert I. Gross
More generally, if we let n denote any
numerator and d any denominator, the fact
that n × d = d × n means that…
n/d × d/n =
(n × d) / (d × n) =
(n × d) / (n × d) = 1
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© 2007 Herbert I. Gross
Illustrating the Division Algorithm
Suppose we want to compute the quotient
2/5 ÷ 3/7.
The algorithm works as follows…
Step 1: Leave the first fraction unchanged.
2/5
Step 2: Replace the “division sign” by the
“times sign”.
© 2007 Herbert I. Gross
2/5 ×
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Step 3: Replace the second fraction by its
reciprocal.
2/5 × 7/3
Step 4: Perform the resulting multiplication
problem.
2/5 × 7/3 =
(2 × 7) / (5 × 3) =
14/15
© 2007 Herbert I. Gross
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Thus the “invert and multiply” algorithm tells us
that dividing a number by a common fraction
means the same thing as multiplying the
number by the reciprocal of the common
fraction.
However, two major question remain, namely...
(1) How did this algorithm come about?
(2) What does the quotient mean?
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© 2007 Herbert I. Gross
To answer the question about how the
algorithm was developed, let’s revisit the
division problem 2/5 ÷ 3/7.
By the definition of division, we are looking for
a number which when multiplied by 3/7 gives
us 2/5 as the product.
We know that if we multiplied 3/7 by 7/3 the
product would be 1; and if we multiply
1 by 2/5, the answer would be 2/5.
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© 2007 Herbert I. Gross
Hence the sequence of steps is:
We first multiply 3/7 by 7/3 to obtain 1 as the
product.
Then we multiply 1 by 2/5 to obtain 2/5 as the
product.
Diagrammatically
3/7
× 7/3
= 1
× 2/5
= 2/ 5
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© 2007 Herbert I. Gross
Looking at the boxed fractions we see that:
2/5 ÷ 3/7 = 7/3 × 2/5 = 2/5 × 7/3 .
Comparing the problem with the answer we
see that…
 We left the first fraction (2/5) alone,
 changed the division symbol ( ÷ ) to a
times symbol ( × ),
 and inverted the second fraction ( 7/3)
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© 2007 Herbert I. Gross
Check
Notice that…
2/5 × 7/3 = 14/ 15
3/7 × 14/15 = 2/5
This illustrates where the “invert and
multiply” rule comes from.
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© 2007 Herbert I. Gross
A Practical Application
To see what the quotient represents, let’s make
up a “real life” problem for which 2/5 ÷ 3/7 gives
us the correct answer.
Suppose an object, which is moving at a
constant speed, travels 2/5 of a mile in 3/7
of a minute. What is its speed in miles per
minute?
Answer: 14/15 miles per minute
( or 14 miles per 15 minutes).
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© 2007 Herbert I. Gross
Solution
“Miles per minute” is a rate; and we compute
the rate by dividing the number of miles an
object travels by the amount of time it took to
travel the distance. Moreover, it makes no
difference whether the numbers are fractions or
whole numbers.
So in this case we see that the number of miles
is 2/5 and the number of minutes is 3/7. The
answer is (2/5 ÷ 3/7) miles per minute.
And as we have already shown, 2/5 ÷ 3/7 =
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14/15.
© 2007 Herbert I. Gross
Other Methods for
Dividing
Common Fractions
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© 2007 Herbert I. Gross
Method 1
* Converting the Fractions to Whole Numbers *
Let’s revisit our solution to the previous
illustrative problem.
 We may again utilize the power of finding
a common denominator. In this case we
are dealing with fifths and sevenths.
5, 10, 15, 20, 25, 30, 35,
35 40
7, 14, 21, 28, 35
35, 42, 49, 56
© 2007 Herbert I. Gross
35 is a common multiple.
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 We know the object moves 2/5 of a mile
every 3/7 of a minute. Hence if the object
moves 3/7 of a minute 35 times, it will
move 2/5 of a mile 35 times.
 2/5 of a mile 35 times (that is 35 × 2/5 miles
or 2/5 of 35 miles) is 14 miles.
 3/7 of a minute 35 times (that is 35 × 3/7
minutes or 3/7 of 35 minutes) is 15 minutes.
 Therefore the object moves 14 miles every
fifteen minutes or 14/15 miles per minute.
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© 2007 Herbert I. Gross
Method 2
* The Generalizing Method *
Method 1 dealt solely with the special example
of an object moving at a constant speed.
However, there are other times when we
might want to compute the quotient
2/5 ÷ 3/7 = 14/15.
 We already know that if we multiply both
numbers in a rate by the same (non zero)
number, we do not change the rate.
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© 2007 Herbert I. Gross
Just as we did in Method 1, observe that 35
is a common multiple of 5 and 7.
 Hence we can immediately convert
2/5 ÷ 3/7 to an equivalent ratio by multiplying
both numbers by 35 to obtain the equivalent
ratio...
(35 × 2/5) ÷ (35 × 3/7) =
14 ÷ 15 =
14/15
© 2007 Herbert I. Gross
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Method 3
* Introducing a Compound Fraction *
Definition
A compound fraction is one in which both the
numerator and denominator are themselves
fractions..
 By the definition of a common fraction,
when we write n/d we assume that n and d
are whole numbers..
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© 2007 Herbert I. Gross
However, mathematicians use fraction
notation to represent division even if the
numbers are not whole numbers.
Thus for example, a mathematician might
write…
2
5
to represent the division problem
3
2/5 ÷ 3/7
7
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© 2007 Herbert I. Gross
The beauty of this notation is that it shares the
same properties as those possessed by
common fractions. For example, the fact that
n × 1 = 1 for any number means that n/1 = n
for any number n (not just whole numbers).
And since we can multiply both numbers in a
division problem by the same non-zero
number without changing the quotient, it
means that we can multiply numerator and
denominator by the same number without
changing the ratio named by the compound
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fractions.
© 2007 Herbert I. Gross
2
Hence start with…
5
We can obtain an equivalent
3
compound fraction whose
7
denominator is 1 by multiplying the numerator
and denominator by 7/3.
2/5 ÷ 3/7 =
2
5
3
7
=
2
7
×
5
3
3 × 7
7
3
2/5 × 7/3
=
1
= 2/5 × 7/3
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© 2007 Herbert I. Gross
Method 4
* The Quotient of Two Quantities
with the Same Units *
This method depends on the fact that we can
cancel the same unit from the numerator and
denominator.
Example
What is the answer to the division problem
6 apples ÷ 2 apples.?
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© 2007 Herbert I. Gross
If you said the answer was 3 apples, you
found the correct answer to a different
problem; namely 6 apples ÷ 2.
 The answer to the question is 3, not 3 apples.
 To see why, remember the definition for
division as “unmultiplying”. That is, 6 ÷ 2 = ( )
means the same thing as ( ) × 2 = 6.
 In this context, 6 apples ÷ 2 apples = ( )
means the same as ( ) × 2 apples = 6 apples.
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© 2007 Herbert I. Gross
 Clearly it takes 3 groups of 2 apples to equal
6 apples, not 3 “apple groups”. In fact
according to our principle for how we multiply
quantities, 3 apples × 2 apples would equal
6 “apple apples” not 6 apples.
With this in mind, we rewrite 2/5 ÷ 3/7 using
common denominators.
2/5 ÷ 3/7 = 14/35 ÷ 15/35 =
14 thirty-fifths
15 thirty-fifths
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© 2007 Herbert I. Gross
Key Point
If 2 quantities are measured in the same unit,
the quotient of these two quantities will be the
same as the quotient of the adjectives.
Stated a different way, the quotient represents
the size of one quantity with respect to the
other.
Example
15 inches ÷ 3 inches = 5 because 15 inches is
5 times as much as 3 inches; that is
5 × 3 inches = 15 inches.
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© 2007 Herbert I. Gross
Important Note
Students often confuse 6 ÷ 2 with 6 ÷ 1/2. As
mentioned previously, when we divide two
adjectives, we are finding the size of one
relative to the size of the other.
In this context, 6 ÷ 2 = 3 means that 6 is 3
times the size of two.
On the other hand, 6 ÷ 1/2 = 12 means that 6
is 12 times the size of one half.
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© 2007 Herbert I. Gross
Example 1
It would take 3 two pound packages of
chocolate to equal 6 one-pound packages.
But it would take 12 half-pound packages of
chocolate to equal 6 one-pound packages.
Example 2
It would take 3 two dollar bills to equal 6 dollars.
But it would take 12 half-dollars to equal
6 dollars.
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© 2007 Herbert I. Gross
Method 5
* The Corn Bread Model *
The corn bread is a visual form of method 4.
More specifically given the problem 2/5 ÷
3/7, we may assume that both fractions are
modifying the corn bread.
That is, the problem becomes…
2/5 of the corn bread ÷ 3/7 of the corn bread.
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© 2007 Herbert I. Gross
Then in our usual way the corn bread has
been pre-sliced into 35 equally sized pieces
(a common multiple of 5 and 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
2/5 of the corn bread = 2/5 of 35 pieces = 14 pieces
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
3/7 of the corn bread = 3/7 of 35 pieces = 15 pieces
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© 2007 Herbert I. Gross
Therefore:
2/5 of the corn bread ÷ 3/7 of the corn bread
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
=
14
15
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
© 2007 Herbert I. Gross
14 pieces ÷ 15 pieces =
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Summary
Although the algorithm for dividing common
fractions might seem a bit “mysterious”, the
important point is that the definition of division
is the same regardless of whether we are
dividing whole numbers or fractions.
The “invert and multiply” algorithm is but one of
several ways to compute the quotient of two
fractions.
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© 2007 Herbert I. Gross
Perhaps the most user-friendly method is to
have two fractions modify a corn bread, and
have the corn bread pre-sliced into equally
sized pieces.
Note
The number of pieces can be any common
multiple of the denominators of the two
fractions.
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© 2007 Herbert I. Gross
Example
If the problem is 4/9 ÷ 5/11, we may think of a
corn bread that is pre-sliced into 99
(that is 9 × 11) equally sized pieces.
Then…
4/9 of the corn bread ÷ 5/11 of the corn bread =
4/9 of 99 pieces ÷ 5/11 of 99 pieces =
44 pieces ÷ 45 pieces =
© 2007 Herbert I. Gross
44/45
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The “invert and multiply” algorithm is simply a
convenient, rote short cut. That is...
4/9 ÷ 5/11 =
4/9 × 11/5 =
44/45
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© 2007 Herbert I. Gross