12. Comparison of Fractions
[RealPlayer
Video] 3 minutes - Comparison of Fractions Size or Magnitude,
and more examples of the use of common denominators in addition and
subtraction.
How raising fractions over a common denominator leads
to direct comparison (and justifies cross multiplication rule
methods for comparison
After reading the following, say which is greater:
four ninths or five elevenths, and find the difference of the larger
minus the smaller.
Example 1. The question which is greater
is often answered by comparing 5×4 = 20 with 6×3 =18. Let use look at
this in more detail. The least common denominator is 12. The
following diagram show both fractions in terms of twenty-fourths: Here 24
= 2× 12
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Here by putting both fractions over the common denominator 4×6= 24, we
see that
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5
6
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=
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20
24
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is more than
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18
24
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=
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3
4
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Therefore
and we can calculate how much more:
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5
6
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-
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3
4
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=
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20
24
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-
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18
24
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=
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2
24
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=
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1
12
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By putting both fractions over the common denominator, the original
comparison can be decided by comparing the over 24 = 4×6
numerators
(i) 20 = 5×4 = (first numerator)×(second denominator)
with
(ii) 18 =6×3 = (first denominator)×(second denominator).
These
These over 6×4 = 24 numerators indicate how many (6×4)ths there are in
the original fractions.
Example 2: The question which is greater
This can be answered by seeing how (13×17)ths there are in each
fraction. We see that
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9
13
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=
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9×17
13×17
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=
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153
13×17
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while
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11
17
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=
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13×11
13×17
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=
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143
13×17
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So the first fraction is greater. It provides 153- 143 = 10 more
(13×17)ths than the second.
For those of you who insist on knowing, 13×17 =221, a number whose
existence we need, but whose value is not required.
The comparison there begins with comparison of the whole number parts. If
those parts are equal, comparison then proceeds with the (proper)
fraction parts. The cross-multiplication or common denominator method
then applies. Note: there is no need to convert the mixed number
into an improper fraction.
A few examples should go here (a to-do).
In process, the question of comparing numbers and fraction given in
decimal form is similar.
- Which is larger 587 or 622? Comparison of the
larger part of each number, the hundred digits gives the answer.
- Which is larger 3.46 or 2.61? Comparison of the whole
number part, the digits in the ones place of each number provides the
answer.
- Which is larger 453.145 or 52.93? Comparison of the
whole number parts of each decimal gives the answer.
Teachers: The comparison of decimals is part
lexicographic. It is fully lexicographic when the decimals are
padded with zeroes on the left to make them have the same number of
decimal places.
Another thought:
In comparing decimals 4352 and 4348, we observe the first
difference occurs in the ten digit place. Here 5 tens or 50 is
greater than 48. Thus the first number 4352 > 4350 > 4348.
Like wise when we compare mixed numbers like
the first number has the greater whole number part. So it is larger than
the second regardless of the relative size of the proper fraction parts
in each.
However in the comparison of
the whole number parts are equal. Thus which is larger depends on
the comparison of the proper fraction parts
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8
15
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=
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24
30
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7
10
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=
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21
30
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Raising terms over a common denominator, the least for instance, or
alternatively the product 150, expresses both fractions as multiplies of
a common unit numerator fraction, over which the comparison is
easy.
Remark: The comparison could be made by converting both mixed numbers
into decimal using a calculator, but that would not develop or strengthen
fraction skills and sense.
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
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