16. Similarities between comparison, addition and subtraction of
fractions
Example 1. Which is greater
Here by putting both fractions over the common denominator 4×6= 24, we
see that
|
5
6
|
=
|
20
24
|
is more than
|
18
24
|
=
|
3
4
|
Therefore
and we can calculate how much more - the smaller subtracted from the
larger is
|
5
6
|
-
|
3
4
|
=
|
20
24
|
-
|
18
24
|
|
|
|
=
|
2
24
|
|
|
|
|
|
=
|
1
12
|
|
|
By putting both fractions over the common denominator, the original
comparison can be decided by comparing the over 24 = 4×6
numerators
Easy Consequence: The raising terms work we done in the comparison
and subtraction can be used to calculate the sum.
|
5
6
|
+
|
3
4
|
=
|
20
24
|
+
|
18
24
|
|
|
|
=
|
38
24
|
|
|
|
|
|
=
|
19
12
|
|
|
|
|
|
=
|
1 +
|
9
12
|
|
|
|
|
=
|
|
|
We are calculating the sum to show that comparison, subtraction and
addition operations are made possible for a pair of fractions (or
several) by raising to the case of like denominators. Remember if
you are asked to do one of the operations, you do not have to do the
others, unless directly asked.
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
|
9
13
|
=
|
9×17
13×17
|
=
|
153
13×17
|
while
|
11
17
|
=
|
13×11
13×17
|
=
|
143
13×17
|
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 for the comparison.
But it is required for the subtraction
|
9
13
|
-
|
11
17
|
=
|
9×17
13×17
|
-
|
13×11
13×17
|
|
|
|
=
|
153
13×17
|
-
|
143
13×17
|
|
|
|
=
|
10
221
|
|
|
Easy Consequence: The raising terms work we done in the comparison
and subtraction can be used to calculate the sum.
|
9
13
|
+
|
11
17
|
=
|
9×17
13×17
|
+
|
13×11
13×17
|
|
|
|
=
|
153
13 ×17
|
+
|
143
13×17
|
|
|
|
=
|
296
221
|
|
|
|
|
|
=
|
1 +
|
75 221
|
|
|
|
|
=
|
|
|
Here 75 = 3 × 25 = 3 × 5^2 has only two prime factors, namely 3 and 5.
Neither factor divides exactly in 221. So the proper fraction
part of the mixed number cannot be simplified further. It just an
accident in this and the previous example that the whole number part of
the mixed number is the number 1.
Again, we are calculating the sum to show that comparison, subtraction
and addition operations are made possible for a pair of fractions (or
several) by raising to the case of like denominators. Remember if
you are asked to do one of the operations, you do not have to do the
others, unless directly asked.
Example 3: Two fractions may be added, compared and subtracted
together using any common denominator. For example, the use of common
denominator 12 = 2×6 = 3×4 leads to
15
6
|
+
|
7
4
|
=
|
30
12
|
+
|
21
12
|
=
|
51
12
|
=
|
4
|
3
12
|
=
|
4
|
1
4
|
the use of common denominator 24 = 4×6 = 6×4 leads to
15
6
|
+
|
7
4
|
=
|
60
24
|
+
|
42
24
|
=
|
102
24
|
=
|
4
|
6
24
|
=
|
4
|
1
4
|
and use of common denominator 36 = 6×6 = 9×4 leads to
15
6
|
+
|
7
4
|
=
|
90
36
|
+
|
63
36
|
=
|
153
36
|
=
|
4
|
9
36
|
=
|
4
|
1
4
|
For all three choices of common denominators, the least and others,
conversion to a like denominator, addition and simplification all lead to
the result 4¼ . But the use of smaller common denominators leads to
smaller numbers in the computation and hence less simplification work in
the end. The use of the smallest or least common denominators
usually gives the most efficient way to add and subtract fractions with
unlike denominators. So try to use the least common denominator. But
some work may be required to find it.
Easy Consequences: Since
15
6
|
=
|
30
12
|
and
|
7
4
|
=
|
21
12
|
we see that the first fraction 15/6 is more than the second
fraction 7/4 by 9 twelfths. We also see that
15
6
|
-
|
7
4
|
=
|
30
12
|
-
|
21
12
|
=
|
9
12
|
=
|
3
4
|
|
Advice and Directions:
In practice, additions, subtractions and comparison may be done with any
convenient common denominator. In the case of comparison, the
product of the common denominators may be best (or not) - it does not
have to be computed - recall example 1.
In the case of addition and subtraction, the use of the smallest or least
common denominator (some work may be needed to find it) leads to smaller
numerators to add or subtract after the original fractions to be added or
subtracted are expressed over a common or like denominator. Beyond
that, when we add or subtract fractions by raising terms to a common
denominator, the resulting fraction with that common denominators is
simplified by lowering terms and/or expressing the result as a mixed
number. We may follow that cosmetic convention because fractions
with smallest possible denominators are supposedly easier to comprehend
or digest than other fractions. For example, we may say one
half instead of three sixths. The efficient calculation of sums and
differences (the result of additions and subtraction) is described in a
following page.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
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See too, the BBC-Belgium story Texting and
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The Logic of Injustice:
How Texas sent
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May 2012, Composition Starting:
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The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
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shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
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Arithmetic
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Algebra
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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
trespassed in the house of the three bears, she found three bowls
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bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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