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6. Improper Fractions and Equivalent Mixed Numbers
The discussion of mixed numbers belongs to the more general
pattern of giving counts and measures with mixed units of counting and/or
measure. Mathematics instruction would be served by a careful
consideration of how to best formulae and unify the underlying ideas. Here is
a first attempt.
Numerals, Fractionals and Quantals
Recall whole numbers may be written in different forms.
Each of those forms is called a numeral. For example
345 = CCCXXXXV in Roman Numerals
= 3 hundred + 4 tens +
5 ones in expanded form
90 = 9 tens = 4 score and 1 ten.
Here we giving number or describing as in terms of groups of hundred,
ten and ones, with the cosmetic convention that there should be 0 to nine
of each multiple of ten.
Decimal form for whole numbers or counts describe the latter in
terms of ones, tens, hundreds and larger multiples or powers of ten. As
a unit for counting the latter are all different. In the early
days of decimal notation, the numbers 1 to 9 were considered simple
numbers because they had single digit decimal forms while numbers 10
onward were considered compound or composite numbers. The number
345 is composed of 3 hundreds, 4 tens and 5
ones.
Whole Numbers and Fractions too may be written in different forms by raising
and lowering terms. Each form may be called a numeral or a fractional. For
an first example, the whole number
| 4 = |
8
2 |
= |
12
3 |
= |
16
4 |
= |
100
25 |
For a second example
| |
3
5 |
= |
6
10 |
= |
30
50 |
= |
9
15 |
Measures too may be expressed in different forms.
4 metres = 40 decimeters = 400 cm = 4000 millimetres
4500 seconds = 75 × 60 seconds =
75 minutes = 1 hour and 15 minutes or 1¼ hours
$3.75 = 3¾ dollars or 375 pennies.
Each form may might called a quantal. We may prefer or like some forms
over another due to various conventions. For example, expressing fractions
in lowest is a convention.
Equivalence: When we do arithmetic with counts, fractions and
quantals, we assume or hope that results (except for form) are
independent of the form in which we write or record or describe whole
numbers, fractions and measures. Different forms of the same number,
fraction or measure are taken to be or assumed to be equivalent not only
for their description, but also in arithmetic. The question of why
or the question of providing a formal justification for this practice is
left for higher studies or research.
Observe, the time 1 hour and 15 minutes is expressed in mixed
units of time measure, that is hours and minutes. In general many
mixed units of measure can be used in calculations. It is a matter
of convention, habit and convenience, which units of size are
employed in measuring, in counting or in describing
fractions.
Fractions with unlike denominators represent multiples of different
unit numerator fractions: one half, one third, one quarter, one fifth,
one sixth and so on. To add, subtract, compare and even divide
fractions, we may express each over a like or common denominator in
order to do the arithmetic. So the form in which an expression is
given or expressed may help calculations.
|
Simplification of Proper Fractions
By convention, a fraction is in simplest form when the numerators and
denominators (tops and bottoms) have not common factors. If they do, the
fraction in question is equivalent to another with smaller numerator and
denominators. By convention, a fraction is in simplest form when its numerator
and denominator are as small as possible.
Simplification of Improper Fractions
In a proper fraction, the numerator and denominators are whole numbers with
the numerator less than the denominator - the top less than the
bottom. If the top is not less than the bottom, the fraction is
improper.
Examples of Proper Fractions
4
5 |
99
100 |
1
7 |
25
50 |
7
12 |
Some are not in simplest form.
Examples of Improper Fractions
45
5 |
100
100 |
15
7 |
130
50 |
70
12 |
Some are not in simplest form - the numerators and denominators have a common
factor (and divisor).
As with proper fractions, simplification may begin with cancellation of
common factors. That will lead to the smallest possible numerators and
denominator fractional expression for the fraction.
45
5 |
= |
9 ×5
5 |
= |
9 |
100
100 |
= |
1 |
|
|
130
50 |
= |
13 ×10
5 ×10 |
= |
13
5 |
Lowering terms by canceling common factors still leads to improper fractions.
Improper Fractions & Mixed Numerals
An improper fraction may be regarded as multiple of a unit-numerator
fraction: one half, one third, one quarter, one fifth and so one. But in those
fractions, the
numerator = a whole number × the denominator + a remainder
where the remainder is less than the denominator.
Identify each group of 2 halves, three thirds,
four quarters and in general N times an N-ths with the whole
number one. Or, we may regard the number one as a group 2 halves,
three thirds, four quarters and in general N times an N-ths.
There in lies the basis for regrouping and so rewriting an improper fraction
as a mixed number - a whole number plus a proper fraction.
Example (A)
The improper fraction
46
11 |
= |
4×11 + 3
11
|
|
|
|
|
= |
4 ×
|
11
11 |
+ |
3
11 |
|
= |
4 ×
|
1 |
+ |
3
11 |
|
= |
4
|
+ |
3
11 |
|
|
= |
4
|
3
11 |
|
|
Thus 46 elevenths may be expressed as mixed counts of ones and elevenths,
where the number of 11-ths is less than 11. So we are rewriting or
re-forming 46 elevenths hundredths as a count of ones and plus a count of
elevenths. That gives a mixed number, a whole number plus a fraction.
Remark 1 - A hazard: The notation for mixed numbers is an
exception to the rule in algebra that two numbers written side by side
indicates a multiplication. Another exception is given by multi-digit decimal
notation.
Example (B)
350
100 |
= |
7×50
2 ×50
|
|
=
|
7
2
|
|
=
|
3×2 + 1
2
|
|
=
|
3 + ½
|
|
= |
3½ |
Now 350 one hundredths
350
100 |
= |
3 ones + one half
|
has the same value as 3 ones (one be a hundred hundreths) and one half - the
latter being 50 one hundredths. So we are rewriting or re-forming 350 hundredths
as a count of ones and halves.
Example (B) Revisited
We may like wise say the improper fraction
350
100 |
= |
3×100 + 50
100
|
= |
3 + ½ |
= |
3½ |
|
|
3 ×
|
100
100 |
+ |
50
100 |
|
|
|
3 ×
|
1 |
+ |
50
100 |
|
|
= |
3
|
+ |
50
100 |
|
|
|
= |
3
|
50
100 |
|
|
|
|
= |
3
|
1
2 |
|
|
|
Here we following the route of converting an improper
fraction (numerator greater than denominator) to a mixed number before
reducing terms in the resulting proper fraction part. . The proper fraction
with a smaller numerator may be easier to factor than the numerator in
the improper fraction.
Algebraic Viewpoint/Description
for reading as part of algebra skill development -
optional reading for now
A Geometric Interpretation or Preliminary
Take N = 23 and d = 5 on first reading.
In measurement we ask how many whole times a shorter length goes into a
longer one. If the longer length is N units and the shorter length has
length d units then the shorter length will go into the longer one say Q whole
times, and there will be r units left over. thus N unit = Q × d units +r
unit where 0 <
r < d.
Here N = 23 and d = 5 on first reading would give Q =4 and r = 3
Since a single unit is one d-th of d units, the r units is the
fraction
of the d unit length divisor Now
| N units |
= |
Q × d units + |
r |
units |
|
|
Q × d units + [ |
r
d |
] × d units |
| |
= |
[Q + [ |
r
d |
] ] × d units |
Hence
| N units ÷
d units |
= |
Q + |
r
d |
Above, write ones in place of units to get
Algebraic Shorthand Description
If N = Q × d +r where 0 <
r < d then
| N ÷
d |
=
|
Qd +r
d
|
= |
Q + |
r
d |
or in fraction notation:
Check: The product of the mixed number
time d is
| { |
Q + |
r
d |
}×d = |
Q × d + ( |
r
d |
) ×d |
| |
|
|
= |
Q × d + |
r × 1 |
| |
|
|
= |
Q × d + |
r |
|
|
|
= |
N |
|
Appendix: Mixed Measures and Mixed Numbers, a common
thread perhaps
Mixed Measures: In describing
lengths of time, we may talk about days, hours, minutes and even
seconds, and by convention all mixed units in this description.
Thus we speak of 2 hours and 15 minutes without convention
requiring a conversion into a large number of minutes, or into a
small mixed number 2¼ of hours. The length
seven quarters of a meter describes the same length as 1.75
meters and 1 meter 75 cm. In describing long thin rectangular, we say
the length is 5 m and the width is 80 cm. For some that description
may be more pleasing than saying 500 cm by 80 cm, or 5 m by 0.8
m. How we describe measures does not affect them but the
numbers in the description depend on the choice of units. The
area of the foregoing rectangular is given by the product of it
dimensions in square meters, in square meters or even as a number
of meter-cm. The latter would be the area of a rectangle with
length one meter and width one cm. The area of the rectangular
is given by three different expressions
A = 5 m × 80cm
A = 5 m × 0.80 m
A = 500 cm × 80 cm
The foregoing leads to three answers: 400 m ×
cm, 4 m2 and 40000 cm2 for the area. Each
has the same value since 1 m2 = 10000 cm2
and 1 m × cm = 100 cm2
There is no harm in using mixed units of measures in evaluating
formulas as long as the unit carried through the steps.
Conversion of the the different units for a measure may be done in any
step or in the original data. What is important is that a
measure be describe as a number of units and not by by a number alone.
In general, arithmetic with measures may be done with mixed unit of
measures alone or multiplied and divided by others. While the form of
a result may vary, its values will not. The following are example of
addition using and keeping mixed units of time measure:
5 hours, 30 minutes + 4 hours, 20 minutes = 9 hours, 50
minutes
2 hours, 50 minutes + 3 hours, 50 minutes = 6 hours, 40
minutes
In the latter, a conversion of 100 minutes into 1 hour, 40 was
don.
An operational mastery of fractions with units will help
Note: While pure mathematics may avoid the carrying of units in
and through calculations by selecting a a consistent system of units
for calculations, the intellectual overhead in selecting that
consisting system and converting all units of measure to it may be
avoided by using and carrying units of measure through
calculations. That is the practice in senior high school and
college courses in chemistry and physics. Moreover, fraction skill
with units present, and manipulations with products and quotients of
units of measure in general, useful for the description of
speed, rates and proportionality constants. There is more
immediate motivation and context to this manipulations with
units of measure as is or multiplied by numbers, than there is to the
combination of monomials in letters w, x, y and z in
products and quotients.
Mixed Numbers: In the early development of decimal
notation, the digits 1 to 9 represent simple numbers while two or
three digit numbers like 42 and 368 represented mixed or
compound numbers. The latter represent the sum of 4 tens and 2
ones, and the sum of 3 hundreds, 6 tens and 8 ones respectively.
So we count in mixed groups: hundreds, tens and ones. Now
the fraction 5 quarter- meters represent a whole number of the
unit one quarter meter. We may write 5 quarter meters as
one meter and one quarter meters. That mixes the unit of length
measure one meter with the unit one quarter meter. Now the mixed
number 4½ stands for 4 wholes and one half a whole. The
mixed units of counting here are ones and one half. Now in counting or
measuring we may find ourselves with 4 wholes, ¾ of a
whole, and ½ a whole. The total count or measure will be 5¼
wholes. The underlying notion here is that we may count and measure
with whole and with multiples of unit-numerator fractions, more easily
written here in word form as one half, one third, one fourth, one
fifth and so on. A mixed number or measure is equal to a whole
number of ones plus a proper fraction: multiples of fractional
units. In general, we add, subtract, multiply and even divide
mixed numbers and measures of ones and units where the units of
counting and measure may be different. The practice of raising
terms for the sake of addition, subtraction or comparison is resembles
the conversion of mixed units of measure into multiples of a common
unit. For example (in a long format chosen to illustrate
ideas)
2
3 |
+ |
3
4 |
= |
2 |
× |
1
3 |
+ |
3 |
× |
1
4 |
|
|
|
|
|
= |
2 |
× |
4
12 |
+ |
3 |
× |
3
12 |
|
Convert one third units and one
quarter wholes
into one twelfth units |
|
|
|
= |
|
|
17
12 |
|
|
|
|
|
Count number of twelfths : 8 = 9 = 17 |
|
|
|
= |
|
1 |
5
12 |
|
|
|
|
|
Convert 12ths into wholes
(that resembles the conversion of more than 10 tenths
in wholes in addition with decimals. |
Remark: Notions of mixed numbers and measures
underlying many arithmetic operations with counts, decimals, fractions
and measures. Clarification of those notions may help us to
decide whether or not, or to what extent we should discuss them in
developing mathematics skills.
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Four Topics
Section Entrance
Fraction Guide
Fractions with Units Guide
Ratios & Fractions Guide
Proportionality Guide
Links
Fraction How-TOs
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2 Fraction Multiplication I
3 Fraction Multiplication II
4 Fraction Multiplication III
5 Equivalent Fractions
6 - Products Algebraically
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9 Fraction Addition I
10. Fraction Addition II
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12. Fraction Addition III
13 Fraction Multiplication IV
14. Fraction Division & Reciprocals
15. Division Formulas Justified
16. Rational Numbers
17. Fraction Webvideos
18 Geometric Notes
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For
Senior
High School & Calculus Students
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<| (o) (o)
|>
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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explain or ease their difficulties. Volume 2 ,Three
Skills for Algebra, in Chapters
8 to 14 & 18 etc, puts words before symbols to
providing the missing words in a way that enrich the
comprehension of all. Those words form the middle part of a algebra
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