Fraction Sense & Operations
See Fractions,
Ratios, Rates, Proportions & Units for a more concrete
approach to the question of what is a fraction and how to do and
justify arithmetic with fractions. The concrete approach uses
line segments and measures to illustrate and develop fraction senses
and operations. The following easy bases the fraction definition
and operations on the division properties of objects - properties that
classify and characterize those objects of which fractions can be
taken.
The division of objects into parts with the same shape, isometric or
congruent geometrically, or into parts of equal value leads to the notion
of unit fractions and multiples of unit fractions. When an object is
divisible it, we take a half, a third, a quarter, a fifth and so
on.
The object may be a pie, a line segment, a region or a solid, or an
amount of money.
Unit Fractions
To be more precise, when N is a whole number, and when an object is
divisible into N parts, isometric or of the same value, each part
provides an Nth of the object. We may describe
that part in writing as
The symbol
may be read as an Nth ,and to coin a phrase, it may be called
an adjective of division. An adjective of division is also an adjective
of quantity if not enumeration.
Special Case:
For convenience, an Nth of an object is the object
itself when N = 1.
Multiples of Unit Fractions
Put N = 5 and M =3 on first reading.
Suppose N is a whole number, and suppose an object is divisible into N
identical or equi-valued parts. Then for each whole number M less than or
equal to N, we make take M of Nths of the
object.the object. Symbolically, that is in writing, we have
or
The two symbols
gives two more adjective of division, adjectives with the same
meaning. We write
to indicate these adjectives of division have the same meaning. They
provide simple fractions. In them the first number M is called the
numerator and the second N is called the denominator. Both
together may be calledterms of the fraction. Mathematicians think
of fractions as ordered pairs.
Proper Fractions and one Improper Fraction
When M < N above, we say
is a proper fraction.
When M is greater than or equal to N, the fraction
is said to be improper. Improper fractions with M > N are not
feasible when there is only one object.
The case M = N gives the first improper fraction
Taking N of the N-ths of an object is equivalent to taking all of the
object physically and geometrically.
In the biological realm, four quarters do make a
whole. For example, an uncut apple will last longer than two
halves. Two halves in this case are not a whole. So there
are exceptions to the notion that all N of the N-ths of an object
equivalent to the whole. Those exceptions may be avoided may making the
cuts virtual instead of real.
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More on Improper Fractions
When several identical objects can be each be divided into N-ths,
isometric or of equal-value, each object contribute N of the N-ths.
So there more than N of the N-ths. That be said, if M is a whole
number, and there sufficiently many objects divided into N-ths, we
can take M of those N-ths in sequence or at random. Thus
symbolically, that is in writing, we have
or
When M is less than N, the fraction
is said to be proper.
When M is greater than or equal to N, the fraction
is said to be improper. Improper fractions with M > N are
not feasible when there is only one object.
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Addition of Fractions - Like Denominators
Suppose one to several objects are divided into N-ths. Then a group
R of these N-ths added to another non-overlapping group T of these N-ths
together provide (R+T) of these N-ths. Symbolically, the foregoing
may be written as
That provides an addition rule for fractions, proper or not, with like
denominators.
In the case where one to several objects are divided into N-ths and
M-ths,a group R of these N-ths added to another non-overlapping group T
of the M-ths may be physically combined together to form a sum
The question of how to represent this sum as fraction follows from the
concepts of equivalence for fractions and/or mixed numbers.
Unit Fraction of Unit Fractions
If an object is divided in R isometric (respectively equi-value)
parts, we may be able to divide each of those parts into say T isometric
(respectively equi-value) parts. The product TR = RT gives the
number of resulting isometric (respectively equi-value) parts. That
foregoing suggests that a T-th of an R-th is an TR-th and an RT-th
part.of the object. Thus the successive division operations
yield the same result as the single division
Moreover as TR = RT due the commutative property for products of whole
numbers, Thus the successive division operations
also has the same result. Symbolically with the times symbol × in place
of the word of, we have
and
The foregoing gives a definition and computational method for the
compound adjective of division provided by a unit fraction of a unit
fraction.
Simple Fractions of Unit Fractions
We now consider the question what is
By definition the latter is
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M
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times
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one T-th
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of
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one R-th
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of
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an
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object
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Since
the foregoing implies
If an object is divided in R isometric (respectively equi-value)
parts, we may be able to subdivide each of those parts into say T
isometric (respectively equi-value) parts and then take M of the
resulting parts The product TR = RT gives the number of resulting
isometric (respectively equi-value) parts. That foregoing suggests that M
times T-th of an R-th is M times a TR-th or RT-th part.of the
object.
Equivalent Fractions - Fractions of equal value
A divisible object can sometimes be divided into N
isometric or equi-value parts in many different ways. From this
point on, we will consider and asssume the case of equ-value
division first and foremost. The case of division into isometric
parts will be met mainly in the form of illustration and not as a support
to the unfolding theory or chains of reason. Here the existence of
a value or measure of a quantity is tacitly assumed.
Recall a unit fraction of a unit fraction is another unit fraction. That
is
when R and T are whole numbers. An R-th part of an object is also an
object. Since T ocurrences of a T-th of an object equals the object, we
have
That is
Now
M
R
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=
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M
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×
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1
R
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=
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M
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×
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T
TR
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|
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=
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M
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×
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(T×
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1
TR
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)
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=
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(M×
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T)
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×
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1
TR
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=
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MT
TR
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Therefore
That is to say M of R-ths of an object is taken to have equal value
to MT of RT-ths of an object in cases where T times an RT-th has
the same value has an R-th. That situation is assumed.
Identification of unlike fractions with physically different
meanings or operations.
The foregiong implies two fractions representing physical different
operations, here M of R-ths of an object and MT of RT-ths of the same
object, are considered to be equivalent and interchangeable in our
value-based calculations or consideration of adjectives of division
(units fractions) and their multiplies (simple fractions).
Assumption. The three simple fractions
of an object give the same value, and so can replace each other.
The equal sign here is use in the sense that all three fractions (applied
to an object) should give the same value. Should more be said
about equivalent (or equi-value) fractions?
Raising Terms, Lowering Terms
Here replacing the first fraction M/R by one of the other is called
raising terms while replacement of one of the others by M/R is
called lowering terms or simplification. Expressing a
fraction A/B in lowest terms or simplest form means finding another
fraction M/N of equal value in which no further lowering of terms is
possible. See or find a discussion of relatively prime whole
numbers.
Addition of Fraction - Unlike Denominators, First Pass
In the case where one to several objects are divided into N-ths and
M-ths,a group R of these N-ths added to another non-overlapping group T
of the M-ths may be physically combined together to form a sum
The question of how to represent this sum as fraction follows from the
concepts of equivalence for fractions and/or mixed numbers. Here we
observe in terms of value, measure or or equivalent fractions
R
N
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=
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RM
NM
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=
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RT×
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1
NM
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and
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T
M
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=
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NT
NM
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=
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NT×
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1
NM
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Hence
R
N
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+
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T
M
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=
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RM×
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1
NM
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+
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NT×
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1
NM
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= (RM+NT)×
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1
NM
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and so
The foregoing uses the common denominator NM in order to express each
fraction in the sum as an equivalent (equi-value) fraction with NM as a
the common or like denominator.
Addition of Fraction - Unlike Denominators, Second Pass
Suppose N = DE and M = FE for some common factor E, a whole number.
Then
R
N
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+
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T
M
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=
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RM+NT
NM
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=
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REF+DET
DEFE
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=
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(RF+DT)× E
DEF× E
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from the distributive, commutative and associative properties of whole
numbers.
Now lowering terms gives
or
The foregoing also follows without lowering terms from the use of
equivalent fractions
R
DE
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=
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RF
DEF
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=
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RF×
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1
DEF
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and
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T
EF
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=
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DT
DEF
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=
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DT×
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1
DEF
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and the rule for addition of fractions with like denominators. Lowering
terms will be possible if E is not the greatest common factor or if the
addition RF+DT results in a common factor with the product DEF. As
a rule of thumb, the most efficient way to add is to use the
greatest common factor E. A logically equivalent way of saying
this, exercise show why, is use the least common denominator.
The choice of greatest common factor E leads to the most
efficient computation in that if E is not the greatest common factor of
N and M then lowering terms is possible after the fractions are
combined into a single term using the rule for addition of fractions
with like denominators.
Products of Simple Fractions
We now consider the question what is
By definition the latter is
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M
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times
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one T-th
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of
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N times one R-th
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of
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an
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object
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So we need find the unit fraction of a simple fraction, that is,
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one T-th
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of
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N times one R-th
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of
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an
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object
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and then multiply by M.
Special Case: The simplest case to visual occurs when N is a
multiple of T. Say N = KT for some whole number K. Then a T-th of
KT objects (here R-ths of another object) is given by K of the
R-ths. Then
1
T
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×
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N
R
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=
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1
T
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×
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KT
R
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=
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K
R
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Second Special Case:
1
T
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of
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N
R
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=
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1
T
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of
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TN
TN
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|
|
|
|
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=
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1
T
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of
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T
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×
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N
TN
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|
|
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=
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N
TN
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General Case:
Now multiplication by M gives the product formula
for a product of simple fractions. Can a more cogent argument be given?
An example
Here 2 thirds of three fifths of a rectangle is computed
graphically.
Three fifths of rectangle are shown shaded in blue
We cut each shaded fifth into thirds
and take two thirds of the three-fifths - see the gray area.
The answer is 6 fifteenths as 15 = 3× 5.
Algebraic Properties.
The properties of arithmetic with whole numbers, that is the associative,
commutative and distributive law for whole numbers, combined with the
above formulas for addition and multiplication, and raising or lowering
terms implie the associative, commutative and distributive laws for
addition and multiplication of fractions. Details are left to the reader.
Division and Multiplication by Reciprocal of Divisor
What is
The product formula
says
The question of many multiples M/T of a fraction N/R gives another
fraction K/S has an answer
Here we observe due to the associative laws for multiplication of
fractions and whole numbers that
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(
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K
S
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×
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R
N
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)
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×
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N
R
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=
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KRN
SNR
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=
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K
S
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as required. So
works.
Definition:
Uniqueness:
Now if P and Q are two fractions with
and
Then
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P =
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(P
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×
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N
R
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)×
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R
N
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=
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K
S
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×
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R
N
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and
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Q =
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(Q
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×
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N
R
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)×
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R
N
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=
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K
S
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×
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R
N
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So Q = P and
is the only answer.
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