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Fractions, also called rational
numbers, are numbers of the
form  , where
a and b are
integers (but b cannot
be zero).
The bottom number is called
the denominator. Think
of it as the denomination:
it tells you what size units
you are talking about—fourths,
fifths, or whatever.
The top number is the numerator.
It tells you how many of those
units you have. For example,
if I have 3 quarters in my
pocket, then I have three-fourths
of a dollar. The denomination
is quarters (fourths), and
I have three of them: 3/4.
Ordinarily
we think of fractions as being
between zero and one, like
3/4 or 2/3. These are called
proper fractions. In
these fractions, the numerator
is smaller than the denominator—but
there is no reason why we
can not have a numerator bigger
than the denominator. Such
fractions are called improper.
What
does an improper fraction
like 5/4 mean? Well, if we
have 5 quarters of something
then we have more than one
whole of that something. In
fact, we have one whole plus
one more quarter (if you have
5 quarters in change, you
have a dollar and a quarter).
One
way of expressing the improper
fraction 5/4 is as the mixed
number  , which
is read as “one and one-fourth.”
This notation is potentially
confusing and is not advised
in algebra.
One
cause of confusion is that
in algebra we use the convention
that multiplication is implied
when two quantities are written
next to each other with no
symbols in between. However,
the mixed number notation
implies addition, not multiplication.
For example,  means
1 plus one-quarter,
and 3 1/2 = 3 + 1/2.
It is possible
to do arithmetic with mixed
numbers by treating the whole
number parts and the fractional
parts separately, but it is
generally more convenient
in algebra to always write
improper fractions. When you
encounter a problem with mixed
numbers, the first thing you
should do is convert them
to improper fractions.
A.
Multiply the integer
part with the bottom of the
fraction part.
B.
Add the result to the
top of the fraction.
The general formula is
- Do
the division to get the
integer part
- Put
the remainder over the old
denominator to get the fractional
part.
Equivalent
fractions are fractions that have the same value, for example
 etc.
Although
all these fractions are written
differently, they all represent
the same quantity. You can
measure a half-cup of sugar
or two quarter-cups of sugar,
or even four eighth-cups of
sugar, and you will still
have the same amount of sugar.
·
Multiply by
a form of One
A fraction
can be converted into an equivalent
fraction by multiplying it
by a form of 1. The number
1 can be represented as a
fraction because any number
divided by itself is equal
to 1 (remember that the fraction
notation means the same thing
as division). In other words,
 etc.
Now
if you multiply a number by
1 it does not change its value,
so if we multiply a fraction
by another fraction that is
equal to 1, we will not be
changing the value of the
original fraction. For example,
In
this case, 2/3 represents
exactly the same quantity
as 4/6, because all we did
was to multiply 2/3 by the
number 1, represented as the
fraction 2/2.
Multiplying
the numerator and denominator
by the same number to produce
an equivalent fraction is
called building up
the fraction.
Numerator
and Denominator Have No Common
Factors
Procedure:
1. Write
out prime factorization of
Numerator and Denominator
2. Cancel
all common factors
This
procedure is just the opposite
of building up a fraction
by multiplying it by a fraction
equivalent to 1.
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A
number is prime
if it has no whole number
factors other than 1
times itself, that is,
the number cannot be
written as a product
of two whole numbers
(except 1 times itself).
Example: 6 is not prime because it can be written
as 2 ´ 3
Example: 7 is prime because the only way to write
it as a product of whole
numbers is 1 ´
7
- The
first few prime numbers
are 2, 3, 5, 7, 11,
13, 17, 19, 23, .
. .
- There
are an infinite number
of prime numbers (the
list goes on forever).
Any
non-prime number can
be decomposed into a
product of prime numbers
Example: 4 = 2 ´
2
Example: 12 = 2 ´
2 ´
3
This
method works well for
larger numbers that
might have many factors.
All you need to do is
think of any two numbers
that multiply to give
your original number,
and write them below
it. Continue this process
for each number until
each branch ends in
a prime number. The
factors of the original
number are the prime
numbers on the ends
of all the branches.
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Example:
Factor the number
60
60 = 2
´ 2 ´
3 ´
5 |
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- Notice
how I started with
the smallest numbers:
first 2’s, then 3’s,
and so on. This is
not required but it
keeps the result nicely
in order.
- If
a number is even,
then it is divisible
by 2.
- If
a number ends in 0
or 5, then it is divisible
by 5.
- If
the digits of a number
add up to a number
divisible by 3, then
the number is divisible
by 3. In this example
15 gives 1 + 5 = 6,
which is divisible
by 3, and therefore
15 is divisible by
3.
- Large
numbers with large
prime factors are
notoriously hard to
factor—it is mainly
just a matter of trial
and error. The public-key
encryption system
for sending secure
computer data uses
very large numbers
that need to be factored
in order to break
the code. The code
is essentially unbreakable
because it would take
an enormous amount
of computer time to
try every possible
prime factor.
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Example:
And
reduce result if needed
If
you don’t reduce the factors
before multiplying, the answer
will have to be reduced.
Example:
Remember
that canceling always leaves
a “1” behind, because you
are really dividing the numerator
and the denominator by the
same number.
·
Add Numerators
when Denominators Are the
Same
·
If the denominators
are not the same, make
them the same by building
up the fractions so that they
both have a common denominator.
·
Any common denominator
will work, but the answer
will have to be reduced if
it is not the Least Common
Denominator.
·
The product
of all the denominators is
always a common denominator
(but not necessarily the Least
Common Denominator).
By
Inspection
The
smallest number that is evenly
divisible by all the denominators
In
General
The
LCD is the product of all
the prime factors of all the
denominators, each factor
taken the greatest number
of times that it appears in
any single denominator.
Factor
the denominators:
Assemble
LCD:
Note
that the three only appears
once, because it is only needed
once to make either
the 12 or the 15:
Now
that you have found the LCD,
multiply each fraction (top
and bottom) by whatever is
needed to build up the denominator
to the LCD:
Then
add the numerators and reduce
if needed (using the LCD does
not guarantee that you won’t
have to reduce):
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