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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.
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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.
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Add Numerators when Denominators Are the Same
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If the denominators are not the same, make them
the same by building up the fractions so that they both have a common denominator.
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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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