You can express any integer as a fraction by simply dividing by 1, or you can express any integer as a fraction by simply choosing a numerator and denominator so that the overall value is equal to the integer.
To factor an integer, simply break the integer down into a group of numbers whose product equals the original number. Don't forget that the number 1 is the factor of every number (normally the number 1 is omitted). Any factor of a number can be divided evenly into that number.
Keep dividing the resulting number by the smallest prime number that will go into the number evenly. Start with "2" if the number is even. Otherwise start with the lowest prime number possible (3, 5, 7, etc.), until you are left with only a Prime Number.
In this example all of the
"2s" are eliminated because there are an equal number of 2s
in both the numerator and denominator. That's what we mean by a fraction
mix that has the value of "1".
The
correct answer for the example above is a reduced fraction that's equal to 3/7.
Here's
another way to look at this same example.
You already know that 2/2 = 1, so...
is the same as
which is equal to 1 x 1 x 1 x 3/7
Therefore, you would
re-write your answer as 3/7.
Equivalent
Fractions
Finding an equivalent
fraction (also called building fractions) is the reverse of reducing the fraction. Instead of searching for the 1 in a fraction
mix so that you can reduce, you insert a 1 and build.
The resulting fraction is called an equivalent fraction.
Try
to remember this one, because you will use it a lot in other homework
assignments.
Here's how you do
it...
Multiplying
the numerator and the denominator by the same number, such as 7, is
the same as multiplying the original fraction by 1 (since 7/7 = 1). It
does not change the value.
Example:
Find
an equivalent fraction for 1/2.
Step
1: Choose any number you wish. Suppose you chose 6.
Step
2: Multiply the numerator and denominator by 6.
Step 3:
Write the equivalent fraction. 1/2 = 6/12
1/2
is equivalent to 6/12. An
equal sign (=) is used to represent equivalent fractions.
Simplifying Improper Fractions
You may remember from other homework assignments that an improper fractions is where the numerator has a greater value than that
of the denominator. So each time you do a math operation on fractions and your answer ends up
as an improper fraction, you must simplify your answer. Because, the
simplified results
will be in the form of a mixed number.
So, to convert
an improper fraction into a mixed number, just divide the numerator by the
denominator. The results will be a whole number part and a fractional part.
Here
is an example...
As
you can see, this is a pretty straightforward operation. But keep in mind that
if there is no remainder, the answer is the WHOLE NUMBER only.
Greatest Common Factor (GCF)
The greatest
common factor of two or more whole numbers is the largest whole number
that divides evenly into each of the numbers. There are two ways
to find the greatest common factor. Remember to follow your homework
instructions, if your teacher asks for a particular method.
The first
method is to list all of the factors of each number, then list the
common factors and choose the largest one.
Example:
Find the GCF of 36 and 54.
The factors
of 36 are 1, 2, 3, 4, 6, 9, 12, 18,
and 36.
The factors
of 54 are 1, 2, 3, 6, 9, 18,
27, and 54.
The common
factors of 36 and 54 are 1, 2, 3, 6, 9, 18
Although the
numbers in bold are all common factors of both 36 and 54, 18
is the greatest common factor.
The second
method is to list the prime factors,
then multiply the common prime factors.
Example:
Let's use the same numbers, 36 and 54.
The prime
factorization of 36 is 2 x 2 x 3 x 3
The prime
factorization of 54 is 2 x 3 x 3 x 3
Notice that
the prime factorizations of 36 and 54 both have one 2
and two 3s in common. So,
we simply multiply these common prime factors to find the greatest
common factor. Like this...
2 x 3 x 3 = 18
Both
methods work!
Least Common Multiple (LCM)
The least
common multiple of two or more non-zero whole numbers is
actually the smallest whole number that is divisible by each of
the numbers. When doing your homework, keep in mind that there are two widely used methods for finding the least
common multiple of a group of numbers.
Method 1
- Simply list the multiples of each number (multiply by 2, 3, 4, etc.)
then look for the smallest number that appears in each list.
Example:
Find the least common multiple for 5, 6, and 15.
Multiples of
5 are 10, 15, 20, 25, 30,
35, 40,...
Multiples of
6 are 12, 18, 24, 30, 36,
42, 48,...
Multiples of
15 are 30, 45, 60, 75,
90,....
Method 2 -
Factor each of the numbers into primes. Then for each different prime
number in all of the factorizations, do the following...
-
Count the
number of times each prime number appears in each of the factorizations.
-
For each prime
number, take the largest of these counts.
-
Write down that
prime number as many times as you counted for it in step 2.
-
The least common
multiple is the product of all the
prime numbers written down.
Example: Find
the least common multiple of 5, 6 and 15.
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