Pre-Algebra: Fractions

This page is designed to help you better understand how to deal with fractions and their uses in Pre-Algebra. Follow any of the links below to go to that section and start understanding fractions.

Lowest Common Multiple
Greatest Common Factor
Multiplication of fractions
Division of fractions
Common Denominators
Quiz on fractions

Lowest Common Multiple

The LCM is something that you will use throughout math. It is especially useful when multiplying and dividing fractions.

This section will help you better understand the LCM and its uses.

Important Things to Remember

When finding an LCM, use only multipliers that are whole numbers. Examples:
```
     4, 8, 43, 104
```
Be aware of all the numbers you are finding an LCM for.

The Tutorial
When finding LCMs, be aware of all the numbers you are finding common multiples of and remember that you can only use whole numbers for multipliers. Also, always be aware of zero, which is not an LCM.

Example

1.  Problem:    Find the LCM of 4 and 5.
    Solution:
    Make a table similar to the following:

    Multiples of 4 | Common Multiples | Multiples of 5
    --------------------------------------------------
           4       |                  |        
           8       |                  |        5
          12       |                  |       10
          16       |                  |       15
          20       |        20        |       20
          ...      |        ...              ...

    20 is a multiple of both numbers.  It is also the first one (lowest of all
    multiples), thereby being the lowest common multiple.

Greatest Common Factor

The GCF is something that you will use throughout your "math experience." It is especially useful when dealing with fractions.

This section will help you better understand how to find and deal with GCFs.

Important Things to Remember

When finding a GCF, use only whole numbers. Examples:
```
     2, 9, 27, 201
```
Be aware of all the numbers you are finding a GCF for.

The Tutorial
When finding GCFs, be aware of all the numbers you are finding common factors of and remember that you can only use whole numbers for factors. When finding a GCF, unlike the LCM, you must list all the factors because you're finding a greatest factor, not a lowest multiple.

Example

1. Problem:    Find the GCF of 8 and 12.
   Solution:
   Make a table similar to the following:

   Factors of 8 | Common Factors | Factors of 12
   ---------------------------------------------
         1      |        1       |       1
         2      |        2       |       2
                |                |       3
         4      |        4       |       4
                |                |       6
         8      |                |
                |                |      12

   4 is a factor of both numbers. It is the largest of the
   factors listed, therefore it is the greatest common
   factor.

Multiplication of Fractions

The multiplication of fractions is one of the more important things you'll learn in math. In fact, it is so important that you need to know how to do it in order to divide fractions, add fractions, and many other things.

This section will help you better understand the important skill of fraction multiplication.

Important Things to Remember

Do not cross-multiply fractions.
Like the multiplication of whole numbers and decimals, you can multiply more than two fractions together in one problem. Example:
```
     1   3   4   12   3
     - * - * - = -- = -
     2   2   5   20   5
```

The Tutorial
When multiplying fractions, multiply the numerator(s) by the numerator(s) and the denominator(s) by the denominator(s). Also, after finding the product of the fractions, be sure to reduce the product to its simplest form (that is one instance of GCF use).

Example

1.  Problem:     3   6
                 - * -
                 4   7
   Solution:    
   3-->6-->18                    Multiply the numerator by the numerator and the
   - * - = --                    denominator by the denominator.
   4-->7-->28

   18/2    9                     Find the GCF of the numerator and denominator
   ---- = --                     and then divide both the numerator and
   28/2   14                     denominator by that number.  The resulting
                                 fraction is the answer.

Division of Fractions

Division of fractions isn't a skill that gets around quite as well as multiplication, but it is very useful!

This section will help you understand how to divide fractions.

Important Things to Remember

Cross-multiplication is involved in the division of fractions.
Flip the second fraction of the two being multiplied at the time upside down to do the problem correctly.
As indicated above, there can be more than two fractions in a division problem involving fractions, but you can only divide 1 fraction by 1 fraction, so you have to do a problem like that in more than one part. Example:
```
     1   3   2   4   2   20   5
     - / - / - = - / - = -- = -
     2  4   5   6   5   12   3
```

The Tutorial

When dividing a fraction by a fraction (remember, a whole number can be written as a fraction (i.e., 4 = 4/1)), flip (take the reciprocal of) the second fraction and then multiply. Be sure to reduce the quotient (simplify the answer).

Example

1.  Problem:     6   2
                 - / -
                 1   3
   Solution:
   6   3                        Take the reciprocal of (flip) the second
   - / -                        fraction.
   1   2

   6-->3-->18                   Multiply the numerators.  Multiply the
   - * - = --                   denominators.
   1-->2--> 2

   18/2   9                     Find the GCF of the numerator and the
   ---- = -                     denominator and divide each by that 
    2/2   1                     number.  Because (9/1) is the same as the
                                whole number 9, the answer is 9.

Common Denominators

To be able to add or subtract fractions from fractions, you need to have the denominators be the same, or common. (This is one of many instances where the ability to muliptly fractions correctly will come in handy.)

This section is designed to help you better understand the process involved in finding a common denominator in order to be able to add and/or subtract a fraction from another number.

Important Things to Remember

To add or subtract a fraction from another number (whole or fractional), the denominator needs to be the same. Example:
```
     1   3                       4   3
     - + -  cannot be done, but  - + - can
     2   8                       8   8
```
When a fraction has a numerator and denominator that are the same number, the fraction is equal to 1. Example:
```
     2
     - = 1
     2
```
Multiplying by 1 does not change a number, even though the form might change. Example:
```
     4   4   2   8
     - = - * - = --
     5   5   2   10
```

The Tutorial

When finding a common denominator so you can add or subtract fractions, you find the LCM of all the denominators of the fractions you are dealing with. Once you've found this number, make the denominators equal this number. To do this, you multiply the denominator and numerator (the denominator is one factor of the LCM) by the correspoding factor of the LCM.

Example

1.  Problem:    4   2
                - + -
                3   5
   Solution:
   15                           The LCM of 3 and 5 is 15.                  

    4     2                     Since the denominators have to equal the LCM,
   --- + ---                    you have to multiply 3 by 5 and 5 by 3.  Now
   3*5   5*3                    both denominators are the same.

    4*5   2*3                   Because you don't want to change the problem
   --- + ---                    in any way, each part of the problem has to
    15   15                     be multiplied by 1 (not one-third or one-
                                fifth as you did in the second step).  To
                                do that, you have to multiply the numera-
                                tor by the same number as you multiplied
                                the denominator by.

   20    6                      Now that you've got the denominators the
   -- + --                      same, you can add the fractions together.
   15   15

   26                           You cannot reduce this fraction, so this
   --                           is the final answer!
   15

Take the quiz on fractions. The quiz is very useful for either review or to see if you've got the topic down.

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