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Mrs. L. Mullane



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Math 9 Notes

 

Rational Numbers


 


A Rational Number is any number that can be written as a fraction with an integer numerator and a non-zero integer denominator.  This includes positive and negative numbers, including terminating and repeating decimals.  A rational number is any number that can be written in the form where m and n are intergers.


 


Examples of rational numbers in decimal form are:  - 0.75, 0.5,, …


Examples of rational numbers in fraction form are: 


 


Facts to know about Rational numbers:

 

  1. Terminating decimals:  When you change a fraction to a decimal and it ends.  Example:



  1. Repeating decimals:  When you change a fraction to a decimal and it goes on and on.  Example:


  1. Repeating decimals can be written as 0.66666666….., or 0. or  .  Use the bar above the number that repeats.
  2. Fraction to a decimal:  you divide the numerator by the denominator.
  3. Numerator:  the top number in the fraction.  Example:   is the numerator.
  4. Denominator:  the bottom number in the fraction.  Example:  is the denominator.
  5. Improper fraction:  is when the numerator is larger than the denominator.  Example:               
  6. Proper fraction:  is when the numerator is smaller than the denominator.  Example:
  7. Mixed number:  is when you have a whole number and a fraction.           Example:


  1. Reduce fractions:  you need to find a number that will divide evenly into both the numerator and denominator evenly.  Example:   can be divided by 2 evenly, for the numerator 2 is 1, and denominator 2 is 5, so the fraction  in lowest terms is       


  1.  Common denominators: when the denominators of the two or more fractions are the same.



What is the Least Common Denominator?

The "Least Common Denominator" is the smallest of all the possible common denominators.

Different Denominators

We can't add fractions with different denominators:


So what do we do? How can they be added?

Answer: We need to make the denominators the same.

Finding a Common Denominator


But what should the new denominator be?

One simple answer is to multiply the current denominators together:

3 × 6 = 18   So instead of having 3 or 6 slices, we will make both of them have 18 slices.

The pizzas now look like this:


Least Common Denominator

That is all fine, but 18 is a lot of slices ... can we do it with fewer slices?

Here is how to find out:


Then find the smallest number that is the same



The answer is 6, and that is the Least Common Denominator.

So let us try using it! We want both fractions to have 6 slices.

  • When we multiply top and bottom   by 2 we get

  • Already has a denominator of 6.


And our question now looks like:



One last step is to simplify the fraction (if possible). In this case 3/6 is simpler as 1/2:



What Did We Do?

The trick was to list the multiples of each denominator, then find the Least Common Multiple

In the previous example the Least Common Multiple of 3 and 6 was 6.

In other words the Least Common Denominator of  and  is 6.

Here are the steps to follow:

  • Find the Least Common Multiple of the denominators (which is called the Least Common Denominator).
  • Change each fraction (using equivalent fractions) to make their denominators the same as the least common denominator
  • Then add (or subtract) the fractions, as we wish!


  1. Adding and Subtracting fractions: 
  • step 1:  make sure the denominators are the same.

  • Step 2:  add or subtract the numerators, put the answer over the denominator.

  • Step 3:  Simplify the fraction (if needed).

     

    If the denominators are different:  then you need to find equivalent fractions that would make the denominators the same: 

    Example:


Step 1: The bottom numbers are different. See how the slices are different sizes?

We need to make them the same before we can continue, because we can't add them like that.

The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2, like this:


Important: you multiply both top and bottom by the same amount,
to keep the value of the fraction the same

Now the fractions have the same bottom number ("6"), and our question looks like this:


The bottom numbers are now the same, so we can go to step 2.

Step 2: Add the top numbers and put them over the same denominator:

                       


Step 3: Simplify the fraction:


A Rhyme To Help You Remember


"If adding or subtracting is your aim,
The bottom numbers must be the same!
"Change the bottom using multiply or divide,
But the same to the top must be applied,
"And don't forget to simplify,
Before its time to say good bye"

Another Example:



Again, the bottom numbers are different (the slices are different sizes)!

The first fraction: by multiplying the top and bottom by 5 we ended up with 5/15 :


The second fraction: by multiplying the top and bottom by 3 we ended up with 3/15 :


The bottom numbers are now the same, so we can go ahead and add the top numbers:

                       

The result is already as simple as it can be, so that is the answer:

Adding Mixed Fractions


Adding and Subtracting Mixed Fractions



To make it easy to add and subtract them, just convert to Improper Fractions first:


I find this is the best way to add mixed fractions:

Example: What is + ?

Convert to Improper Fractions:     and

Common denominator of 4:   stays the same, = (multiply top and bottom by 2)

Now Add:

Convert back to Mixed Fractions:


Subtracting Mixed Fractions

Just follow the same method, but subtract instead of add:

Example: What is ?

Convert to Improper Fractions:  =    and =

Common denominator of 12:  =   and =

Now Subtract: =

Convert back to Mixed Fractions: = 


  1. Multiply fractions:  First multiply the numerators, then multiply the denominators, and then simplify if needed.  Example:   =


Hint:  It is better you to reduce fractions if possible before you multiply or you’ll have large numbers to deal with.  

Example:    


If you simplified first then you have:  


 

Fractions and whole numbers: 


Make the whole number a fraction, by putting it over 1.


                                    5 =


Then continue as before.

          Example:    =

          Mixed Fractions: 

Convert to improper fractions, multiply the fractions, and simplify.

                        Example: 

Think of Pizzas.



First, convert the mixed fraction (1 3/8) to an improper fraction (11/8):


Now multiply that by 3:



And, lastly, convert to a mixed fraction (only because the original fraction was in that form):


And this is what it looks like in one line:


 


Another Example: What is  ?


Do the steps from above:


  1. convert to Improper Fractions
  2. Multiply the Fractions
  3. convert the result back to Mixed Fractions




  1. Dividing Fraction:  Turn the second fraction upside down, then multiply.


 


There are 3 Simple Steps to Divide Fractions:


Step 1. Turn the second fraction (the one you want to divide by) upside down
(this is now a reciprocal).

Step 2. Multiply the first fraction by that reciprocal

Step 3. Simplify the fraction (if needed)


Example: 



Step 1. Turn the second fraction upside down (it becomes a reciprocal):




Step 2. Multiply the first fraction by that reciprocal:


                                               


Step 3. Simplify the fraction: 



How Many?


A question like 20 divided by 5 is asking "how many 5s in 20?" (= 4)


So divided by is asking "how many ’s in "


 is really asking:  How many  in ?


 


Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?



So now you can see why =3


 


Fractions and Whole Numbers


Make the whole number a fraction, by putting it over 1.


Then continue as before.


Example:


Make 5 into:


                       



Step 1. Turn the second fraction upside down (the reciprocal):  
Step 2. Multiply the first fraction by that reciprocal: 
Step 3. Simplify the fraction:


The fraction is already as simple as it can be.


Why Turn the Fraction Upside Down?


Because dividing is the opposite of multiplying!



But for DIVISION we:

  • divide by the top number
  • multiply by the bottom number

Example: dividing by 5/2 is the same as multiplying by 2/5


So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.


 http://www.mathworksheets4kids.com/test-practice/fractions/index.html


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