Mrs. L. Mullane 
Math 9 NotesRational Numbers
A Rational Number is any number that can be written as a fraction with an integer numerator and a nonzero 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:
What is the Least Common Denominator?The "Least Common Denominator" is the smallest of all the possible common denominators. Different DenominatorsWe 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 DenominatorBut 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 DenominatorThat 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.
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:
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, 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 Rememberâ™« "If adding or subtracting is your aim, 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 FractionsAdding and Subtracting Mixed FractionsTo 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 FractionsJust 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: =
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:
There are 3 Simple Steps to Divide Fractions:
Example:
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 NumbersMake the whole number a fraction, by putting it over 1. Then continue as before. Example:Make 5 into:
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:
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.
