Adding+and+Subtracting+Fractions

Adding Fractions
You can add fractions easily if the bottom number (the //denominator//) is the same: Another example:
 * 1/4 || + || 1/4 ||  || 2/4 ||   || simplyfied to 1/2 ||
 * (One-Quarter) ||  || (One-Quarter) ||   || (Two-Quarters) ||   || (One-Half) ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-1-4.jpg width="120" height="120"]] || + || [[image:http://www.mathsisfun.com/images/fractions/pie-1-4.jpg width="120" height="120"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pie-2-4.jpg width="120" height="120"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-1-2.jpg width="120" height="120"]] ||
 * 5/8 || + || 1/8 ||  || 6/8 ||   || simplyfied to 3/4 ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-5-8.jpg width="120" height="120"]] || + || [[image:http://www.mathsisfun.com/images/fractions/pie-1-8.jpg width="120" height="120"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pie-6-8.jpg width="120" height="120"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-3-4.jpg width="120" height="120"]] ||

Adding Fractions with Different Denominators
But what if the **denominators** (the bottom numbers) are not the same? As in this example:

You must //somehow// make the denominators the same. In this case it is easy, because we know that 1/4 is the same as 2/8. The sum now becomes:
 * 3/8 || + || 1/4 || = || ? ||  ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-3-8.jpg width="120" height="120"]] || + || [[image:http://www.mathsisfun.com/images/fractions/pie-1-4.jpg width="120" height="120"]] || = || [[image:http://www.mathsisfun.com/images/fractions/pie-huh.jpg width="120" height="120"]] ||  ||

In that example it was easy to make the denominators the same, but it can be harder ... so you may need to use either of these methods: to make them the same (they both work, use whichever you prefer). =What is a Denominator?= //It shows how many equal parts the item is divided into// ||
 * 3/8 || + || 2/8 || = || 5/8 ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-3-8.jpg width="120" height="120"]] || + || [[image:http://www.mathsisfun.com/images/fractions/pie-2-8.jpg width="120" height="120"]] || = || [[image:http://www.mathsisfun.com/images/fractions/pie-5-8.jpg width="120" height="120"]] ||
 * [|Least Common Denominator], or
 * [|Common Denominator]
 * [[image:http://www.mathsisfun.com/images/fraction.gif width="309" height="131" caption="fraction"]] || //The **denominator** is the bottom number in a fraction.//

Fractions with Different Denominators
You **can't** add fractions with different denominators: So what do you do? How can they be added? Answer: You need to make the **denominators the same**.
 * || 1/3 || + || 1/6 || = || ? ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-3.jpg width="111" height="111" caption="1/3"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-huh.jpg width="120" height="120" caption="1"]] ||

Common Denominator
But what should the new denominator be? ==One simple answer is to multiply the current denominators together: 3 x 6 =18. BUT whatever we do to the denominator we must do to the numerator. For 1/3 we multiplied 3 by 6 so we must multiple 1 x6. For 1/6 we multiplied 6 x 3 so we must multiply 1 x 3.==

So instead of having 3 or 6 slices, we will make **both** of them have **18 slices**. The pizzas now look like this. //(Read more about common denominator on my wiki page on fraction // =Least Common Denominator= That is all fine, but 18 is a lot of slices ... can you 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: 1/3 becomes 2/6 and 1/6 remains the same. And that is what the Least Common Denominator is all about. It lets you add (or subtract) fractions using the least number of slices.
 * || 6/18 || + || 3/18 || = || 9/18 ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-6-18.jpg width="114" height="114" caption="1/3"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-18.jpg width="114" height="113" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pizza-9-18.jpg width="114" height="114" caption="1"]] ||
 * 1/3 || List the multiples of 3: ||  || 3, 6, 9, 12, 15, 18, 21, ... ||
 * 1/6 || List the multiples 6: ||  || 6, 12, 18, 24, ... ||
 * || multiples of 3: ||  || 3, **6**, 9, 12, 15, 18, 21, ... ||
 * || multiples 6: ||  || **6**, 12, 18, 24, ... ||
 * || 2/6 || + || 1/6 || = || 3/6 ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-2-6.jpg width="112" height="112" caption="2/6"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-6.jpg width="112" height="111" caption="3/6"]] ||

=Using Least Common Multipl//e// //The smallest (non-zero) number that is a multiple of two or more numbers.// =

The multiples of a number are what you get when you **multiply it by other numbers** (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table. Here are some examples:


 * The multiples of **3** are: **3, 6, 9, 12, 15, 18, 21, etc** ... ||
 * The multiples of **12** are: **12, 24, 36, 48, 60, 72, etc...** ||

What is a "Common Multiple" ?
When you list the multiples of two (or more) numbers, and find the **same value in both lists**, then that is a //**common multiple**// of those numbers. For example, when you write down the multiples of **4** and **5**, the //common// multiples are those that are found in both lists:


 * The multiples of 4 are: 4,8,12,16,**20**,24,28,32,36,**40**,44,... ||
 * The multiples of 5 are: 5,10,15,**20**,25,30,35,**40**,45,50,... ||

Notice that **20** and **40** appear in both lists? So, the common multiples of 4 and 5 are: **20, 40,** (and 60, 80, etc ..., too) = = =What is the "Least Common Multiple" ?=

It is simply the **smallest** of the common multiples.
eg The **Least** Common Multiple of 4 and 5 is **20**.

Finding the Least Common Multiple
It is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.

Example: Find the least common multiple for 3 and 5:
The multiples of 3 are **//3, 6, 9, 12, 15, ...//**,

and the multiples of 5 are **//5, 10, 15, 20, ...//**, like this: As you can see on this number line, the first time the multiples match up is 15. **Answer: 15**

Example: Find the least common multiple for 4, 6, and 8

 * Multiples of 4 are: 4, 8, 12, 16, 20, **24**, 28, 32, 36, ...

Multiples of 6 are: 6, 12, 18, **24**, 30, 36, ...

Multiples of 8 are: 8, 16, **24**, 32, 40, .... So, **24** is the least common multiple of 4, 6 and 8. || =Here are the steps to follow when adding or subtracting fractions with the same denominators:=

> 1/4 + 1/4 > **Step 2**. Add the top numbers and put the answer over the same denominator : > ** 1 ** **+** ** 1 ** **=** ** 2 ** = 2 > ** 4 ** > **Step 3**. Simplify the fraction: > ** 2 ** **=** ** 1 ** > ** 4 ** ** 2 ** > Watch the following video for an over view of this. > media type="youtube" key="p3jZq7zCzPw" height="315" width="420" > And also how to add mixed fractions with a common denominator > media type="youtube" key="614dVlsDlV4" height="315" width="420"
 * Step 1: Make sure the bottom numbers (the denominators ) are the same
 * Step 2: Add the top numbers (the numerators ). Put the answer over the same denominator.
 * Step 3: Simplify the fraction (if needed).
 * ==Example with denominators the same:==
 * Step 1**. The bottom numbers are already the same. Go straight to step 2.

BUT WHAT IF THE DENOMINATORS ARE NOT THE SAME?

 * ** 1 ** || **+** || ** 1 ** ||
 * ** 3 ** ||^  || ** 6 ** ||
 * 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 this:
 * || 1/3 || + || 1/6 || = || ? ||  ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-3.jpg width="111" height="111" caption="1/3"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-huh.jpg width="120" height="120" caption="1"]] ||   ||

The number "6" is twice as big as "3", so to make the bottom numbers the same I can multiply the top and bottom of the first fraction by **2**, like this:
 * × 2 ||
 * [[image:http://www.mathsisfun.com/images/left-up-over-arrow.gif width="75" height="25"]] ||


 * ** 1 ** || **=** || ** 2 ** ||
 * ** 3 ** ||^  || ** 6 ** ||

Now the first fraction has the same bottom number ("6"), and our question looks like this: The bottom numbers (the denominators) are the same, so we can go to step 2.
 * [[image:http://www.mathsisfun.com/images/left-under-over-arrow.gif width="75" height="25"]] ||
 * × 2 ||
 * || 2/6 || + || 1/6 ||  ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-2-6.jpg width="112" height="112" caption="2/6"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   ||

In picture form it looks like this:
 * Step 2**: Add the top numbers and put them over the same denominator:
 * 2 || + || 1 ||  || 2 + 1 ||   || 3 ||
 * 6 ||^  || 6 ||^   || 6 ||^   || 6 ||
 * || 2/6 || + || 1/6 || = || 3/6 ||  ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-2-6.jpg width="112" height="112" caption="2/6"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-6.jpg width="112" height="111" caption="3/6"]] ||   ||


 * Step 3**: Simplify the fraction:
 * 3 || = || 1 ||
 * 6 ||^  || 2 ||

Example: What is 1/6 + 7/15 ?
The Denominators are 6 and 15: So the **Least Common Multiple** of 6 and 15 is **30**. = Note: what you do to the bottom of the fraction, you must also do to the top = When you multiply 6 **×** 5 you get 30, and when you multiply 15 **×** 2 you also get 30:
 * || multiples of 6: ||  || 6, 12, 18, 24, **30**, 36, ... ||
 * || multiples 15: ||  || 15, **30**, 45, 60, ||
 * ||  || **× 5** ||   ||
 * [[image:http://www.mathsisfun.com/images/left-up-over-arrow.gif width="75" height="25"]] ||
 * 1 || = || 5 ||
 * 6 ||^  || 30 ||
 * [[image:http://www.mathsisfun.com/images/left-under-over-arrow.gif width="75" height="25"]] ||
 * || **× 5** ||  ||   || and ||   ||   || **× 2** ||   ||
 * [[image:http://www.mathsisfun.com/images/left-up-over-arrow.gif width="75" height="25"]] ||
 * 7 || = || 14 ||
 * 15 ||^  || 30 ||
 * [[image:http://www.mathsisfun.com/images/left-under-over-arrow.gif width="75" height="25"]] ||
 * || **× 2** ||  ||   ||

Now we can easily do the addition by adding the top number. 5/30 + 14/30 = 19/30 The attached video clip gives a good explanation for this media type="youtube" key="MEo0rSZ4O3w" height="315" width="560" =Subtracting Fractions=

There are 3 simple steps to subtract fractions

 * Step 1. Make sure the bottom numbers (the denominators) are the same
 * Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.
 * Step 3. Simplify the fraction.

Example 1:

 * 3 || – || 1 ||
 * 4 ||^  || 4 ||
 * Step 1**. The bottom numbers are already the same. Go straight to step 2.
 * Step 2**. Subtract the top numbers and put the answer over the same denominator:


 * 3 || – || 1 ||  || 3 – 1 ||   || 2 ||
 * 4 ||^  || 4 ||^   || 4 ||^   || 4 ||


 * Step 3**. Simplify the fraction:
 * 2 || = || 1 ||
 * 4 ||^  || 2 ||

Example 2:

 * 1 || – || 1 ||
 * 2 ||^  || 6 ||
 * 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** subtract them like this:
 * || 1/2 || - || 1/6 || = || ? ||  ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-2.jpg width="111" height="111" caption="1/3"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-huh.jpg width="120" height="120" caption="1"]] ||   ||

To make the bottom numbers the same, multiply the top and bottom of the first fraction ( 1/2 ) by **3** like this:
 * × 3 ||
 * [[image:http://www.mathsisfun.com/images/left-up-over-arrow.gif width="75" height="25"]] ||


 * 1 || = || 3 ||
 * 2 ||^  || 6 ||

And now our question looks like this: The bottom numbers (the denominators) are the same, so we can go to step 2.
 * [[image:http://www.mathsisfun.com/images/left-under-over-arrow.gif width="75" height="25"]] ||
 * × 3 ||
 * || 3/6 || - || 1/6 ||  ||   ||   ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-6.jpg width="112" height="112" caption="2/6"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pie-blank.jpg width="120" height="120" caption="1"]] ||   ||

In picture form it looks like this:
 * Step 2**. Subtract the top numbers and put the answer over the same denominator:
 * 3 || – || 1 ||  || 3 – 1 ||   || 2 ||
 * 6 ||^  || 6 ||^   || 6 ||^   || 6 ||
 * || 3/6 || - || 1/6 || = || 2/6 ||  ||
 * || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-6.jpg width="112" height="112" caption="2/6"]] ||  || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] ||   || [[image:http://www.mathsisfun.com/images/fractions/pizza-2-6.jpg width="112" height="111" caption="3/6"]] ||   ||
 * Step 3**. Simplify the fraction:

For an overview of this watch the following video media type="youtube" key="T8Lo_cOW0As" height="315" width="420" and for a complete recap of adding and subtracting fractions: media type="youtube" key="ZhWtNmNLhfw" height="315" width="420" =Adding and Subtracting Mixed Fractions= this is the best way to add mixed fractions:
 * 2 || = || 1 ||
 * 6 ||^  || 3 ||
 * convert them to [|Improper Fractions]
 * then add them (using [|Addition of Fractions])
 * then convert back to Mixed Fractions:

Example: What is 2 3/4 + 3 1/2 ?
Convert to Improper Fractions: 2 3/4 = 11/4  3 1/2 = 7/2  Common denominator of 4: 11/4 stays as 11/4 7/2 becomes 14/4 Now Add: 11/4 + 14/4 = 25/4 Convert back to Mixed Fractions: 25/4 = 6 1/4

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

Example: What is 15 3/4 - 8 5/6 ?
Convert to Improper Fractions: 15 3/4 = 63/4  8 5/6 = 53/6  Common denominator of 12: 63/4 becomes 189/12 53/6 becomes 106/12 Now Subtract: 189/12 - 106/12 = 83/12 Convert back to Mixed Fractions: 83/12 = 6 11/12