Multiplying+and+Dividing+Fractions

=Multiplying Fractions= //Multiply the tops, multiply the bottoms.// 1. Multiply the top numbers (the //numerators//). 2. Multiply the bottom numbers (the //denominators//). 3. Simplify the fraction if needed. ||  ||
 * ===There are 3 simple steps to multiply fractions===

Example 1

 * 1 || × || 2 ||
 * 2 ||^  || 5 ||
 * Step 1**. Multiply the top numbers:
 * 1 || × || 2 || = || 1 × 2 || = || 2 ||
 * 2 ||^  || 5 ||^   ||   ||^   ||   ||
 * Step 2**. Multiply the bottom numbers:
 * 1 || × || 2 || = || 1 × 2 || = || 2 ||
 * 2 ||^  || 5 ||^   || 2 × 5 ||^   || 10 ||
 * 1 || × || 2 || = || 1 × 2 || = || 2 ||
 * 2 ||^  || 5 ||^   || 2 × 5 ||^   || 10 ||
 * 2 ||^  || 5 ||^   || 2 × 5 ||^   || 10 ||

(If you are unsure of the last step see [| Equivalent Fractions])
 * Step 3**. Simplify the fraction:
 * 2 || = || 1 ||
 * 10 ||^  || 5 ||
 * 10 ||^  || 5 ||

With Pen and Paper
And here is how to do it with a pen and paper (press the play button):

Example 2

 * 1 || × || 9 ||
 * 3 ||^  || 16 ||
 * Step 1**. Multiply the top numbers:
 * 1 || × || 9 || = || 1 × 9 || = || 9 ||
 * 3 ||^  || 16 ||^   ||   ||^   ||   ||
 * Step 2**. Multiply the bottom numbers:
 * 1 || × || 9 || = || 1 × 9 || = || 9 ||
 * 3 ||^  || 16 ||^   || 3 × 16 ||^   || 48 ||
 * 1 || × || 9 || = || 1 × 9 || = || 9 ||
 * 3 ||^  || 16 ||^   || 3 × 16 ||^   || 48 ||
 * 3 ||^  || 16 ||^   || 3 × 16 ||^   || 48 ||

=Multiplying Mixed Fractions= //("Mixed Fractions" are also called "Mixed Numbers")// To multiply [|Mixed Fractions]:
 * Step 3**. Simplify the fraction:
 * 9 || = || 3 ||
 * 48 ||^  || 16 ||
 * 48 ||^  || 16 ||
 * 1) convert to [|Improper Fractions]
 * 2) [|Multiply the Fractions]
 * 3) convert the result back to Mixed Fractions

Example
What is 1 3/8 × 3 ? Think of Pizzas. First, convert the mixed fraction (1 3/8 ) to an an improper fraction ( 11/8 ): 1 lot of 8, plus the 3 eighths = 8+3 = 11 eighths. ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-1-1.jpg width="120" height="120"]][[image:http://www.mathsisfun.com/images/fractions/pie-3-8.jpg width="120" height="120"]] || 1 3/8 is 1 pizza and 3 eighths of another pizza. ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-8-8.jpg width="120" height="120"]][[image:http://www.mathsisfun.com/images/fractions/pie-3-8.jpg width="120" height="120"]] || Cut the whole pizza into eighths and how many eighths do you have in total?

Now multiply that by 3:

|| 1 3/8 × 3 = 11/8 × 3/1 = 33/8
 * [[image:http://www.mathsisfun.com/images/fractions/pie-8-8.jpg width="120" height="120"]][[image:http://www.mathsisfun.com/images/fractions/pie-3-8.jpg width="120" height="120"]]

You have 33 eighths. || And, lastly, convert to a mixed fraction (only because the original fraction was in that form): || 33 eighths is 4 whole pizzas (4×8=32) and 1 eighth left over. || And this is what it looks like in one line: 1 3/8 × 3 = 11/8 × 3/1 = 33/8 = 4 1/8
 * [[image:http://www.mathsisfun.com/images/fractions/pie-8-8.jpg width="120" height="120"]][[image:http://www.mathsisfun.com/images/fractions/pie-8-8.jpg width="120" height="120"]]

Another Example:
What is 1 1/2 x 2 1/5 ? If you know how to go from Mixed Fraction to Improper Fractions and back again it is easy ...

Step, by step it is:
Convert both to improper fractions 1 1/2 × 2 1/5 = 3/2 × 11/5 [|Multiply the fractions] (multiply the top numbers, multiply bottom numbers): 3/2 × 11/5 = (3 × 11)/(2 × 5) = 33/10 Convert to a mixed number 33/10 = **3 3/10 ** If you are clever you can do it all in one line like this: 1 1/2 × 2 1/5 = 3/2 × 11/5 = 33/10 = **3 3/10 **

One More Example:
What is 3 1/4 x 3 1/3 ? Convert both to improper fractions 3 1/4 × 3 1/3 = 13/4 × 10/3 Multiply 13/4 × 10/3 = 130/12 Convert to a mixed number (and [|simplify]): 130/12 = **10 10/12 ** = **10 5/6 ** Once again, here it is in one line: 3 1/4 × 3 1/3 = 13/4 × 10/3 = 130/12 = 10 10/12 = **10 5/6 **

This One Has Negatives
I was asked to solve this one: -1 5/9 × -2 1/7 = ? So, my first step was to convert Mixed to Improper Fractions: 1 5/9 = 9/9 + 5/9 = 14/9

2 1/7 = 14/7 + 1/7 = 15/7 Then multiply the Improper Fractions //(Note: [|negative times negative gives positive])// : -14/9 × -15/7 = -14×-15 / 9×7 = 210/63

I then decided to simplify next, first by 7 (because I noticed that 21 and 63 are both multiples of 7), then again by 3 (but I could have done it in one step by dividing by 21): 210/63 = 30/9 = 10/3

Finally convert to a Mixed Fraction (because that was the style of the question): 10/3 = (9+1)/3 = 9/3 + 1/3 = 3 1/3

=Dividing Fractions= //Turn the second fraction upside down, then just multiply.//

There are 3 Simple Steps to Divide Fractions:
(this is now a [|reciprocal]). ||
 * Step 1. Turn the second fraction //(the one you want to divide by)// upside-down
 * Step 2. [|Multiply] the first fraction by that reciprocal

Step 3. [|Simplify] the fraction (if needed) ||  ||   ||

Example 1

 * 1 || ÷ || 1 ||
 * 2 ||^  || 6 ||
 * 2 ||^  || 6 ||

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


 * 1 || becomes || 6 ||
 * 6 ||^  || 1 ||
 * 6 ||^  || 1 ||

Step 2. Multiply the first fraction by that **reciprocal**:

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

With Pen and Paper
And here is how to do it with a pen and paper (press the play button):

Does it make sense?
You can change a question like "What is 20 divided by 5?" into "How many 5s fit into 20?" In the same way our fraction question can become: Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?
 * Does || 1 || ÷ || 1 || really equal **3** ? ||
 * ^  || 2 ||^   || 6 ||^   ||
 * ^  || 2 ||^   || 6 ||^   ||
 * 1 || ÷ || 1 || [[image:http://www.mathsisfun.com/images/style/right-arrow.gif width="46" height="46" caption="becomes"]] || How many || 1 || in || 1 || ? ||
 * 2 ||^  || 6 ||^   ||^   || 6 ||^   || 2 ||^   ||
 * 2 ||^  || 6 ||^   ||^   || 6 ||^   || 2 ||^   ||
 * How many || [[image:http://www.mathsisfun.com/images/fractions/pizza-1-6.jpg width="112" height="111" caption="1/6"]] || in || [[image:http://www.mathsisfun.com/images/fractions/pizza-3-6.jpg width="112" height="111" caption="3/6"]] || ? ||  || **Answer: 3** ||


 * So now you can see that ||  || 1 || ÷ || 1 || = **3** ||   || really does makes sense! ||
 * ^  ||^   || 2 ||^   || 6 ||^   ||^   ||^   ||
 * ^  ||^   || 2 ||^   || 6 ||^   ||^   ||^   ||

Example 2

 * 1 || ÷ || 1 ||
 * 8 ||^  || 4 ||
 * 8 ||^  || 4 ||

Step 1. Turn the second fraction upside-down (the **reciprocal**):


 * 1 || becomes || 4 ||
 * 4 ||^  || 1 ||
 * 4 ||^  || 1 ||

Step 2. Multiply the first fraction by that **reciprocal**:


 * 1 || × || 4 || = || 1 × 4 || = || 4 ||
 * 8 ||^  || 1 ||^   || 8 × 1 ||^   || 8 ||
 * 8 ||^  || 1 ||^   || 8 × 1 ||^   || 8 ||

Step 3. Simplify the fraction:

And that is all you have to do.
 * 4 || = || 1 ||
 * 8 ||^  || 2 ||
 * 8 ||^  || 2 ||

But maybe you want to know **why** we do it this way ...

Well ... what Does a Fraction Do?
So you:
 * A fraction says to: ||  ||   ||
 * * multiply by the top number
 * divide by the bottom number ||  || [[image:http://www.mathsisfun.com/numbers/images/fraction-multiply-divide.gif width="135" height="61"]] ||
 * [[image:http://www.mathsisfun.com/images/fractions/pie-3-4.jpg width="120" height="120"]] || Example: **3/4** means to cut into 4 pieces, and then take 3 of those.
 * divide by 4
 * multiply by 3 ||

Example: **3/4** of 20 is:
20 divided by 4, then times 3 = (20/4) × 3 = 5 × 3 = **15** Or you could multiply before dividing: 20 times 3, then divide by 4 = (20 × 3) / 4 = 60/4 = **15** Either way gets the same result

Dividing
But when you **DIVIDE** by a fraction, you are asked to do the **opposite of multiply** ... So you:
 * **divide** by the top number
 * **multiply** by the bottom number

Example: dividing by **5/2** is the same as multiplying by **2/5**
Because: Dividing by 5, then Multiplying by 2 is the same as Multiplying by 2, then Dividing by 5 So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.