Mixed Fractions

Mixed Fractions
(Also called "Mixed Numbers") | | A Mixed Fraction is a whole number and a proper fraction combined. such as 1 3/4. | 1 3/4 | | | (one and three-quarters) | | |
Examples
2 3/8 | 7 1/4 | 1 14/15 | 21 4/5 |
See how each example is made up of a whole number and a proper fraction together? That is why it is called a "mixed" fraction (or mixed number).
Names
We can give names to every part of a mixed fraction:

Three Types of Fractions
There are three types of fraction:

Mixed Fractions or Improper Fractions
You can use either an improper fraction or a mixed fraction to show the same amount.
For example 1 3/4 = 7/4, as shown here: 1 3/4 | | 7/4 | | = | |
Converting Mixed Fractions to Improper Fractions
To convert a mixed fraction to an improper fraction, follow these steps:

| * Multiply the whole number part by the fraction's denominator. * Add that to the numerator * Write that result on top of the denominator. | Example: Convert 3 2/5 to an improper fraction.
Multiply the whole number by the denominator:
3 × 5 = 15
Add the numerator to that:
15 + 2 = 17
Then write that down above the denominator, like this: 17 | | 5 |
Converting Improper Fractions to Mixed Fractions
To convert an improper fraction to a mixed fraction, follow these steps:

| * Divide the numerator by the denominator. * Write down the whole number answer * Then write down any remainder above the denominator. | Example: Convert 11/4 to a mixed fraction.
Divide:
11 ÷ 4 = 2 with a remainder of 3
Write down the 2 and then write down the remainder (3) above the denominator (4), like this: 2 | 3 | | | | 4 | When to Use Improper Fractions or Mixed Fractions
For everyday use, people understand mixed fractions better:
Example: It is easier to say "I ate 2 1/4 sausages", than "I ate 9/4 sausages"
But for mathematics improper fractions are actually better than mixed fractions.
Because mixed fractions can be confusing when you write them down in a formula (are the two parts supposed to be added or multipled?): Mixed Fraction: | What is: | 1 + 2 1/4 | | ? | | Is it: | 1 + 2 + 1/4 | | = 3 1/4 ? | | Or is it: | 1 + 2 × 1/4 | | = 1 1/2 ? | | | | | | Improper Fraction: | What is: | 1 + 9/4 | | ? | | It is: | 4/4 + 9/4 = 13/4 | | |
Adding and Subtracting Mixed Fractions | Quick Definition: A Mixed Fraction is a whole number and a fraction combined, such as 1 3/4. | 1 3/4 | | (one and three-quarters) | |
To make it easy to add and subtract them, just convert to Improper Fractions first: | Quick Definition: An Improper fraction has a top number larger than or equal to the bottom number,

such as 7/4 or 4/3

(It is "top-heavy") | 7/4 | | (seven-fourths or seven-quarters) | | Adding Mixed Fractions
I find this is the best way to add mixed fractions: * 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
(by multiplying top and bottom by 2)
Now Add:
11/4 + 14/4 = 25/4
Convert back to Mixed Fractions:
25/4 = 6 1/4
When you get more experience you can do it faster like this:
Example: What is 3 5/8 + 1 3/4
Convert them to improper fractions:
3 5/8 = 29/8
1 3/4 = 7/4
Make same denominator: 7/4 becomes 14/8 (by multiplying top and bottom by 2)
And add:
29/8 + 14/8 = 43/8 = 5 3/8 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
Multiplying Mixed Fractions
("Mixed Fractions" are also called "Mixed Numbers")
To multiply Mixed Fractions: 1. convert to Improper Fractions 2. Multiply the Fractions 3. convert the result back to Mixed Fractions
Example: What is 13/8 × 3 ?
Think of Pizzas. | 1 3/8 is 1 pizza and 3 eighths of another pizza. |
First, convert the mixed fraction (1 3/8) to an an improper fraction (11/8): | Cut the whole pizza into eighths and how many eighths do you have in total?1 lot of 8, plus the 3 eighths = 8+3 = 11 eighths. |

Now multiply that by 3:

| 1 3/8 × 3 = 11/8 × 3/1 = 33/8
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
Another Example: What is 11/2 x 21/5 ?
Do the steps from above: 1. convert to Improper Fractions 2. Multiply the Fractions 3. convert the result back to Mixed Fractions

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 31/4 x 31/3 ?
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: What is -15/9 × -21/7 ?
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 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: 1 | ÷ | 1 | | | | 2 | | 6 |

Step 1. Turn the second fraction upside-down (it becomes a reciprocal): 1 | becomes | 6 | | | | 6 | | 1 |

Step 2. Multiply the first fraction by that reciprocal: 1 | × | 6 | = | 1 × 6 | = | 6 | | | | | | | | 2 | | 1 | | 2 × 1 | | 2 |
Step 3. Simplify the fraction: 6 | = | 3 | | | | 2 | | |
With Pen and Paper
And here is how to do it with a pen and paper (press the play button):
A Trick to Help You
Try to rewrite the question the other way around ...
You can rewrite a division question like 20 divided by 5 into "how many 5s in 20?" (=4)
So you can also rewrite 1/2 divided by 1/6 into "how many 1/6s in 1/2" (=3) Further Explanation ...
When you divide, you are cutting something into equal shares. 1 | ÷ | 1 | is really asking: | | | | | 2 | | 6 | |

How many | 1 | in | 1 | ? | | | | | | | 6 | | 2 | | Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"? How many | | in | | ? | | Answer: 3 | So now you can see why | | 1 | ÷ | 1 | = 3 | | | | | | | | | 2 | | 6 | |

Another Example: 1 | ÷ | 1 | | | | 8 | | 4 |

Step 1. Turn the second fraction upside-down (the reciprocal): 1 | becomes | 4 | | | | 4 | | 1 |

Step 2. Multiply the first fraction by that reciprocal: 1 | × | 4 | = | 1 × 4 | = | 4 | | | | | | | | 8 | | 1 | | 8 × 1 | | 8 |

Step 3. Simplify the fraction: 4 | = | 1 | | | | 8 | | 2 |
To help you remember:
♫ "Dividing fractions, as easy as pie,
Flip the second fraction, then multiply."
"And don't forget to simplify,
Before it's time to say goodbye" ♫ Fractions and Whole Numbers
What about division with fractions and whole numbers?
Make the whole number a fraction, by putting it over 1. Example: 5 is also | 5 | | | | 1 |
Then continue as before.
Example:
2 | ÷ | 5 | | | | 3 | | |
Make 5 into 5/1 : 2 | ÷ | 5 | | | | 3 | | 1 |

Step 1. Turn the second fraction upside-down (the reciprocal): 5 | becomes | 1 | | | | 1 | | 5 |

Step 2. Multiply the first fraction by that reciprocal: 2 | × | 1 | = | 2 × 1 | = | 2 | | | | | | | | 3 | | 5 | | 3 × 5 | | 15 |

Step 3. Simplify the fraction:
The fraction is already as simple as it can be. Answer = | 2 | | | | 15 | Example: 3 | ÷ | 1 | | | | | | 4 |
Make 3 into 3/1 : 3 | ÷ | 1 | | | | 1 | | 4 |

Step 1. Turn the second fraction upside-down (the reciprocal): 1 | becomes | 4 | | | | 4 | | 1 |

Step 2. Multiply the first fraction by that reciprocal: 3 | × | 4 | = | 3 × 4 | = | 12 | | | | | | | | 1 | | 1 | | 1 × 1 | | 1 |

Step 3. Simplify the fraction: 12 | = | 12 | | | | 1 | | | Remember the "Trick to Help You" ...
You can rewrite a division question like "20 divided by 5" into "how many 5s in 20" (=4)
So you can also rewrite "3 divided by ¼" into "how many ¼s in 3" (=12) Why Turn the Fraction Upside Down?
Because dividing is the opposite of multiplying! A fraction says to: | | | * multiply by the top number * divide by the bottom number | | |
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.