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There are 3 simple steps to multiply fractions

1. Multiply the top numbers (the numerators).

2. Multiply the bottom numbers (the denominators).

3. Simplify the fraction if needed.

Example:1/2 × 2/5

Step 1. Multiply the top numbers:

1/2 × 2/5  =  1 × 2   =  2

Step 2. Multiply the bottom numbers:

1/2 × 2/5  =  1 × 2/2 × 5  =  2/10

Step 3. Simplify the fraction:

2/10 = 1/5

With Pizza

Here you can see it with pizza ...

Do you see that half of two-fifths is two-tenths?
Do you also see that two-tenths is simpler as one-fifth?

With Pen and Paper

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

Another Example:1/3 × 9/16

Step 1. Multiply the top numbers:

1/3 × 9/16  =  1 × 9   =  9

Step 2. Multiply the bottom numbers:

1/3 × 9/16  =  1 × 9/3 × 16  =  9/48

Step 3. Simplify the fraction:

9/48 = 3/16

(This time we simplified by dividing both top and bottom by 3)

The Rhyme

♫ "Multiplying fractions: no big problem,
Top times top over bottom times bottom.
"And don't forget to simplify,
Before it's time to say goodbye" ♫

Fractions and Whole Numbers

What about multiplying 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:2/3  ×  5

Make 5 into 5/1 :

2/3 × 5/1

Now just go ahead as normal.

Multiply tops and bottoms:

2/3 × 5/1  =  2 × 5/3 × 1  =  10/3

The fraction is already as simple as it can be.

Answer = 10/3

Or you can just think of the whole number as being a "top" number:

Example:3 × 2/9

Multiply tops and bottoms:

3  × 2/9  =  3 × 2/9  =  6/9

Simplify:

6/9 = 2/3

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(Also called "Mixed Numbers")

1/3 4
(one and three-quarters)

A Mixed Fraction is a whole number and a proper fraction combined.

Such as 1/3 4

Examples

2 3/8 | 7  7/1 4 | 1  14/15 | 2 1/4 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

We 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 (3) by the fraction's denominator (5):

3 × 5 = 15

Add the fraction's numerator (2) to that:

15 + 2 = 17

Then put that 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 answerThen write down any remainder above the denominator.

Example: Convert 1 1/4 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 we write them in a formula: should the two parts be added or multiplied?

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 |   |

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7/4
(seven-fourths or seven-quarters)

An Improper Fraction has a top number larger than (or equal to) the bottom number.

It is "top-heavy"

More Examples

3/2 | 7/3 | 16/15 | 15/15 | 100/5

See how the top number is bigger than (or equal to) the bottom number?
That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions).

Three Types of Fractions

There are three types of fraction:

Fractions

A Fraction (such as ⁷/₄) has two numbers:

Numerator Denominator

The top number (the Numerator) is the number of parts we have.
The bottom number (the Denominator) is the number of parts the whole is divided into.

Example: ⁷/₄ means:

We have 7 parts

Each part is a quarter (¹/₄) of a whole

So we can define the three types of fractions like this:

Proper Fractions: | The numerator is less than the denominator
Examples: ¹/₃, ³/₄, ²/₇
|
Improper Fractions: | The numerator is greater than (or equal to) the denominator
Examples: ⁴/₃, ¹¹/₄, ⁷/₇
|
Mixed Fractions: | A whole number and proper fraction together
Examples: 1 ¹/₃, 2 ¹/₄, 16 ²/₅

Improper Fraction

So an improper fraction is a fraction where the top number (numerator) is greater than or equal to the bottom number (denominator): it is top-heavy.

4/4

Can be Equal

What about when the numerator is equal to the denominator? For example ⁴/₄ ?

Well it is the same as a whole, but it is written as a fraction, so most people agree it is a type of improper fraction.

Improper Fractions or Mixed Fractions

We can use either an improper fraction or a mixed fraction to show the same amount.

For example 13/4 = 7/4, as shown here:

13 4 |   | 7 4 | = |

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 answerThen 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

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 numeratorThen write the 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 that to the numerator:

15 + 2 = 17

Then write that result above the denominator, like this:

17/5

Are Improper Fractions Bad ?

NO, they aren't bad!

For mathematics they are actually better than mixed fractions. Because mixed fractions can be confusing when we write them in a formula: should the two parts be added or multiplied?

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 = 1 3/4 |   |

But, 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"

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|   | A Proper Fraction has a top number

less than its bottom number
3/8
(Three-Eighths)

Examples

3/8 | 1/4 | 14/15 | 4/5

See how the top number is smaller than the bottom number in each example? That makes it a Proper Fraction.

Three Types of Fractions

There are three types of fraction:

Fractions

A Fraction (such as ³/₈) has two numbers:

Numerator Denominator

The top number is the Numerator, it is the number of parts you have.
The bottom number is the Denominator, it is the number of parts the whole is divided into.

Example: ³/₄ means:

We have 3 parts

Each part is a quarter (¹/₄) of a whole

So we can define the three types of fractions like this:

Proper Fractions: | The numerator is less than the denominator
Examples: ¹/₃, ³/₄, ²/₇
|
Improper Fractions: | The numerator is greater than (or equal to) the denominator
Examples: ⁴/₃, ¹¹/₄, ⁷/₇
|
Mixed Fractions: | A whole number and proper fraction together
Examples: 1 ¹/₃, 2 ¹/₄, 16 ²/₅

Proper Fractions

So, a proper fraction is just a fraction where the numerator (the top number) is less than the denominator (the bottom number). Here are some examples of proper fractions:

|

|

¹/₂ | ¹/₄ | ³/₈
(One-Half) | (One-Quarter) | (Three-Eighths)

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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"

Example 3:

1/3 + 1/5

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

1/3 | + | 1/5 | = | ? |   |  |   | |   | |   |

But let us try dividing them into smaller sizes that will each be the same:

5/15 | + | 3/15 |   |   |   |  |   | |   | |   |

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

× 5

1/3 |   =   | 5/15

× 5

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

× 3

1/5 |   =   | 3/15

× 3

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

5/15 | + | 3/15 | = | 8/15 |   |  |   | |   | |   |

The result is already as simple as it can be, so that is the answer: 8/15

Making the Denominators the Same

In the previous example how did we know to cut them into ¹/₁₅ths to make the denominators the same? We simply multiplied the two denominators together (3 × 5 = 15).

Read about the two main ways to make the denominators the same here:

Common Denominator Method, or the

Least Common Denominator Method

They both work, use which one you prefer!

Example: Cupcakes

You want to make and sell cupcakes:

A friend can supply the ingredients, if you give them ¹/₃of sales

And a market stall costs ¹/₄ of sales

How much is that altogether?

We need to add ¹/₃ and ¹/₄

1 | + | 1 | = | ?
3 | 4 | ?

First make the bottom numbers (the denominators) the same.

Multiply top and bottom of ¹/₃ by 4:

1 × 4 | + | 1 | = | ?
3 × 4 | 4 | ?

And multiply top and bottom of ¹/₄ by 3:

1 × 4 | + | 1 × 3 | = | ?
3 × 4 | 4 × 3 | ?

Now do the calculations:

4 | + | 3 | = | 4+3 | = | 7
12 | 12 | 12 | 12

Answer: 7/12 of sales go in ingredients and market costs.

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There are 3 Simple Steps to add fractions:

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 denominator

Step 3: Simplify the fraction (if needed)

Example 1:

1/4 + 1/4

Step 1. The bottom numbers (the denominators) are already the same. Go straight to step 2.

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

1/4 + 1/4 = 1 + 1/4 = 2/4

Step 3. Simplify the fraction:

2/4 = 1/2

In picture form it looks like this:

1/4 | + | 1/4 | = | 2/4 | = | 1/2 |   | |   | |   |

... and do you see how 2/4 is simpler as 1/2 ? (see Equivalent Fractions.)

Example 2:

1/3 + 1/6

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

1/3 | + | 1/6 | = | ? |   |  |   | |   | |   |

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:

× 2

1/3 |   =   | 2/6

× 2

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:

2/6 | + | 1/6 |   |   |   |  |   | |   | |   |

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:

2/6 + 1/6 = 2 + 1/6 = 3/6

In picture form it looks like this:

2/6 | + | 1/6 | = | 3/6 |   |  |   | |   | |   |

Step 3: Simplify the fraction:

3/6 = 1/2

In picture form the whole answer looks like this:

2/6 | + | 1/6 | = | 3/6 | = | 1/2 |   | |   | |   |