A fraction represents equal parts of a whole number, When we divide a number into equal parts, each part is called a fraction. It is represented in the form of \(\frac { a }{ b} \), where a is numerator and b is denominator. Refer to the further modules to know about different properties applicable to fractional numbers such as Associative, Commutative, Identity, Inverse, Distributive, etc.

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### Multiplication of Fractional Numbers

Multiplication of fractional numbers starts with the multiplication of the given numerators followed by multiplication of the denominators. Then, the resultant fraction can be simplified further and reduced to its lowest terms to make easy calculations. Multiplication of fractions is not the same as addition or subtraction of fractions, where the denominator should be the same. Here any two fractions without the same denominator can also be multiplied.

### Properties of Multiplication of Fractional Numbers

**Property 1: Associative Property**

This property states that when three fractional numbers are multiplied, the order will not change the product.

It is represented by

(\(\frac { a}{ b} \) * \(\frac { c }{ d} \) ) *\(\frac { e }{ f} \) = \(\frac { a }{ b} \) * (\(\frac { c}{ d} \) *\(\frac { e}{ f} \))

**Example:** ( \(\frac { 2 }{ 7} \) * \(\frac { 7}{ 5} \) ) * \(\frac { 4}{ 3} \) = \(\frac { 2 }{ 7} \) * ( \(\frac { 7 }{ 5} \) * \(\frac { 4 }{ 3} \) )

\(\frac { 8 }{ 15} \) = \(\frac { 15}{ b} \)

L.H.S = R.H.S

Here even if you interchange the place of fractional numbers, the product will be the same.

**Property 2: Commutative Property**

This property states that when two fractional numbers are multiplied, the order of multiplying numbers will not change the product’s value.

It is represented by

\(\frac { a }{ b} \)*\(\frac { c }{ d} \) = \(\frac { c}{ d} \)*\(\frac { a }{ b} \)

**Example:** \(\frac { 1 }{ 4} \) * \(\frac { 2 }{ 5} \) = \(\frac { 2 }{ 5} \) * \(\frac { 1 }{ 4} \)

\(\frac { 1 }{ 10} \) = \(\frac { 1 }{ 10} \)

L.H.S = R.H.S

Multiplication of fractions will follow the commutative property.

**Property 3: Identity Property**

This property states that when a fractional number is multiplied by 1, then the result will be the same number.

It is represented by \(\frac { a }{ b} \) * 1 = \(\frac { a }{ b} \).

**Example:** \(\frac { 4 }{ 3} \) *1 = \(\frac { 4 }{ 3} \).

Multiplication of fractions will follow the Identity property.

**Property 4: Inverse Property**

This property states that when a fractional number is multiplied with the inverse of that number that is equal to one.

It is represented by

\(\frac { a }{ b} \) * \(\frac { b }{ a} \) = 1

where a and b are not zero.

**Example :**

\(\frac { 2}{ 5} \) * \(\frac { 5}{ 2} \) = 1

1 = 1

L.H.S = R.H.S

Multiplication of fractions will follow the Inverse property.

**Property 5: Distributive Property of Multiplication over Addition**

This property states that multiplication can be distributed over addition. Grouping can be done in any manner.

It is represented by

\(\frac { a}{ b} \) * (\(\frac { c}{ d} \) + \(\frac { e}{ f} \) ) = (\(\frac { a}{ b} \) * \(\frac { c}{ d} \) ) + (\(\frac { a}{ b} \) * \(\frac { e}{ f} \) )

**Example:**

\(\frac { 3}{ 5} \) *(\(\frac { 2}{ 3} \) + \(\frac { 1}{ 5} \) ) = (\(\frac { 3}{ 5} \) *\(\frac { 2}{ 3} \) ) + (\(\frac { 3}{ 5} \) *\(\frac { 1}{ 5} \) )

\(\frac { 13}{ 25} \) = \(\frac { 13}{ 25} \)

L.H.S = R.H.S

Multiplication of fractions will follow the distributive property.

**Property 6: Zero property**

This property states that when a fractional number is multiplied by zero, then the result will be zero.

It is represented by

\(\frac { a}{ b} \) * 0 = 0

**Example:** \(\frac { 3}{ 4} \) * 0 = 0

0 = 0

L.H.S = R.H.S

Multiplication of fractions will follow the zero property.

### Multiplication of Fractional Numbers Examples

**Example 1: **

There was \(\frac { 5}{ 8} \) of a pie left in the fridge. Daniel ate \(\frac { 1}{ 5} \) of the leftover pie. How much of a pie did he have?

**Solution:**

Given

Pie left in the fridge = \(\frac { 5}{ 8} \)

Pie ate by daniel =\(\frac { 1}{ 5} \)

Pie he have = Pie left in the fridge * Pie ate by daniel

= \(\frac { 5}{ 8} \) * \(\frac { 1}{ 5} \)

= \(\frac { 5}{ 40} \)

= \(\frac { 1}{ 8} \)

Therefore, he have \(\frac { 1}{ 8} \) of pie.

**Example 2: **

According to a recipe, each batch of pancake mix can make 12 pancakes. Kathy is making 3 batches for a brunch party. If each batch needs \(\frac { 7}{ 12} \) cups of milk, how much milk does she need in total?

**Solution: **

Given

Each batch of pancake mix can make = 12 pancakes

Kathy is making batches for brunch party = 3

Each batch needs cups of milk = \(\frac { 7}{ 12} \)

Total Amount of milk she need = kathy is making batches for brunch party * Each batch needs cups of milk

= 3 * \(\frac { 7}{ 12} \)

= \(\frac { 21}{ 12} \)

= \(\frac { 7}{ 4} \)

Hence, Total Amount of milk she need = \(\frac { 7}{ 4} \)