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# Multiplying Fractions

by Ron Kurtus (revised 18 February 2009)

** Multiplying fractions** simply consists of multiplying the numerators (top numbers) by each other and the denominators (bottom numbers) by each other. If you want to multiply a whole number by a fraction, you must first convert the whole number to a fraction and then proceed. When multiplying fractions and mixed numbers, you must first convert the mixed number to a fraction and then proceed.

Questions you may have include:

- How do you multiply two fractions?
- How do you multiply a fraction and whole number?
- How do you multiply a fraction and a mixed number?

This lesson will answer those questions.

## Multiply two fractions

It is easy to multiple two fractions together. You just multiply the top numbers (numerators) by each other and the bottom numbers (denominators) by each other.

### Example

Consider the multiplying the fractions:

2/7 × 3/5

Multiply top numbers or numerators (**2** and **3**) together and bottom numbers or denominators (**7** and **5**) together:

(2 × 3) / (7 × 5) =6/35

You can apply this technique to multiply a series of fractions.

## Multiply whole number with fraction

You can multiply a whole number and a fraction by considering the whole number as a form of fraction. For example, the number **5** can be considered the fraction **5/1** (**5** divided by **1** equals **5**). Thus, you follow the same technique as in multiply two fractions together.

### Example

Consider the multiplying a whole number by a fraction:

5 × 3/7

Change the whole number to a fraction.

5/1 × 3/7

Multiply numerators together and denominators together.

(5 × 3)/(1 × 7) =15/7

Notethat15/7is considered an improper fraction. The proper form for your fraction answer is that the numerator is smaller than the denominator.

Divide **15** by **7** to reduce your answer a mixed number.

15 ÷ 7 = 2 1/7

### Another example

Consider the multiplying a fraction by a whole number:

1/2 × 9

Change whole number to fraction:

1/2 × 9/1

Multiply numerators together and denominators together:

(1 × 9)/(2 × 1) =9/2

Reduce improper fraction to a mixed number:

9 ÷ 2 = 4 1/2

## Multiply with mixed numbers

When you multiply a fraction times a mixed number, you must first convert the mixed number into a fraction and then perform the multiplication.

### First convert mixed number

A mixed number is the sum of a whole number and a fraction: **5 1/2 = 5 + 1/2**.

You can convert a mixed number into a fraction by first converting the whole number into a fraction over **1**. For example: **5 = 5/1**.

In order to complete the addition of **5/1 + 1/2**, both of the denominators must be the same. In other words, in this case, you want to have both denominators equal to **2**.

You can achieve this by multiplying **5/1 × 2/2 = 10/2**. (Note that **2/2 = 1**, so you are just multiplying by **1**.)

Thus:

5 1/2 = 10/2 + 1/2 = 11/2

Instead of going through those steps, you can shortcut the operation for **5 1/2** by simply multiplying the **5** by the denominator of the fraction (**2**) and then adding to the numerator of the fraction (**1**). Thus, **(5 × 2) + 1 = 11** and **5 1/2 = 11/2**.

#### Another example

To convert **7 2/3** to a mixed number, change the whole number **7** to a fraction over **1**.

7/1 + 2/3

Multiply the whole number fraction by **3/3**:

7/1 × 3/3 = 21/3

Add the fractions:

21/3 + 2/3 = 23/3

#### Shorter method

Once you get good at it, you can use the shortcut method of multiplying the whole number by the denominator of the fraction and adding to the numerator:

7 2/3 =(7 × 3 + 2)/3 =

You always do multiplication before addition within the parentheses:

(21 + 3)/3 =23/3

### Multiply fraction and mixed number

It doesn't matter whether you multiply the fraction times the mixed number or the mixed number times the fraction.

3/5 × 6 2/7 =

Change the mixed number to a fraction:

3/5 × (6 × 7 + 2)/7 =

3/5 × (42 + 2)/7 =

Multiply the fractions:

3/5 × 44/7 =

(3 × 44)/(5 × 7)

132/35

Since **132** is greater than **35**, you need to reduce the fraction to a proper fraction or mixed number. Thus, by dividing **132 ÷ 35 = 3 21/35**.

### Multiply two mixed numbers

A similar process is followed when you multiply two mixed numbers.

2 1/2 × 3 2/3 =

Change both mixed numbers to fractions:

(2 × 2 + 1)/ = 5/2

(3 × 3 + 2)/ = 11/3

5/2 × 11/3 =

55/6=

9 1/6

## Summary

Multiplying fractions consists of multiplying the numerators by each other and the denominators by each other. In multiplying a whole number by a fraction, you must first convert the whole number to a fraction and then proceed. When multiplying fractions and mixed numbers, you must first convert the mixed number to a fraction and then proceed.

Be amazed

## Resources and references

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## Where are you now?

## Multiplying Fractions