# Dividing Fractions Has A Weird Rule

Dividing fractions can be a little tricky. It’s the only operation that requires using the **reciprocal**. Using the reciprocal simply means you **flip** the fraction over, or **invert it**.

**For example**, the reciprocal of **2/3** is** 3/2**.

After we give you the division rule, we will show you WHY you have to use the reciprocal in the first place.

**But for now…**

### Here’s the Rule for Division

To divide, **convert** the fraction division process to a multiplication process by using the following steps.

- Change the “÷” (division sign) to “x” (multiplication sign) and invert the number
**to the right**of the sign. - Multiply the numerators.
- Multiply the denominators.
- Re-write your answer in its simplified or reduced form, if needed

Once you complete **Step #1** for dividing fractions, the problem actually changes from **division to multiplication**.

### Example 1: Dividing Fractions by Fractions

### 1/2 ÷ 1/3 = 1/2 x 3/1

### 1/2 x 3/1 = 3/2

### Simplified Answer is 1 1/2

### Example 2: Dividing Fractions by Whole Numbers

### 1/2 ÷ 5 = 1/2 ÷ 5/1

### (Remember to convert

whole numbers to fractions, FIRST!)

### 1/2 ÷ 5/1 = 1/2 x 1/5

### 1/2 x 1/5 = 1/10

### Example 3: Dividing Whole Numbers by Fractions

### 6 ÷ 1/3 = 6/1 ÷ 1/3

### (Remember to convert

whole numbers to fractions, FIRST!)

### 6/1 ÷ 1/3 = 6/1 x 3/1

### 6/1 x 3/1 = 18/1 = 18

Now that’s all there is to it. The **main things** you have to remember when you divide is to convert whole numbers to fractions first, then invert the fraction** to the right** of the division sign, and change the sign to multiplication.

The **“divisor”** has some **other considerations** you should keep in mind…

**Special Notes!**

- Remember to only invert the divisor.
- The divisor’s numerator or denominator
**can not**be “zero”. - Convert the operation to multiplication
**BEFORE**performing any cancellations.

## Want to practice dividing fractions?

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I promised to try to **explain why** the rule requires inverting the divisor.

Here goes..

### Why Dividing Fractions Requires Inverting The Divisor

Let’s use our simple example to actually validate this strange Rule for division.

If you really think about it, we are dividing a **fraction by a fraction**, which forms what is called a **“complex fraction”**. It actually looks like this…

When working with complex fractions, what we want to do first is **get rid of the denominator** **(1/3)**, so we can work this problem easier.

You may recall that any number multiplied by its reciprocal is equal to 1. And since, **1/3 x 3/1 = 1**, we can use the reciprocal property of 1/3, **(3/1)**, to make the value of the denominator equal to 1.

You might also recall that whatever we do to the fraction’s denominator, we **must** also do to its numerator, so as not to change the overall fraction “value”.

**So let’s multiply both the numerator and denominator by 3/1…**

**Which gives us…**

### Here’s what happened…

By multiplying the numerator and denominator of our complex fraction by 3/1, we were then able to use the reciprocal property of a fraction to eliminate the denominator. Actually, without our helpful **Rule**, we would have to use all of the steps above.

So, the **Rule** for dividing fractions really saves us a lot of steps!

Now that’s the simplest explanation I could come up with for **WHY** and **HOW** we end up with a **Rule** that says whenever we divide fractions, we **must invert the divisor**!