Julie can't seem to escape her math homework. Once she finishes with the multiplication of fractions, she is on to reciprocals.

"I don't understand the use of these at all," she tells her sister Cali.

"You don't think so now, but wait until you divide fractions. Then reciprocals are very useful," Cali explains.

"What am I going to do with this one?" Julie asks.

She shows her sister the textbook.

What is the reciprocal of this fraction? Do you know how to get a product of 1?

**This Concept has all of the necessary information for writing reciprocals. Pay attention and we will come back to this problem at the end of the Concept.**

### Guidance

There are first steps to everything. You will be learning how to divide fractions very soon, in fact, this will begin in the next Concept. But before we dive into the mechanics of dividing fractions, let’s think about some division facts. This will cover some of these "first steps".

**We know that division is the opposite of multiplication**, in fact we could say that multiplication is the ** inverse operation** of division.

**What is an inverse operation?**

**An inverse operation is the opposite operation.** The word “inverse” is a fancy way of saying opposite. If the opposite of addition is subtraction, then subtraction is the inverse operation of addition. We can also say that division is the inverse of multiplication.

**What do inverse operations have to do with dividing fractions?**

**Well, when we divide fractions, we need to perform the inverse operation. To divide a fraction, we have to multiply by the** *reciprocal***of the second fraction.**

**What is a reciprocal?**

A ** reciprocal** is the inverse or opposite form of a fraction. When we change the division to its inverse, multiplication, we also change the second fraction to its reciprocal. We can make any fraction a reciprocal by simply flipping the numerator and the denominator.

**The reciprocal of four-fifths is five-fourths. We simply flipped the numerator and the denominator of the fraction to form its reciprocal.**

**Notice that if we multiply a fraction and it’s reciprocal that the product is 1.**

We will begin dividing fractions in the next Concept, but for right now it is important that you understand that a reciprocal is the inverse of a fraction and know how to write a reciprocal of a fraction.

Try a few of these on your own. Write a reciprocal for each fraction.

#### Example A

**Solution: **

#### Example B

**Solution: **

#### Example C

**Solution: **

Now back to Julie and the reciprocals. Here is the original problem once again.

Julie can't seem to escape her math homework. Once she finishes with the multiplication of fractions, she is on to reciprocals.

"I don't understand the use of these at all," she tells her sister Cali.

"You don't think so now, but wait until you divide fractions. Then reciprocals are very useful," Cali explains.

"What am I going to do with this one?" Julie asks.

She shows her sister the textbook.

What is the reciprocal of this fraction?

To find a product of one, we have to multiply this fraction by its reciprocal.

**This is our answer.**

### Vocabulary

- Inverse Operation
- opposite operation. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

- Reciprocal
- the inverse of a fraction. We flip a fraction’s numerator and denominator to write a reciprocal. The product of a fraction and its reciprocal is one.

### Guided Practice

Here is one for you to try on your own.

Write a reciprocal for the fraction .

**Answer**

To write a reciprocal, we simply "flip" the fraction so that the denominator becomes the numerator and the numerator becomes the denominator.

**Our answer is .**

### Video Review

Khan Academy: Reciprocal of a Mixed Number

### Practice

Directions: Write reciprocals of the following fractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.