Lily says she has an unbeatable math trick. She bets the class that she can guess everyone's final number at the end of the trick. She says, "Pick any whole number. Add 3. Multiply the sum by 5. Then, subtract 10. Multiply that number by the reciprocal. I bet I can guess your final number." Why is Lily so sure she will always guess correctly?

In this concept, you will learn how to write and identify reciprocal fractions.

### Identifying and Writing Reciprocal Fractions

Before you dive into the mechanics of dividing fractions, let’s think about some division facts. Division is the opposite of multiplication. Multiplication and division are **inverse operations**. The word “inverse” means opposite. Addition and subtraction are also inverse operations.

When dividing with fractions, the rule is to multiply by the reciprocal of the divisor.

A **reciprocal** is the inverse or opposite form of a fraction. You find the reciprocal of any number by flipping the numerator and the denominator.

Here is an example.

\begin{align*}\frac{4}{5} \overrightarrow{} \frac{5}{4}\end{align*}

Flip the numerator and the denominator. The reciprocal of four-fifths is five-fourths.

Here is another example.

\begin{align*}\frac{1}{2} \overrightarrow{} \frac{2}{1}\end{align*}

Look what happens when you multiply a fraction by its reciprocal.

\begin{align*}\frac{1}{2} \times \frac{2}{1} = \frac{2}{2} = 1\end{align*}

The product of any number and its reciprocal is always 1.

Identifying the reciprocal of a fraction is important to learning how to divide fractions.

### Examples

#### Example 1

Earlier, you were given a problem about Lily's riddle.

Lily bets the class that she can guess everyone's final number at the end of her riddle: "Pick any whole number. Add 3. Multiply the sum by 5. Then, subtract 10. Multiply that number by the reciprocal. I bet I can guess your final number." Look at the last part of the riddle, "multiply that number by the reciprocal." By now you know that any number multiplied by its reciprocal will always be 1.

Here is an example. If you start with the number 5, The number after subtracting 10 will be 30.

\begin{align*}5+3=8\times 5 = 40 - 10 = 30\end{align*}

The reciprocal of 30 is \begin{align*}\frac{1}{30}\end{align*}

\begin{align*}30\times \frac{1}{30}= 1\end{align*}

Lily is sure because no matter the starting number, the last number will always be 1.

#### Example 2

Write a reciprocal for the fraction \begin{align*}\frac{5}{7}\end{align*}

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

\begin{align*}\frac{5}{7}\overrightarrow{}\frac{7}{5}\end{align*}

The reciprocal is \begin{align*}\frac{7}{5}\end{align*}

#### Example 3

Identify the reciprocal of \begin{align*}\frac{1}{4}\end{align*}

Flip the numerator and the denominator.

\begin{align*}\frac{1}{4}\overrightarrow{}\frac{4}{1}\end{align*}

The reciprocal is \begin{align*} \frac{4}{1}\end{align*}

#### Example 4

Identify the reciprocal of \begin{align*}\frac{4}{7}\end{align*}

Flip the numerator and the denominator.

\begin{align*}\frac{4}{7}\overrightarrow{}\frac{7}{4}\end{align*}

The reciprocal is \begin{align*} \frac{7}{4}\end{align*}

#### Example 5

Identify the reciprocal of \begin{align*}\frac{2}{5}\end{align*}

Flip the numerator and the denominator.

\begin{align*}\frac{2}{5}\overrightarrow{}\frac{5}{2}\end{align*}

The reciprocal is \begin{align*} \frac{5}{2}\end{align*}

### Review

Identify the reciprocals

- \begin{align*} \frac{1}{2}\end{align*}
12 - \begin{align*} \frac{2}{3}\end{align*}
23 - \begin{align*} \frac{4}{5}\end{align*}
45 - \begin{align*} \frac{11}{12}\end{align*}
1112 - \begin{align*} \frac{8}{9}\end{align*}
89 - \begin{align*} \frac{9}{10}\end{align*}
910 - \begin{align*} \frac{12}{13}\end{align*}
- \begin{align*} \frac{11}{2}\end{align*}
- \begin{align*} \frac{14}{6}\end{align*}
- \begin{align*} \frac{8}{3}\end{align*}
- \begin{align*} \frac{9}{4}\end{align*}
- \begin{align*} \frac{11}{7}\end{align*}
- \begin{align*} \frac{15}{4}\end{align*}
- \begin{align*} \frac{18}{7}\end{align*}
- \begin{align*} \frac{21}{8}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.7.