What if you had two numbers like \begin{align*}\frac{6}{5}\end{align*}

### Watch This

CK-12 Foundation: 0206S Dividing Rationals

### Try This

For more practice dividing fractions, try the game at http://www.aaamath.com/div66ox2.htm or the one at http://www.mathplayground.com/fractions_div.html.

### Guidance

An **identity element** is a number which, when combined with a mathematical operation on a number, leaves that number unchanged. For example, the **identity element** for addition and subtraction is **zero**, because adding or subtracting zero to a number doesn’t change the number. And zero is also what you get when you add together a number and its opposite, like 3 and -3.

**Multiplicative Inverses**

The **inverse operation** of addition is subtraction—when you add a number and then subtract that same number, you end up back where you started. Also, adding a number’s opposite is the same as subtracting it—for example, \begin{align*}4 + (-3)\end{align*}

Multiplication and division are also inverse operations to each other—when you multiply by a number and then divide by the same number, you end up back where you started. Multiplication and division also have an identity element: when you multiply or divide a number by **one**, the number doesn’t change.

Just as the **opposite** of a number is the number you can add to it to get zero, the **reciprocal** of a number is the number you can multiply it by to get one. And finally, just as adding a number’s opposite is the same as subtracting the number, multiplying by a number’s reciprocal is the same as dividing by the number.

The reciprocal of a number \begin{align*}x\end{align*}**multiplicative inverse**. Any number times its own multiplicative inverse equals one, and the multiplicative inverse of \begin{align*}x\end{align*}

To find the multiplicative inverse of a rational number, we simply ** invert the fraction**—that is, flip it over. In other words:

The multiplicative inverse of \begin{align*}\frac{a}{b}\end{align*}

You’ll see why in the following exercise.

#### Example A

*Find the multiplicative inverse of each of the following.*

a) \begin{align*}\frac{3}{7}\end{align*}

b) \begin{align*}\frac{4}{9}\end{align*}

c) \begin{align*}3\frac{1}{2}\end{align*}

d) \begin{align*}-\frac{x}{y}\end{align*}

e) \begin{align*}\frac{1}{11}\end{align*}

**Solution**

a) When we invert the fraction \begin{align*}\frac{3}{7}\end{align*}

b) Similarly, the inverse of \begin{align*}\frac{4}{9}\end{align*}

c) To find the multiplicative inverse of \begin{align*}3\frac{1}{2}\end{align*}

d) Don’t let the negative sign confuse you. The multiplicative inverse of a negative number is also negative! Just ignore the negative sign and flip the fraction as usual.

The multiplicative inverse of \begin{align*}-\frac{x}{y}\end{align*}

e) The multiplicative inverse of \begin{align*}\frac{1}{11}\end{align*}

Look again at the last example. When we took the multiplicative inverse of \begin{align*}\frac{1}{11}\end{align*}

**Divide Rational Numbers**

Earlier, we mentioned that multiplying by a number’s reciprocal is the same as dividing by the number. That’s how we can divide rational numbers; to divide by a rational number, just multiply by that number’s reciprocal. In more formal terms:

\begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.\end{align*}

#### Example B

*Divide the following rational numbers, giving your answer in the* **simplest form**.

a) \begin{align*}\frac{1}{2} \div \frac{1}{4}\end{align*}

b) \begin{align*}\frac{7}{3} \div \frac{2}{3}\end{align*}

c) \begin{align*}\frac{x}{2} \div \frac{1}{4y}\end{align*}

d) \begin{align*}\frac{11}{2x} \div \left ( -\frac{x}{y} \right )\end{align*}

**Solution**

a) Replace \begin{align*}\frac{1}{4}\end{align*} with \begin{align*}\frac{4}{1}\end{align*} and multiply: \begin{align*}\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2\end{align*}.

b) Replace \begin{align*}\frac{2}{3}\end{align*} with \begin{align*}\frac{3}{2}\end{align*} and multiply: \begin{align*}\frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3}{3 \cdot 2} = \frac{7}{2}\end{align*}.

c) \begin{align*}\frac{x}{2} \div \frac{1}{4y} = \frac{x}{2} \times \frac{4y}{1} = \frac{4xy}{2} = \frac{2xy}{1} = 2xy\end{align*}

d) \begin{align*}\frac{11}{2x} \div \left ( -\frac{x}{y} \right ) = \frac{11}{2x} \times \left ( -\frac{y}{x} \right ) = -\frac{11y}{2x^2} \end{align*}

**Solve Real-World Problems Using Division**

**Speed, Distance and Time**

An object moving at a certain **speed** will cover a fixed **distance** in a set **time**. The quantities speed, distance and time are related through the equation \begin{align*}\text{Speed} = \frac{\text{Distance}}{\text{Time}}\end{align*}.

#### Example C

*Anne runs a mile and a half in a quarter hour. What is her speed in miles per hour?*

**Solution**

We already have the distance and time in the correct units (miles and hours), so we just need to write them as fractions and plug them into the equation.

\begin{align*}\text{Speed} = \frac{1\frac{1}{2}}{\frac{1}{4}} = \frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times \frac{4}{1} = \frac{3 \cdot 4}{2 \cdot 1} = \frac{12}{2} = 6\end{align*}

Anne runs at 6 miles per hour.

Watch this video for help with the Examples above.

CK-12 Foundation: Dividing Rationals

### Vocabulary

The **multiplicative inverse** of a number is the number which produces 1 when multiplied by the original number. The multiplicative inverse of \begin{align*}x\end{align*} is the reciprocal \begin{align*}\frac{1}{x}\end{align*}. To find the multiplicative inverse of a fraction, simply **invert the fraction**: \begin{align*}\frac{a}{b}\end{align*} inverts to \begin{align*}\frac{b}{a}\end{align*}.

To divide fractions, invert the divisor and multiply: \begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\end{align*}.

An object moving at a certain **speed** will cover a fixed **distance** in a set **time**. The quantities speed, distance and time are related through the equation \begin{align*}\text{Speed} = \frac{\text{Distance}}{\text{Time}}\end{align*}.

### Guided Practice

*Divide the following rational numbers, giving your answer in the* **simplest form**.

a) \begin{align*}\frac{3}{10} \div \frac{7}{5}\end{align*}

b) \begin{align*}\frac{9x}{5} \div \frac{9}{5}\end{align*}

**Solution**

a) Replace \begin{align*}\frac{7}{5}\end{align*} with \begin{align*}\frac{5}{7}\end{align*} and multiply: \begin{align*}\frac{3}{10} \times \frac{5}{7} = \frac{15}{70} =\frac{3}{10}\end{align*}.

b) Replace \begin{align*}\frac{9}{5}\end{align*} with \begin{align*}\frac{5}{9}\end{align*} and multiply: \begin{align*}\frac{9x}{5} \times \frac{5}{9} = \frac{45x}{45} = x\end{align*}.

### Explore More

For 1-5, find the multiplicative inverse of each of the following.

- 100
- \begin{align*}\frac{2}{8}\end{align*}
- \begin{align*}-\frac{19}{21}\end{align*}
- 7
- \begin{align*}-\frac{z^3}{2xy^2}\end{align*}

For 6-10, divide the following rational numbers. Write your answer in the simplest form.

- \begin{align*}\frac{5}{2} \div \frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{2} \div \frac{7}{9}\end{align*}
- \begin{align*}\frac{5}{11} \div \frac{6}{7}\end{align*}
- \begin{align*}\frac{1}{2} \div \frac{1}{2}\end{align*}
- \begin{align*}-\frac{x}{2} \div \frac{5}{7}\end{align*}