What if you had two numbers like \begin{align*}\frac{6}{5}\end{align*} and \begin{align*}\frac{2}{5}\end{align*}? How could you divide the first one by the second one so that your answer was in simplest form? After completing this Concept, you'll be able to solve division problems like this one.

### 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*} is the same as \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*} is also called the **multiplicative inverse**. Any number times its own multiplicative inverse equals one, and the multiplicative inverse of \begin{align*}x\end{align*} is written as \begin{align*}\frac{1}{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*} is \begin{align*}\frac{b}{a}\end{align*}, as long as \begin{align*}a \neq 0\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*}, we get \begin{align*}\frac{7}{3}\end{align*}. Notice that if we multiply \begin{align*}\frac{3}{7} \cdot \frac{7}{3}\end{align*}, the 3’s and the 7’s both cancel out and we end up with \begin{align*}\frac{1}{1}\end{align*}, or just 1.

b) Similarly, the inverse of \begin{align*}\frac{4}{9}\end{align*} is \begin{align*}\frac{9}{4}\end{align*}; if we multiply those two fractions together, the 4’s and the 9’s cancel out and we’re left with 1. That’s why the rule “invert the fraction to find the multiplicative inverse” works: the numerator and the denominator always end up canceling out, leaving 1.

c) To find the multiplicative inverse of \begin{align*}3\frac{1}{2}\end{align*} we first need to convert it to an improper fraction. Three wholes is six halves, so \begin{align*}3\frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\end{align*}. That means the inverse is \begin{align*}\frac{2}{7}\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*} is \begin{align*}-\frac{y}{x}\end{align*}.

e) The multiplicative inverse of \begin{align*}\frac{1}{11}\end{align*} is \begin{align*}\frac{11}{1}\end{align*}, or simply 11.

Look again at the last example. When we took the multiplicative inverse of \begin{align*}\frac{1}{11}\end{align*} we got a whole number, 11. That’s because we can treat that whole number like a fraction with a denominator of 1. Any number, even a non-rational one, can be treated this way, so we can always find a number’s multiplicative inverse using the same method.

**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*}.

### Practice

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*}