2.6: Division of Rational Numbers

Difficulty Level: At Grade Created by: CK-12
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What if you had two numbers like 65\begin{align*}\frac{6}{5}\end{align*} and 25\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.

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, 4+(3)\begin{align*}4 + (-3)\end{align*} is the same as 43\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 x\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 x\begin{align*}x\end{align*} is written as 1x\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 ab\begin{align*}\frac{a}{b}\end{align*} is ba\begin{align*}\frac{b}{a}\end{align*}, as long as a0\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) 37\begin{align*}\frac{3}{7}\end{align*}

b) 49\begin{align*}\frac{4}{9}\end{align*}

c) 312\begin{align*}3\frac{1}{2}\end{align*}

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

e) 111\begin{align*}\frac{1}{11}\end{align*}

Solution

a) When we invert the fraction 37\begin{align*}\frac{3}{7}\end{align*}, we get 73\begin{align*}\frac{7}{3}\end{align*}. Notice that if we multiply 3773\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 11\begin{align*}\frac{1}{1}\end{align*}, or just 1.

b) Similarly, the inverse of 49\begin{align*}\frac{4}{9}\end{align*} is 94\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 312\begin{align*}3\frac{1}{2}\end{align*} we first need to convert it to an improper fraction. Three wholes is six halves, so 312=62+12=72\begin{align*}3\frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\end{align*}. That means the inverse is 27\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 xy\begin{align*}-\frac{x}{y}\end{align*} is yx\begin{align*}-\frac{y}{x}\end{align*}.

e) The multiplicative inverse of 111\begin{align*}\frac{1}{11}\end{align*} is 111\begin{align*}\frac{11}{1}\end{align*}, or simply 11.

Look again at the last example. When we took the multiplicative inverse of 111\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:

ab÷cd=ab×dc.\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) 12÷14\begin{align*}\frac{1}{2} \div \frac{1}{4}\end{align*}

b) 73÷23\begin{align*}\frac{7}{3} \div \frac{2}{3}\end{align*}

c) x2÷14y\begin{align*}\frac{x}{2} \div \frac{1}{4y}\end{align*}

d) 112x÷(xy)\begin{align*}\frac{11}{2x} \div \left ( -\frac{x}{y} \right )\end{align*}

Solution

a) Replace 14\begin{align*}\frac{1}{4}\end{align*} with 41\begin{align*}\frac{4}{1}\end{align*} and multiply: 12×41=42=2\begin{align*}\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2\end{align*}.

b) Replace 23\begin{align*}\frac{2}{3}\end{align*} with 32\begin{align*}\frac{3}{2}\end{align*} and multiply: 73×32=7332=72\begin{align*}\frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3}{3 \cdot 2} = \frac{7}{2}\end{align*}.

c) x2÷14y=x2×4y1=4xy2=2xy1=2xy\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) 112x÷(xy)=112x×(yx)=11y2x2\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.

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.

1. 100
2. \begin{align*}\frac{2}{8}\end{align*}
3. \begin{align*}-\frac{19}{21}\end{align*}
4. 7
5. \begin{align*}-\frac{z^3}{2xy^2}\end{align*}

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

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

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Dividend In a division problem, the dividend is the number or expression that is being divided.
divisor In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression $152 \div 6$, 6 is the divisor and 152 is the dividend.
identity element An identity element is a value which, when combined with an operation on another number, leaves that other number unchanged. The identity element for addition is zero, the identity element for multiplication is one.
Multiplicative Inverse The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).
Quotient The quotient is the result after two amounts have been divided.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
reciprocal The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.

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