5.8: Dividing Fractions

Suppose that you have four pizzas and each of the pizzas has been sliced into eight equal slices. Therefore, each slice of pizza represents \(\frac{1}{8}\) of a whole pizza.

Figure \(\PageIndex{1}\): One slice of pizza is \(\frac{1}{8}\) of one whole pizza.

Now for the question: How many one-eighths are there in four? This is a division statement. To find how many one-eighths there are in 4, divide 4 by \(\frac{1}{8}\). That is,

Number of one-eighths in four = 4 ÷ \(\frac{1}{8}\).

On the other hand, to find the number of one-eights in four, Figure \(\PageIndex{1}\) clearly demonstrates that this is equivalent to asking how many slices of pizza are there in four pizzas. Since there are 8 slices per pizza and four pizzas,

Number of pizza slices = 4 · 8.

The conclusion is the fact that 4 ÷ \(\frac{1}{8}\) is equivalent to 4 · 8. That is,

\[\begin{align*} 4 ÷ \frac{1}{8} &= 4 \cdot 8 \\[4pt] &= 32. \end{align*}\]

Therefore, we conclude that there are 32 one-eighths in 4.

Reciprocals

Multiplicative Inverse Property

Let \(\dfrac{a}{b}\) be any fraction. The number \(\dfrac{b}{a}\) is called the multiplicative inverse or reciprocal of \(\dfrac{a}{b}\). The product of reciprocals is 1.

\[ \frac{a}{b} \cdot \frac{b}{a} = 1\nonumber \]

Note: To find the multiplicative inverse (reciprocal) of a number, simply invert the number (turn it upside down).

For example, the number \(\frac{1}{8}\) is the multiplicative inverse (reciprocal) of 8 because

\[ 8 \cdot \frac{1}{8} = 1.\nonumber \]

Note that 8 can be thought of as \(\frac{8}{1}\). Invert this number (turn it upside down) to find its multiplicative inverse (reciprocal) \(\frac{1}{8}\).

Example \(\PageIndex{1}\)

Find the reciprocals of: (a) \(\dfrac{2}{3}\) and (b) 12.

Solution

1. Because \( \frac{2}{3} \cdot \frac{3}{2} = 1\), the reciprocal of \(\dfrac{2}{3}\) is \(\dfrac{3}{2}\).
2. Because \( 12 \cdot \left(  \frac{1}{12} \right) = 1\), the reciprocal of 12 is \(\dfrac{1}{12}\). Again, note that we simply inverted the number 12 (understood to equal \(\frac{12}{1}\)) to get its reciprocal \(\frac{1}{12}\).

Try It \(\PageIndex{1}\)

Find the reciprocals of: (a) \(\dfrac{3}{7}\) and (b) 15

Answer

(a) \(\dfrac{7}{3}\), (b) \(\dfrac{1}{15}\)

Division

Recall that we computed the number of one-eighths in four by doing this calculation:

\[ \begin{align*} 4 ÷ \frac{1}{8} &= 4 · 8 \\[4pt] &= 32.\end{align*}\]

Note how we inverted the divisor (second number), then changed the division to multiplication. This motivates the following definition of division.

Division Definition

If \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are any fractions, then

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}.\nonumber \]

That is, we invert the divisor (second number) and change the division to multiplication. Note: We like to use the phrase “invert and multiply” or "multiply by the reciprocal" as a memory aid for this definition.

Example \(\PageIndex{2}\)

Divide \(\dfrac{1}{2}\) by \(\dfrac{3}{5}\).

Solution

To divide \(\dfrac{1}{2}\) by \(\dfrac{3}{5}\), invert the divisor (second number), then multiply.

\[ \begin{align*} \frac{1}{2} \div \frac{3}{5} &= \frac{1}{2} \cdot \frac{5}{3} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{5}{6} ~&& \textcolor{red}{ \text{ Multiply.}} \end{align*}\]

Try It \(\PageIndex{2}\)

Divide: \( \dfrac{2}{3} \div \dfrac{10}{3} \)

Answer

\(\dfrac{1}{5}\)

Example \(\PageIndex{3}\)

Simplify the following expressions: (a) 3 ÷ \(\dfrac{2}{3}\) and (b) \(\dfrac{4}{5}\) ÷ 5.

Solution

In each case, invert the divisor (second number), then multiply.

1. Note that 3 is understood to be \(\dfrac{3}{1}\).

\[ \begin{align*} 3 \div \frac{2}{3} &= \frac{3}{1} \cdot \frac{3}{2} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{9}{2} ~ && \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \end{align*} \]

2. Note that 5 is understood to be \(\dfrac{5}{1}\).

\[ \begin{align*} \frac{4}{5} \div 5 &= \frac{4}{5} \cdot \frac{1}{5} ~ && \textcolor{red}{ \text{ Invert the divisor (second number).}} \\[4pt] &= \frac{4}{25} ~ && \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \end{align*}\]

Try It \(\PageIndex{3}\)

Divide: \( \dfrac{15}{7} \div 5 \)

Answer

\(\dfrac{3}{7}\)

After inverting, you may need to factor and cancel.

Example \(\PageIndex{4}\)

Divide \(\dfrac{6}{35}\) by \(\dfrac{33}{55}\).

Solution

Invert, multiply, factor, and cancel common factors.

\[ \begin{aligned}  \frac{6}{35} \div \frac{33}{55} =  \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ =  \frac{6 \cdot 55}{35 \cdot 33} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ =  \frac{(2 \cdot 3) \cdot (5 \cdot 11)}{(5 \cdot 7) \cdot (3 \cdot 11)} ~ & \textcolor{red}{ \text{ Factor numerators and denominators.}} \\ =  \frac{2 \cdot \cancel{3} \cdot \cancel{5} \cdot \cancel{11}}{ \cancel{5} \cdot 7 \cdot \cancel{3} \cdot \cancel{11}} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ =  \frac{2}{7} ~ & \textcolor{red}{ \text{ Remaining factors.}} \end{aligned}\nonumber \]

Try It \(\PageIndex{4}\)

Divide: \( \dfrac{6}{15} \div \left( \dfrac{42}{35} \right) \)

Answer

\(\dfrac{1}{3}\)

Of course, you can also choose to factor numerators and denominators in place, then cancel common factors.