2.3: 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 1/8 of a whole pizza.

Figure \(\PageIndex{1}\): One slice of pizza is 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 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 ÷ (1/8) is equivalent to 4 · 8. That is,

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

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

Reciprocals

The number 1 is still the multiplicative identity for fractions.

Multiplicative Identity Property

Let a/b be any fraction. Then,

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

The number 1 is called the multiplicative identity because the identical number is returned when you multiply by 1.

Next, if we invert 3/4, that is, if we turn 3/4 upside down, we get 4/3. Note what happens when we multiply 3/4 by 4/3.

The number 4/3 is called the multiplicative inverse or reciprocal of 3/4. The product of reciprocals is always 1.

Multiplicative Inverse Property

Let a/b be any fraction. The number b/a is called the multiplicative inverse or reciprocal of 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 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 8/1. Invert this number (turn it upside down) to find its multiplicative inverse (reciprocal) 1/8.

Example \(\PageIndex{1}\)

Find the multiplicative inverses (reciprocals) of: (a) 2/3, (b) −3/5, and (c) −12.

Solution

a) Because

\[ \frac{2}{3} \cdot \frac{3}{2} = 1,\nonumber \]

the multiplicative inverse (reciprocal) of 2/3 is 3/2.

b) Because

\[ - \frac{3}{5} \cdot \left( - \frac{5}{3} \right) = 1,\nonumber \]

the multiplicative inverse (reciprocal) of −3/5 is −5/3. Again, note that we simply inverted the number −3/5 to get its reciprocal −5/3.

c) Because

\[ -12 \cdot \left( - \frac{1}{12} \right) = 1, \nonumber \]

the multiplicative inverse (reciprocal) of −12 is −1/12. Again, note that we simply inverted the number −12 (understood to equal −12/1) to get its reciprocal −1/12.

Exercise \(\PageIndex{1}\)

Find the reciprocals of: (a) −3/7 and (b) 15

Answer

(a) −7/3, (b) 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 a/b and 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” as a memory aid for this definition.

Example \(\PageIndex{2}\)

Divide 1/2 by 3/5.

Solution

To divide 1/2 by 3/5, invert the divisor (second number), then multiply.

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

Exercise \(\PageIndex{2}\)

Divide:

\[ \frac{2}{3} \div \frac{10}{3}\nonumber \]

Answer

1/5

Example \(\PageIndex{3}\)

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

Solution

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

a) Note that 3 is understood to be 3/1.

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

b) Note that 5 is understood to be 5/1.

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

Exercise \(\PageIndex{3}\)

Divide:

\[ \frac{15}{7} \div 5\nonumber \]

Answer

\(\frac{3}{7}\)

After inverting, you may need to factor and cancel, as we learned to do in Section 4.2.

Example \(\PageIndex{4}\)

Divide −6/35 by 33/55.

Solution

Invert, multiply, factor, and cancel common factors.

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

Note that unlike signs produce a negative answer.

Exercise \(\PageIndex{4}\)

Divide:

\[ \frac{6}{15} \div \left( - \frac{42}{35} \right)\nonumber \]

Answer

-1/3

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

 

Exercises

Find the reciprocal of the given number.

1. 16/5

3. −17

5. −15/16

7. 30

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Determine which property of multiplication is depicted by the given identity.

19. \( \frac{−19}{12} \cdot 1 = \frac{−19}{12}\)

25. \(− \frac{4}{1} \cdot \left( − \frac{1}{4} \right) = 1\)

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Divide the fractions and simplify your result.

33. \(\frac{8}{23} \div \frac{−6}{11}\)

34. \(\frac{−10}{21} \div \frac{−6}{5}\)

45. \(\frac{5}{9} \div \frac{4}{3}\)

56. \(\frac{−2}{7} \div \frac{−8}{7}\)

57. \(\frac{20}{17} \div 5\)

66. \(−6 \div \frac{−21}{8}\)

67. \(\frac{3}{4} \div (−9)\)

 

 

 

 

 

Answers

1. \(\frac{5}{16}\)

3. \(− \frac{1}{17}\)

5. \(− \frac{16}{15}\)

7. \(\frac{1}{30}\)

19. multiplicative identity property

25. multiplicative inverse property

33. \(− \frac{44}{69}\)

34. \(\frac{25}{63}\)

45. \( \frac{5}{12}\)

56. \(\frac{1}{4}\)

57. \(\frac{4}{17}\)

66. \(\frac{16}{7}\)

67. \(− \frac{1}{12}\)