Learn Dividing Fractions with Mixed Numbers and Reciprocal Concepts

Division is a process of sharing a quantity equally. In everyday life, quantities can be expressed as fractions instead of whole numbers. For example, a carpenter may need to cut a

5 \frac{1}{4} -

foot piece of wood into

1 \frac{1}{3} -

foot pieces. Here, the carpenter needs to divide the two fractions to find how many equal-length pieces they would get.

5 \frac{1}{4} \div 1 \frac{1}{3} = ?

In this lesson, similar problems will be solved to explain how to divide fractions. Additionaly, dividing fractions will be associated with multiplying fractions. Then they will be modeled by using visual fraction models.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Rational Numbers
Fraction
Mixed Numbers
Multiplying Fractions

Explore

Numbers With a Product of $1$

The applet shows the multiplication of two numbers whose product is

1 .

What should be the second number?

$product of random fractions$

Describe the relationship between the numbers.

Discussion

Reciprocals

Two numbers are reciprocals, or multiplicative inverses, of each other when their product is the multiplicative identity. For example, the reciprocal of

9

\frac{1}{9}

because their product is

1 .

9 \cdot \frac{1}{9} = 1

The reciprocal of a number

a

can be found by dividing

1

a .

Number a Reciprocal \frac{1}{a}

Shortcuts exist to find the reciprocals of specific types of numbers such as natural numbers, integer numbers, fractions, and decimals.

Type	Reciprocal	Example
Natural number $a$	$\frac{1}{a}$	The reciprocal of $2$ is $\frac{1}{2} .$
Integer numbers $a,$ $a \neq = 0$	$\frac{1}{a}$	The reciprocal of $- 6$ is $- \frac{1}{6} .$
Fraction $\frac{a}{b},$ $b \neq = 0$	$\frac{b}{a}$	The reciprocal of $\frac{3}{2}$ is $\frac{2}{3} .$
Decimal $a$	$\frac{1}{a}$	The reciprocal of $0.2$ is $\frac{1}{0 . 2} .$

Finding the reciprocal of a mixed number is like finding the reciprocal of a fraction. However, the mixed number must first be written as an improper fraction.

Discussion

Dividing Fractions

Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

Here,

b,

c,

and

d

are not

0 .

The division of two fractions can then be considered as a multiplication of two fractions. Consider the following division of two fractions.

\frac{1 2}{2 5} \div \frac{3}{5}

The quotient can be found in three steps.

Multiply by the Reciprocal of the Divisor

The division of fractions begins by keeping the first fraction as is. Then change the division sign with the multiplication sign and write the reciprocal of the second fraction. Note that the reciprocal of a fraction is found by switching the numerator and denominator of the fraction.

Multiply the Fractions

The result is now a multiplication of two fractions. The product of the fractions is equal to the product of the numerators divided by the product of the denominators.

\frac{1 2}{2 5} \cdot \frac{5}{3}

MultFrac

Multiply fractions

\frac{1 2 \cdot 5}{2 5 \cdot 3}

Multiply

\frac{6 0}{7 5}

Simplfy if Possible

The resulting fraction can be simplified because

60

and

75

have a common factor.

6075 = 2^{2} \cdot 3 \cdot 5 = 3 \cdot 5^{2}

The greatest common factor of the numbers is

3 \cdot 5 = 15 .

Simplify the fraction by

15 .

\frac{6 0}{7 5}

ReduceFrac

$\frac{a}{b} = \frac{a / 1 5}{b / 1 5}$

\frac{6 0 / 1 5}{7 5 / 1 5}

SimpQuot

Simplify quotient

\frac{4}{5}

The divison expression is equal to

\frac{4}{5} .

The same steps above are also used when dividing a fraction by a whole number. This is because every whole number can be thought of as a fraction with a denominator of

1 .

Example

Breaking a Piece of Wood Into Equal Pieces

Tearrik was gifted an heirloom by his grandfather. It is a handmade kimono.

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Tearrik wants to make a box to hold this beautiful kimono. He plans to cut a piece of wood that is $5$ feet long. He wants the cuts to create equal parts.

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a How many

\frac{4}{5} -

foot pieces can he cut from the original piece of wood?

b There is one piece of wood remaining. What is its length?

Hint

a Divide the length of the wood by the length of a smaller piece.

b Use the answer from Part A.

Solution

a Tearrik plans to cut the piece of wood into parts that are each

\frac{4}{5}

feet long.

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He wants to know the number of smaller pieces of wood he can produce from the larger piece of wood. Divide the length of the larger piece of wood by the length desired for the smaller pieces to determine that number.

5 \div \frac{4}{5}

Dividing a whole number by a fraction is the same as multiplying that whole number by the reciprocal of the fraction. Recall the fact that all whole numbers are fractions whose denominator is

1 .

5 \div \frac{4}{5}

Rewrite

Rewrite $5$ as $\frac{5}{1}$

\frac{5}{1} \div \frac{4}{5}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{5}{1} \cdot \frac{5}{4}

MultFrac

Multiply fractions

\frac{5 \cdot 5}{1 \cdot 4}

Multiply

\frac{2 5}{4}

The fraction solved for is an improper fraction. Write it as a mixed number for making sense of how many pieces of wood there are.

\frac{2 5}{4}

WriteSum

Write as a sum

\frac{2 4 + 1}{4}

WriteSumFrac

Write as a sum of fractions

\frac{2 4}{4} + \frac{1}{4}

CalcQuot

Calculate quotient

6 + \frac{1}{4}

Rewrite

Rewrite $6 + \frac{1}{4}$ as $6 \frac{1}{4}$

6 \frac{1}{4}

The quotient is

6 \frac{1}{4} .

This means that Tearrik can produce

6

smaller pieces of

\frac{4}{5}

feet each from the original

5 -

foot piece.

b In Part A, dividing

5

into

\frac{4}{5}

was found to be

6 \frac{1}{4} .

5 \div \frac{4}{5} = 6 \frac{1}{4}

This finding is interpreted as Tearrik getting six

\frac{4}{5} -

foot pieces. The remaining piece is

\frac{1}{4}

of a

\frac{4}{5} -

foot piece. The length of the remaining piece of wood can be found by multiplying these fractions.

\frac{1}{4} \times \frac{4}{5}

MultFrac

Multiply fractions

\frac{1 \cdot 4}{4 \cdot 5}

CancelCommonFac

Cancel out common factors

\frac{1 \cdot 4}{4 \cdot 5}

SimpQuot

Simplify quotient

\frac{1}{5}

The length of the remaining piece of wood is a

\frac{1}{5}

of one foot.

Alternative Solution

Use a Diagram

A diagram can be used to model the division of $5$ by $\frac{4}{5} .$ Divide each foot of the $5 -$ foot-long wood into $5$ equal pieces.

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Notice that each of the smaller parts represents a $\frac{1}{5}$ of a foot. Then determine how many of the $\frac{4}{5} -$ foot-long pieces are contained within the wood.

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There are $6$ of them. The length of the remaining part is a $\frac{1}{5}$ of a foot. Note that the remaining part is also $\frac{1}{4}$ of $\frac{4}{5} .$ This confirms that the result found algebraically is correct.

Example

Distance Between House and Paint Shop

Tearrik is excited about making the box. A problem arises, however. He realizes a bit of paint would look cool but he does not have any in his home. Tearrik is full of energy and starts to run to the nearest paint shop.

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Tearrik runs $\frac{2}{3}$ of the way from the garage to the nearest paint shop.

a If Tearriks runs

\frac{3}{5}

miles, find the distance between his home and the shop.

b Tearrik buys

\frac{3}{4}

gallons of blue paint. He pours the paint evenly into

6

cups. How many gallons of paint did he put in each cup?

Hint

a Think about this question;

\frac{2}{3}

of what number is

\frac{3}{5} ?

b Divide the number of gallons by

6 .

Solution

a The distance Tearrik ran is given. He ran

\frac{2}{3}

of the way to the paint shop. This distance is equal to

\frac{3}{5}

of a mile.

The distance from the house to the paint shop is missing. That distance can be determined by finding

t w o - t h i r d s

w h a t

n u m b e r

t h r e e

f i f t h s

\frac{2}{3} of what number is \frac{3}{5} ?

This question can be mathematically expressed as follows.

\frac{2}{3} \times ? = \frac{3}{5}

Now, this multiplication problem can be written as a division problem.

\frac{2}{3} \times ? = \frac{3}{5} \Leftrightarrow \frac{3}{5} \div \frac{2}{3} = ?

The quotient of this division represents the distance to the paint shop. Consider that dividing a fraction by a fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.

\frac{3}{5} \div \frac{2}{3}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{3}{5} \cdot \frac{3}{2}

MultFrac

Multiply fractions

\frac{3 \cdot 3}{5 \cdot 2}

Multiply

\frac{9}{1 0}

The distance to the paint shop is

\frac{9}{1 0}

miles.

b The amount of blue paint Tearrik bought is a given,

\frac{3}{4}

gallons. He poured that amount evenly into

6

cups. The diagram illustrates the total amount of paint and the unknown amount per cup.

Dividing

\frac{3}{4}

6

gives how many gallons of paint each cup contains.

\frac{3}{4} \div 6

This is a division of a fraction by a whole number. That means the whole number should be written as a fraction to calculate the quotient.

\frac{3}{4} \div \frac{6}{1}

Now, the steps performed when dividing two fractions can be followed.

\frac{3}{4} \div \frac{6}{1}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{3}{4} \cdot \frac{1}{6}

MultFrac

Multiply fractions

\frac{3 \cdot 1}{4 \cdot 6}

Multiply

\frac{3}{2 4}

The number three is a common factor between the denominator and numerator of the obtained fraction. This fact can be used to simplify the fraction.

\frac{3}{2 4} \frac{3}{2 4} = \frac{3 \cdot 1}{3 \cdot 8} ⇕ = \frac{1}{8}

This means that Tearrik pours

\frac{1}{8}

gallon into each cup.

Pop Quiz

Dividing Fractions

The applet shows random divisions involving fractions. Find the corresponding quotient of the given division. Simplify the answer to its lowest terms. If the answer is a whole number, write it as a fraction with a denominator of $1 .$

$quotient of random fractions$

Discussion

Dividing Mixed Numbers

A division of fractions involving mixed numbers requires first writing the mixed numbers as improper fractions. Next, the same steps performed when dividing proper fractions can be followed. For example, consider the division of the following mixed numbers.

3 \frac{1}{5} \div 2 \frac{2}{1 5}

First, the mixed numbers in this expression will be written as improper factions. Recall that a mixed number

a \frac{b}{c}

is equal to

\frac{a \cdot c + b}{c} .

3 \frac{1}{5} \div 2 \frac{2}{1 5}

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{3 \cdot 5 + 1}{5} \div \frac{2 \cdot 1 5 + 2}{1 5}

Multiply

\frac{1 5 + 1}{5} \div \frac{3 0 + 2}{1 5}

AddTerms

Add terms

\frac{1 6}{5} \div \frac{3 2}{1 5}

Now the usual steps can be used to find the quotient of these two fractions. The division sign is changed to a multiplication sign. Then the second fraction is replaced with its reciprocal.

\frac{1 6}{5} \div \frac{3 2}{1 5}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{1 6}{5} \cdot \frac{1 5}{3 2}

▼

Evaluate

MultFrac

Multiply fractions

\frac{1 6 \cdot 1 5}{5 \cdot 3 2}

SplitIntoFactors

Split into factors

\frac{1 6 \cdot 5 \cdot 3}{5 \cdot 1 6 \cdot 2}

CancelCommonFac

Cancel out common factors

\frac{1 6 \cdot 5 \cdot 3}{5 \cdot 1 6 \cdot 2}

SimpQuot

Simplify quotient

\frac{3}{2}

The given quotient is

\frac{3}{2},

which can be rewritten as

1 \frac{1}{2} .

Example

Finding the Width of the Piece of Wood

Tearrik now has $6$ pieces of wood. Each piece has a length of $\frac{4}{5}$ feet. The total area of the pieces is $1 \frac{3}{5}$ square feet.

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What is the width of each piece?

Hint

Multiply $\frac{4}{5}$ by $6$ to find the total length.

The formula for the area of a rectangle is the width times the length.

How should the formula be written so that the width can be found?

Solution

Start by finding the length of the greater rectangle. The result of multiplying the length of a small wood piece by

6,

because there are

6

pieces, will give the length of the rectangle.

\frac{4}{5} \times 6

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{4 \cdot 6}{5}

Multiply

\frac{2 4}{5}

The length of the rectangle is

\frac{2 4}{5}

feet.

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Recall the formula for the area of a rectangle. It is the rectangle's width times its length.

Area = Width \times Length

Here, the area and length of the rectangle are already known. Its width is what needs to be found. At this point of the process, it is helpful to rearrange the formula to isolate the width to one side.

Width = Area \div Length

The width can then be calculated using the known values.

Width = 1 \frac{3}{5} \div \frac{2 4}{5}

The expression on the right-hand side is a division of a mixed number by a fraction. The mixed number should be converted into an improper fraction.

1 \frac{3}{5} \div \frac{2 4}{5}

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{1 \cdot 5 + 3}{5} \div \frac{2 4}{5}

MultByOne

$a \cdot 1 = a$

\frac{5 + 3}{5} \div \frac{2 4}{5}

AddFrac

Add fractions

\frac{8}{5} \div \frac{2 4}{5}

Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.

\frac{8}{5} \div \frac{2 4}{5}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{8}{5} \cdot \frac{5}{2 4}

MultFrac

Multiply fractions

\frac{8 \cdot 5}{5 \cdot 2 4}

SplitIntoFactors

Split into factors

\frac{8 \cdot 5}{5 \cdot 8 \cdot 3}

CancelCommonFac

Cancel out common factors

\frac{8 \cdot 5}{5 \cdot 8 \cdot 3}

SimpQuot

Simplify quotient

\frac{1}{3}

The width of the rectangle is

\frac{1}{3}

of a foot. This also represents the width of each small rectangle.

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Example

Comparing the Time Spent on Completing the Box

Tearrik realizes that he cannot create a box as he imagined. He asks his mom for help. Together, they cut two of the $\frac{4}{5} -$ foot-long pieces of wood into squares. They manage to form a box by putting the pieces together. After that, they painted the box to match the kimono.

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They spent

1 \frac{5}{6}

hours constructing the box, and they spent

1 \frac{1}{1 2}

hours painting it. How many times longer did it take to construct the box than it did to paint it? Write the answer as a mixed number.

Hint

Think about the following question. $1 \frac{1}{1 2}$ times what number is $1 \frac{5}{6} ?$

Express the question mathematically. Can it be written as a division problem?

Solution

The question asks to compare the time it took to create the box versus the time it took to paint it.

Construction Time 1 \frac{5}{6} Painting Time 1 \frac{1}{1 2}

This requires to find a number that is equal to

1 \frac{5}{6}

when multiplied by

1 \frac{1}{1 2} .

1 \frac{1}{1 2} times what number is 1 \frac{5}{6} ? ⇓ 1 \frac{1}{1 2} \times ? = 1 \frac{5}{6}

This multiplication problem can be written as a division problem.

1 \frac{1}{1 2} \times ? = 1 \frac{5}{6} \Leftrightarrow 1 \frac{5}{6} \div 1 \frac{1}{1 2} = ?

The answer can now be found by dividing the mixed numbers. First convert the mixed numbers into improper fractions.

1 \frac{5}{6} \div 1 \frac{1}{1 2}

▼

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{1 \cdot 6 + 5}{6} \div \frac{1 \cdot 1 2 + 1}{1 2}

MultByOne

$a \cdot 1 = a$

\frac{6 + 5}{6} \div \frac{1 2 + 1}{1 2}

AddTerms

Add terms

\frac{1 1}{6} \div \frac{1 3}{1 2}

Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.

\frac{1 1}{6} \div \frac{1 3}{1 2}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{1 1}{6} \cdot \frac{1 2}{1 3}

MultFrac

Multiply fractions

\frac{1 1 \cdot 1 2}{6 \cdot 1 3}

SplitIntoFactors

Split into factors

\frac{1 1 \cdot 6 \cdot 2}{6 \cdot 1 3}

CancelCommonFac

Cancel out common factors

\frac{1 1 \cdot 6 \cdot 2}{6 \cdot 1 3}

SimpQuot

Simplify quotient

\frac{1 1 \cdot 2}{1 3}

Multiply

\frac{2 2}{1 3}

The result should be given as a mixed number.

\frac{2 2}{1 3}

▼

Write fraction as a mixed number

WriteSum

Write as a sum

\frac{1 3 + 9}{1 3}

WriteSumFrac

Write as a sum of fractions

\frac{1 3}{1 3} + \frac{9}{1 3}

CalcQuot

Calculate quotient

1 + \frac{9}{1 3}

Rewrite

Rewrite $1 + \frac{9}{1 3}$ as $1 \frac{9}{1 3}$

1 \frac{9}{1 3}

It took

1 \frac{9}{1 3}

times longer to create the box than it did to paint it.

Pop Quiz

Dividing Mixed Numbers

The applet shows a division expression that involves at least one mixed number. Find the indicated quotient. Simplify the answer. If the answer is a whole number, write it as a fraction with a denominator of $1 .$

Closure

Is It Possible to Divide by Zero?

Another important characteristic about division should be discussed before ending this lesson. Think of division expressions where the divisor is zero. For example,

\frac{5}{0} .

What does this expression equal?

\frac{5}{0} = ?

This division is considered undefined or not possible. That is because there is no number that equals

5

when multiplied by zero.

? \cdot 0 = 5 \times

Remember, division indicates how many times the denominator fits into the numerator. In this example, no matter how many zeros are tried to fit in

5,

the number

5

will never be reached.

Dividing by $0$ then becomes impossible.

Extra

The Consequences of Dividing by Zero

Suppose dividing by zero was defined. Then, the logic below would be accepted as true. Let

a

and

b

be any real numbers.

a = b

SubEqn

$LHS - b = RHS - b$

a - b = 0

DivEqn

$LHS / (a - b) = RHS / (a - b)$

\frac{a - b}{a - b} = \frac{0}{a - b}

SimpQuot

Simplify quotient

1 = 0

This is a contradiction because

1

is not equal to

0 .

This contradiction resulted from supposing that dividing by zero is defined. As a result, the statement is false. Dividing by zero is undefined.

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Catch-Up and Review

Hint

Solution

Alternative Solution

Hint

Solution

Hint

Solution

Hint

Solution

Extra