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

Explore

## Numbers With a Product of

The applet shows the multiplication of two numbers whose product is What should be the second number?
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 is because their product is
The reciprocal of a number can be found by dividing by

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 The reciprocal of is
Integer numbers The reciprocal of is
Fraction The reciprocal of is
Decimal The reciprocal of is
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.

Here, and are not The division of two fractions can then be considered as a multiplication of two fractions. Consider the following division of two fractions.
The quotient can be found in three steps.
1
Multiply by the Reciprocal of the Divisor
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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.
2
Multiply the Fractions
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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.
3
Simplfy if Possible
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The resulting fraction can be simplified because and have a common factor.
The greatest common factor of the numbers is Simplify the fraction by
The divison expression is equal to
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
Example

## Breaking a Piece of Wood Into Equal Pieces

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

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

External credits: textures.com
a How many 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 feet long.
External credits: textures.com
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.
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
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.
The quotient is This means that Tearrik can produce smaller pieces of feet each from the original foot piece.
b In Part A, dividing into was found to be
This finding is interpreted as Tearrik getting six foot pieces. The remaining piece is of a foot piece. The length of the remaining piece of wood can be found by multiplying these fractions.
The length of the remaining piece of wood is a of one foot.

### Alternative Solution

Use a Diagram

A diagram can be used to model the division of by Divide each foot of the foot-long wood into equal pieces.

External credits: textures.com

Notice that each of the smaller parts represents a of a foot. Then determine how many of the foot-long pieces are contained within the wood.

External credits: textures.com

There are of them. The length of the remaining part is a of a foot. Note that the remaining part is also of 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.

External credits: macrovector

Tearrik runs of the way from the garage to the nearest paint shop.

a If Tearriks runs miles, find the distance between his home and the shop.
b Tearrik buys gallons of blue paint. He pours the paint evenly into cups. How many gallons of paint did he put in each cup?

### Hint

b Divide the number of gallons by

### Solution

a The distance Tearrik ran is given. He ran of the way to the paint shop. This distance is equal to of a mile.
The distance from the house to the paint shop is missing. That distance can be determined by finding of is .
This question can be mathematically expressed as follows.
Now, this multiplication problem can be written as a division problem.
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.
The distance to the paint shop is miles.
b The amount of blue paint Tearrik bought is a given, gallons. He poured that amount evenly into cups. The diagram illustrates the total amount of paint and the unknown amount per cup.
Dividing by gives how many gallons of paint each cup contains.
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.
Now, the steps performed when dividing two fractions can be followed.
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.
This means that Tearrik pours 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

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.
First, the mixed numbers in this expression will be written as improper factions. Recall that a mixed number is equal to
Write mixed number as a fraction
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.
Evaluate
The given quotient is which can be rewritten as
Example

## Finding the Width of the Piece of Wood

Tearrik now has pieces of wood. Each piece has a length of feet. The total area of the pieces is square feet.

External credits: textures.com
What is the width of each piece?

### Hint

Multiply by 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 because there are pieces, will give the length of the rectangle.