3. Divide Fractions
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Chapter 3
3.
Divide Fractions
This lesson offers a comprehensive guide to dividing fractions, a fundamental skill in mathematics. It covers the concept of reciprocals, which are numbers that, when multiplied together, yield the number one. For example, the reciprocal of 9 is 1/9. The lesson also explains how to handle mixed numbers, which are numbers that have both a whole number and a fraction part. Additionally, it touches on the mathematical rule that division by zero is undefined, providing a logical explanation for this. These concepts are essential for students studying mathematics and for anyone who needs to perform complex calculations in daily life, such as cooking or construction.
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| | 12 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Divide Fractions
Slide of 12
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 541-foot piece of wood into 131-foot pieces. Here, the carpenter needs to divide the two fractions to find how many equal-length pieces they would get.
541÷131=?
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 1
The applet shows the multiplication of two numbers whose product is 1. 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 9 is 91 because their product is 1.
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.
9⋅91=1
The reciprocal of a number a can be found by dividing 1 by a. NumberaReciprocala1
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 | a1 | The reciprocal of 2 is 21. |
| Integer numbers a, a=0 | a1 | The reciprocal of -6 is -61. |
| Fraction ba, b=0 | ab | The reciprocal of 23 is 32. |
| Decimal a | a1 | The reciprocal of 0.2 is 0.21. |
Discussion
Dividing Fractions
Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.
ba÷dc=ba⋅cd
2512÷53
The quotient can be found in three steps.
1
Multiply by the Reciprocal of the Divisor
expand_more 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
expand_more 3
Simplfy if Possible
expand_more The resulting fraction can be simplified because 60 and 75 have a common factor.
6075=22⋅3⋅5=3⋅52
The greatest common factor of the numbers is 3⋅5=15. Simplify the fraction by 15.
The divison expression is equal to 54. 
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 5 feet long. He wants the cuts to create equal parts.
External credits: textures.com
a How many 54-foot pieces can he cut from the original piece of wood?
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b There is one piece of wood remaining. What is its length?
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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 54 feet long.
External credits: textures.com
5÷54
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÷54
Rewrite
Rewrite 5 as 15
15÷54
DivFracByFracDiv
ba÷dc=ba⋅cd
15⋅45
MultFrac
Multiply fractions
1⋅45⋅5
Multiply
Multiply
425
425
WriteSum
Write as a sum
424+1
WriteSumFrac
Write as a sum of fractions
424+41
CalcQuot
Calculate quotient
6+41
Rewrite
Rewrite 6+41 as 641
641
b In Part A, dividing 5 into 54 was found to be 641.
5÷54=641
This finding is interpreted as Tearrik getting six 54-foot pieces. The remaining piece is 41 of a 54-foot piece. The length of the remaining piece of wood can be found by multiplying these fractions. 41×54
MultFrac
Multiply fractions
4⋅51⋅4
CancelCommonFac
Cancel out common factors
4⋅51⋅4
SimpQuot
Simplify quotient
51
Alternative Solution
Use a DiagramA diagram can be used to model the division of 5 by 54. Divide each foot of the 5-foot-long wood into 5 equal pieces.
External credits: textures.com
Notice that each of the smaller parts represents a 51 of a foot. Then determine how many of the 54-foot-long pieces are contained within the wood.
External credits: textures.com
There are 6 of them. The length of the remaining part is a 51 of a foot. Note that the remaining part is also 41 of 54. 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 32 of the way from the garage to the nearest paint shop.
a If Tearriks runs 53 miles, find the distance between his home and the shop.
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b Tearrik buys 43 gallons of blue paint. He pours the paint evenly into 6 cups. How many gallons of paint did he put in each cup?
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Hint
a Think about this question; 32 of what number is 53?
b Divide the number of gallons by 6.
Solution
a The distance Tearrik ran is given. He ran 32 of the way to the paint shop. This distance is equal to 53 of a mile.
32 of what number is 53?
This question can be mathematically expressed as follows.
32 × ?=53
Now, this multiplication problem can be written as a division problem.
32 × ?=53⇔53÷32=?
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.
53÷32
DivFracByFracDiv
ba÷dc=ba⋅cd
53⋅23
MultFrac
Multiply fractions
5⋅23⋅3
Multiply
Multiply
109
b The amount of blue paint Tearrik bought is a given, 43 gallons. He poured that amount evenly into 6 cups. The diagram illustrates the total amount of paint and the unknown amount per cup.
43÷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.
43÷16
Now, the steps performed when dividing two fractions can be followed.
43÷16
DivFracByFracDiv
ba÷dc=ba⋅cd
43⋅61
MultFrac
Multiply fractions
4⋅63⋅1
Multiply
Multiply
243
243243=3⋅83⋅1⇕=81
This means that Tearrik pours 81 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.
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.
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.
The given quotient is 23, which can be rewritten as 121.
351÷2152
First, the mixed numbers in this expression will be written as improper factions. Recall that a mixed number acb is equal to ca⋅c+b.
351÷2152
▼
Write mixed number as a fraction
516÷1532
516÷1532
DivFracByFracDiv
ba÷dc=ba⋅cd
516⋅3215
▼
Evaluate
MultFrac
Multiply fractions
5⋅3216⋅15
SplitIntoFactors
Split into factors
5⋅16⋅216⋅5⋅3
CancelCommonFac
Cancel out common factors
5⋅16⋅216⋅5⋅3
SimpQuot
Simplify quotient
23
Example
Finding the Width of the Piece of Wood
Tearrik now has 6 pieces of wood. Each piece has a length of 54 feet. The total area of the pieces is 153 square feet.
External credits: textures.com
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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.
The length of the rectangle is 524 feet. 
Recall the formula for the area of a rectangle. It is the rectangle's width times its length.
Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
The width of the rectangle is 31 of a foot. This also represents the width of each small rectangle. 
External credits: textures.com
Area=Width×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÷Length
The width can then be calculated using the known values.
Width=153÷524
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. 153÷524
▼
Write mixed number as a fraction
58÷524
58÷524
DivFracByFracDiv
ba÷dc=ba⋅cd
58⋅245
MultFrac
Multiply fractions
5⋅248⋅5
SplitIntoFactors
Split into factors
5⋅8⋅38⋅5
CancelCommonFac
Cancel out common factors
5⋅8⋅38⋅5
SimpQuot
Simplify quotient
31
External credits: textures.com
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 54-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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Hint
Think about the following question. 1121 times what number is 165?
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.
Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
The result should be given as a mixed number.
It took 1139 times longer to create the box than it did to paint it.
Construction Time165Painting Time1121
This requires to find a number that is equal to 165 when multiplied by 1121.
1121 times what number is 165?⇓1121 × ?=165
This multiplication problem can be written as a division problem.
1121 × ?=165⇔165÷1121=?
The answer can now be found by dividing the mixed numbers. First convert the mixed numbers into improper fractions.
165÷1121
▼
Write mixed number as a fraction
611÷1213
611÷1213
DivFracByFracDiv
ba÷dc=ba⋅cd
611⋅1312
MultFrac
Multiply fractions
6⋅1311⋅12
SplitIntoFactors
Split into factors
6⋅1311⋅6⋅2
CancelCommonFac
Cancel out common factors
6⋅1311⋅6⋅2
SimpQuot
Simplify quotient
1311⋅2
Multiply
Multiply
1322
1322
▼
Write fraction as a mixed number
WriteSum
Write as a sum
1313+9
WriteSumFrac
Write as a sum of fractions
1313+139
CalcQuot
Calculate quotient
1+139
Rewrite
Rewrite 1+139 as 1139
1139
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, 05. What does this expression equal?
05= ?
This division is considered undefined or not possible. That is because there is no number that equals 5 when multiplied by zero.
?⋅0=5 ×
Remember, division indicates how many times the denominator fitsinto 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 ZeroSuppose dividing by zero was defined. Then, the logic below would be accepted as true. Let a and b be any real numbers.
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.
Divide Fractions
Exercises
Diego finds a diagram that explains why dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
| Steps | Explanation | 32÷54 |
|---|---|---|
| Step I | Rewrite it! | 5432 |
| Step II | Multiply the numerator and denominator by the reciprocal of △. | 54×?32×? |
| Step III | Simplify the denominator. | ◊32×45 |
| Step IV | Simplify the fraction. | ◯×45 |
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We want to complete the given steps. They will demonstrate why we multiply by the reciprocal when we divide fractions. We will consider the following expression. 2/3 ÷ 4/5 The first step is given. It tells us to write the given quotient as a fraction with the dividend in the numerator and the divisor in the denominator. 2/3÷ 4/5=23/45 In the next step, we need to multiply both numerator and denominator by a reciprocal of some fraction. Note that this fraction will be the divisor, 45. Multiplying by the reciprocal of 45 will let us simplify the denominator in the next step. The reciprocal of 45 is 54 because 45* 54=1. Let's write this step down. 23/45 = 23* 54/45* 54 We can simplify the denominator now. We will use the fact that a fraction multiplied by its reciprocal equals 1. 23* 54/45* 54=23* 54/1 The final step is simplifying the fraction. Remember that a fraction is equal to the expression in its numerator when its denominator is 1. Knowing that, we can complete the last step. 23* 54/1= 2/3* 5/4 All these steps led to the fact that 14÷ 38= 14* 83. This shows us why this equation is correct.
| Steps | Explanation | 2/3 ÷ 4/5 |
|---|---|---|
| Step I | Rewrite it! | 23/45 |
| Step II | Multiply the numerator and denominator by the reciprocal of 45. | 23* 54/45* 54 |
| Step III | Simplify the denominator. | 23* 54/1 |
| Step IV | Simplify the fraction. | 2/3 * 5/4 |
We can now match the shapes with the numbers. ccc △ → 4/5 & & □ → 5/4 [1.1em] ◊ → 1 & & ◯ → 2/3
Let's find the result of the division. Remember that the product of two fractions is equal to the product of the numerators over the product of the denominators.
The result of the division of 23 by 45 is 56.
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Divide Fractions
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