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 19. 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.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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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 5 14-foot piece of wood into 1 13-foot pieces. Here, the carpenter needs to divide the two fractions to find how many equal-length pieces they would get.
5 14 ÷ 1 13 = ? 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.
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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?
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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 19 because their product is 1. 9*1/9=1 The reciprocal of a number a can be found by dividing 1 by a.
Number &Reciprocal a &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 | 1/a | The reciprocal of 2 is 12. |
| Integer numbers a, a≠0 | 1/a | The reciprocal of -6 is - 16. |
| Fraction a/b, b≠0 | b/a | The reciprocal of 32 is 23. |
| Decimal a | 1/a | The reciprocal of 0.2 is 10.2. |
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Discussion
Dividing Fractions
Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.
a/b ÷ c/d = a/b * 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. 12/25 ÷ 3/5 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. 60 & = 2^2 * 3 * 5 75 & = 3 * 5^2
The greatest common factor of the numbers is 3 *5 =15. Simplify the fraction by 15.
The divison expression is equal to 45.
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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 45-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 45 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. 5 ÷ 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 ÷ 4/5
Rewrite
Rewrite 5 as 5/1
5/1 ÷ 4/5
DivFracByFracDiv
a/b÷c/d=a/b*d/c
5/1 * 5/4
MultFrac
Multiply fractions
5 * 5/1 * 4
Multiply
Multiply
25/4
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.
25/4
WriteSum
Write as a sum
24+1/4
WriteSumFrac
Write as a sum of fractions
24/4 + 1/4
CalcQuot
Calculate quotient
6 + 1/4
Rewrite
Rewrite 6+1/4 as 6 14
6 14
The quotient is 6 14. This means that Tearrik can produce 6 smaller pieces of 45 feet each from the original 5-foot piece.
b
In Part A, dividing 5 into 45 was found to be 6 14.
5 ÷ 4/5 = 6 14 This finding is interpreted as Tearrik getting six 45-foot pieces. The remaining piece is 14 of a 45-foot piece. The length of the remaining piece of wood can be found by multiplying these fractions.
1/4 * 4/5
MultFrac
Multiply fractions
1* 4/4 * 5
CancelCommonFac
Cancel out common factors
1* 4/4 * 5
SimpQuot
Simplify quotient
1/5
The length of the remaining piece of wood is a 15 of one foot.
Alternative Solution
Use a DiagramA diagram can be used to model the division of 5 by 45. 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 15 of a foot. Then determine how many of the 45-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 15 of a foot. Note that the remaining part is also 14 of 45. This confirms that the result found algebraically is correct.
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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 23 of the way from the garage to the nearest paint shop.
a
If Tearriks runs 35 miles, find the distance between his home and the shop.
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b
Tearrik buys 34 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; 23 of what number is 35?
b
Divide the number of gallons by 6.
Solution
a
The distance Tearrik ran is given. He ran 23 of the way to the paint shop. This distance is equal to 35 of a mile.
The distance from the house to the paint shop is missing. That distance can be determined by finding two-thirds of what number is three fifths. 23 of what number is 35? This question can be mathematically expressed as follows. 2/3 * = 3/5 Now, this multiplication problem can be written as a division problem. 2/3 * = 3/5 ⇔ 3/5 ÷ 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.
3/5 ÷ 2/3
DivFracByFracDiv
a/b÷c/d=a/b*d/c
3/5 * 3/2
MultFrac
Multiply fractions
3 * 3/5 * 2
Multiply
Multiply
9/10
The distance to the paint shop is 910 miles.
b
The amount of blue paint Tearrik bought is a given, 34 gallons. He poured that amount evenly into 6 cups. The diagram illustrates the total amount of paint and the unknown amount per cup.
Dividing 34 by 6 gives how many gallons of paint each cup contains. 3/4 ÷ 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. 3/4 ÷ 6/1 Now, the steps performed when dividing two fractions can be followed.
3/4 ÷ 6/1
DivFracByFracDiv
a/b÷c/d=a/b*d/c
3/4 * 1/6
MultFrac
Multiply fractions
3 * 1/4 * 6
Multiply
Multiply
3/24
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. 3/24 &= 3*1/3*8 &⇕ 3/24 &= 1/8 This means that Tearrik pours 18 gallon into each cup.
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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.
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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 15 ÷ 2 215 First, the mixed numbers in this expression will be written as improper factions. Recall that a mixed number a bc is equal to a* c +bc.
3 15 ÷ 2 215
▼
Write mixed number as a fraction
MixedToFrac
a bc=a* c+b/c
3 * 5 + 1/5 ÷ 2 * 15+2/15
Multiply
Multiply
15 + 1/5 ÷ 30+2/15
AddTerms
Add terms
16/5 ÷ 32/15
16/5 ÷ 32/15
DivFracByFracDiv
a/b÷c/d=a/b*d/c
16/5 * 15/32
▼
Evaluate
MultFrac
Multiply fractions
16 * 15/5 * 32
SplitIntoFactors
Split into factors
16 * 5 * 3/5 * 16 * 2
CancelCommonFac
Cancel out common factors
16 * 5 * 3/5 * 16 * 2
SimpQuot
Simplify quotient
3/2
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Example
Finding the Width of the Piece of Wood
Tearrik now has 6 pieces of wood. Each piece has a length of 45 feet. The total area of the pieces is 1 35 square feet.
External credits: textures.com
What is the width of each piece?
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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 245 feet.
External credits: textures.com
Recall the formula for the area of a rectangle. It is the rectangle's width times its length. 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= 1 35 ÷ 24/5 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 35÷ 24/5
▼
Write mixed number as a fraction
8/5 ÷ 24/5
Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
8/5 ÷ 24/5
DivFracByFracDiv
a/b÷c/d=a/b*d/c
8/5 * 5/24
MultFrac
Multiply fractions
8 * 5/5 * 24
SplitIntoFactors
Split into factors
8 * 5/5 * 8 * 3
CancelCommonFac
Cancel out common factors
8 * 5/5 * 8 * 3
SimpQuot
Simplify quotient
1/3
The width of the rectangle is 13 of a foot. This also represents the width of each small rectangle.
External credits: textures.com
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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 45-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.
They spent 1 56 hours constructing the box, and they spent 1 112 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.
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Hint
Think about the following question. 1 112 times what number is 1 56?
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 & & Painting Time 1 56 & & 1 112
This requires to find a number that is equal to 1 56 when multiplied by 1 112.
1 112 times what number is 1 56? ⇓ 1 112 * = 15/6
This multiplication problem can be written as a division problem.
1 112 * = 1 56 ⇔ 1 56 ÷ 1 112 =
The answer can now be found by dividing the mixed numbers. First convert the mixed numbers into improper fractions.
1 56÷ 1 112
▼
Write mixed number as a fraction
11/6 ÷ 13/12
Remember that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
11/6 ÷ 13/12
DivFracByFracDiv
a/b÷c/d=a/b*d/c
11/6 * 12/13
MultFrac
Multiply fractions
11 * 12/6 * 13
SplitIntoFactors
Split into factors
11 * 6 * 2/6 * 13
CancelCommonFac
Cancel out common factors
11 * 6 * 2/6 * 13
SimpQuot
Simplify quotient
11* 2/13
Multiply
Multiply
22/13
The result should be given as a mixed number.
22/13
▼
Write fraction as a mixed number
WriteSum
Write as a sum
13+9/13
WriteSumFrac
Write as a sum of fractions
13/13 + 9/13
CalcQuot
Calculate quotient
1 + 9/13
Rewrite
Rewrite 1+9/13 as 1 913
1 913
It took 1 913 times longer to create the box than it did to paint it.
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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.
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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, 50. What does this expression equal?
5/0 = ?
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 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 ZeroSuppose 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).
a-b/a-b=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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Divide Fractions
Exercises
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Recall that the reciprocal of a non-zero number is 1 divided by that number. In addition to that, their product results in the multiplicative identity of 1.
| Non-zero Number | Reciprocal | Product |
|---|---|---|
| a | 1/a | a * 1/a=1 |
However, we want to find the reciprocal of a fraction. There is a straightforward way to find it. We switch the numerator and denominator of the fraction.
| Fraction | Reciprocal |
|---|---|
| a/b | b/a |
We can now find the reciprocal of 2 5 using this information. We will switch its numerator and denominator. cc Fraction & Reciprocal [0.5em] 2/5 & 5/2 We can test our results by checking if the product of these numbers is 1. Let's do it!
This confirm that the reciprocal of 25 is 52.
We will find the reciprocal of the number 7. Recall that the reciprocal of a natural number is 1 divided by that number.
| Natural Number | Reciprocal |
|---|---|
| a | 1/a |
Now that we have this information, we can write the reciprocal of 7. cc Number & Reciprocal [0.5em] 7 & 1/7 Again, we can check our result by seeing if the product of these numbers is 1.
This confirm that the reciprocal of 7 is 17.
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Solution
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Consider that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 1/7÷9/14=1/7 * 14/9 We will now multiply the fractions. Remember that the product of two fractions equals the product of the numerators over the product of the denominators.
We can now simplify the resulting fraction. In this case, 7 is the greatest common factor of 14 and 63. Using this fact, we can rewrite the numerator and denominator of the resulting fraction.
The given division is equal to 29. 1/7÷9/14=2/9
Let's begin by rewriting the expression so that all of the numbers are fractions before we evaluate the expression. We can write any whole number as a fraction with a denominator of 1.
Next, recall that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 3/4÷4/1=3/4 * 1/4 We multiply the numerators with each other and the denominators with each other.
This fraction is in simplest form. The result of the division is then 316. 3/4÷4=3/16
Again, let's start by rewriting the expression so that all of the numbers are fractions.
We can now multiply the first fraction by the reciprocal of the second fraction.
The result of this division is 10. 2÷1/5=10
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Solution
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We will first rewrite the numbers in the expression as an improper fractions.
Next, we multiply the first fraction by the reciprocal of the second fraction to find the quotient.
Note that 12 is equal to 2 * 6. This means that we can simplify the numerator and denominator before we multiply.
The quotient is 52, which is an improper fraction. Let's write it as a mixed number.
The result of the division is then 2 12. 6 ÷ 2 25=2 12
We can find the given quotient by following the same steps we followed in the previous part. Let's first rewrite the numbers.
We can now multiply 125 by the reciprocal of 61, which is 16.
The given quotient is 25. 2 25÷ 6 =2/5
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We want to divide a mixed number by a fraction. 3 13 ÷ 5/6 We will start by rewriting the mixed number as an improper fraction.
Recall that dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. Let's do it!
The result of dividing these fractions is 4.
We are given a division of a fraction into a mixed number.
14/16 ÷ 1 34
Our first step in finding this quotient is to convert the mixed number into an improper fraction. Let's do it!
The next step is to multiply the first fraction by the reciprocal of the second fraction.
The quotient is equal to 12.
In this case, we are asked to find the division of two mixed numbers.
4 39 ÷ 1 79
Let's start by converting the mixed numbers into improper fractions.
Next, we will multiply the first fraction by the reciprocal of the second fraction.
The result is an improper fraction. We will write it as a mixed number.
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Solution
The diagram models the division of two numbers.
Which of the following expressions is modeled by the diagram?
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Let's only focus on the blue region. We can see that there are three units. Each of these three units is divided into three small parts.
We see that the blue region represents 2 23. We know that the diagram is used to model a division. Let's review what division means.
Division |- This operation represents the process of calculating how many times one quantity is contained within another quantity.
We see that two parts sized one and one-third fit within in a part sized two and two-thirds. We will note this on the given diagram.
Considering the definition of division and the diagram, we can say that the dividend is 2 23 and the divisor is 1 13. Then we can write the following expression. 2 23 ÷ 1 13 Note that the quotient is equal to 2 because two units of the size one and one-third can fit inside two and two-thirds.
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Divide Fractions
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