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.
| Student Learning Objectives: |
|---|
|
| | 12 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Divide Fractions
Slide of 12
In this lesson uses visual models to explore how to divide fractions and examine how it is related to multiplication.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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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, 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
There are shortcuts to finding the reciprocals of specific types of numbers.
| 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 cannot equal 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 Keep the first fraction as is. Change the division sign with the multiplication sign and write the reciprocal of the second fraction 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. Reduce the fraction by 15.
The division expression is equal to 45.
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Example
Breaking a Piece of Wood Into Equal Pieces
Tearrik wants to cut a 5-foot long plank of wood into equal parts.
External credits: textures.com
a
How many full 45-foot pieces can he cut from the original board?
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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 plank by the length of a smaller piece.
b
Use the answer from Part A.
Solution
a
To find how many pieces of wood of length 45-foot Tearrik can cut from a 5-foot plank, we divide 5 by 45.
5 ÷ 4/5 When we divide a whole number by a fraction, we multiply the whole number by the reciprocal of the fraction. Recall that all whole numbers are fractions with a denominator of 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
To see how many full pieces of wood we can cut, we can rewrite the improper fraction of 254 as a mixed number.
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 cut 6 full pieces of length 45-foot from the original 5-foot plank of wood.
b
In Part A, we found that we can cut 6 14 pieces from the 5-foot plank. The 14 in the mixed number represents a partial piece. We can find its length by multiplying 14 by 45-foot.
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 15 of a foot.
Alternative Solution
Use a DiagramTo divide 5 by 45, we can also use a diagram. Let's first divide each foot of the 5-foot plank into 5 equal parts.
External credits: textures.com
Each smaller part represents 15 of a foot. Now we can count how many groups of four parts we can make. The leftover part represents the length of the non-full 45-foot piece.
External credits: textures.com
We found that Tearrik can cut the plank into 6 pieces of length 45-foot, with a leftover 15-foot piece. This matches the answer from the main solution and confirms that we are correct.
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Example
Camping Out
Tearrik hikes 35 mile up the trail to his campsite. This is 23 the length of the entire trail.
a
How long is the entire trail to the campsite?
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b
At the campsite, Tearrik divides 34 gallon of water evenly among 6 bottles. How many gallons of water go in each bottle?
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Hint
a
Two-thirds of what number is 35?
b
Divide the amount of water by 6.
Solution
a
We know that 35 mile makes 23 of the distance Tearrik has to hike to his campsite.
2/3 * Trail = 3/5mi To find the length of the trail, we can divide both sides by 23. Trail = 3/5 ÷ 2/3mi Now we can divide the two fractions. Remember that we can do this by multiplying the first fraction by the reciprocal of the second.
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 trail to the campsite is 910 mile long.
b
To find how much water Tearrik poured into each bottle, we divide 3/4 of a gallon of water by 6, the number of bottles he filled.
3/4 ÷ 6 To find this quotient, we first rewrite 6 as a fraction. 3/4 ÷ 6/1 Now we follow the same steps as before when dividing two fractions.
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
Finally, we can simplify 324 by dividing both its numerator and denominator by 3. 3 ÷ 3/24 ÷ 3 &= 1/8 Tearrik poured 18 gallon of water into each bottle.
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Pop Quiz
Dividing Fractions
Find the quotient of fractions. Simplify the answer if possible. 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
To perform a division with mixed numbers, start by rewriting the mixed numbers as improper fractions. Then, follow the same steps as when dividing fractions. Consider the following example. 3 15 ÷ 2 215 First, rewrite the mixed numbers 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 has 6 pieces of wood with a length of 45 foot. 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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Hint
Multiply the fraction 45 by 6 to find the total length. The formula for the area of a rectangle is the width times the length.
Solution
First, we can find the length of the plank of wood by multiplying the length of one small piece by 6.
The length of the plank of wood is 245 feet.
External credits: textures.com
We can find the width of the piece of wood using the formula for the area of a rectangle. Area = Width * Length [0.9em] 1 35ft^2 = Width * 24/5ft Now let's rearrange the formula. If we divide both sides by 245, we get the width. For simplicity, we can leave out the units for now. 1 35 ÷ 24/5 = Width We need to divide a mixed number by a fraction. To do this, we start by rewriting the mixed number as an improper fraction.
1 35÷ 24/5
▼
Write mixed number as a fraction
8/5 ÷ 24/5
Next, we find the quotient of two fractions by multiplying the first fraction by the reciprocal of the second.
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 plank of wood is 13 of a foot. This also represents the width of each small cut.
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Example
Comparing the Times Spent on Completing the Box
Tearrik cuts two of his 45-foot-long pieces of wood into squares. He uses these squares and the remaining four cut pieces of wood to make a box. After that, he paints the box.
He spent 1 56 hours building the box and 1 112 hours painting it. How many times longer did it take him to build the box than it did to paint it? Write the answer as a mixed number.
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Hint
Find the ratio of the construction time to the painting time.
Solution
To find how many times longer it took Tearrik to build the box than to paint it, we divide the construction time by the painting time.
Construction Time/Painting Time = 1 56 ÷ 1 112
To find this ratio, we start by rewriting the mixed numbers as improper fractions.
1 56÷ 1 112
▼
Write mixed number as a fraction
11/6 ÷ 13/12
Next, we can find the quotient. 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
Finally, we can write our answer 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 as long to build the box as it did to paint it.
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Pop Quiz
Dividing Mixed Numbers
Find the indicated quotient. Simplify the answer if possible. 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?
Before we end the lesson, let's consider division by 0. For example, what would we expect the quotient of 50 to be?
5/0 = ?
This expression is considered undefined or impossible. 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 we add together to try to get 5, we will never reach the number 5.
|
Dividing by 0 then becomes impossible. |
Extra
The Consequences of Dividing by ZeroSuppose dividing by 0 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 0 is defined. Instead, we showed why dividing by 0 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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Solution
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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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