Learn Dividing Fractions with Mixed Numbers and Reciprocal Concepts

Multiply fractions

12* 5/25*3

60/75

Simplfy if Possible

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.

60/75

ReduceFrac

a/b=.a /15./.b /15.

60/15/75/15

Simplify quotient

4/5

The division expression is equal to 45.

The same steps above are also used when dividing a fraction by a whole number, as 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 wants to cut a 5-foot long plank of wood into equal parts.

How many 45-foot pieces can he cut from the original board?

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

Hint

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

Use the answer from Part A.

Solution

Tearrik wants to cut the plank of wood into 45-foot lengths.

We want to find how many smaller lengths he can cut from the plank. Let's divide the length of the larger piece by the length of each smaller piece. 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 that all whole numbers are fractions with a denominator of 1.

5 ÷ 4/5

Rewrite

Rewrite 5 as 5/1

5/1 ÷ 4/5

a/b÷c/d=a/b*d/c

5/1 * 5/4

Multiply fractions

5 * 5/1 * 4

25/4

We got an improper fraction. Let's write it as a mixed number to see how many full 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 45-foot lengths each from the original 5-foot plank of wood.

In Part A, we found that dividing 5 by 45 is 6 14.

5 ÷ 4/5 = 6 14 The partial piece is 14 of 45-foot long. We can find this length by multiplying the fractions.

1/4 * 4/5

Multiply fractions

1* 4/4 * 5

Cancel out common factors

1* 4/4 * 5

Simplify quotient

1/5

The length of the remaining piece of wood is a 15 of one foot.

Alternative Solution

Use a Diagram

A diagram can be used to model the division of 5 by 45. Divide each foot of the 5-foot plank into 5 equal pieces.

Each of the smaller parts represents a 15 of a foot. Let's count how many groups of four parts we can make.

We can make 6 groups of 4 smaller sections. The length of the leftover part is 15 of a foot, which is also 14 of 45. This confirms that the result we found algebraically is correct.

Example

Camping Out

Tearrik hikes 35 mile up the trail to his campsite. This is 23 the length of the entire trail.

How long is the entire trail to the campsite?

At the campsite, Tearrik divides 34 gallon of water evenly among 6 bottles. How many gallons of water go in each bottle?

Hint

Two-thirds of what number is 35?

Divide the amount of water by 6.

Solution

We know that Tearrik hiked 23 of the trail to his campsite. The distance he hiked was 35 of a mile.

We do not know the distance from the beginning of the trail to the campsite. We can find the distance 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 We can rewrite this multiplication as a division problem. 2/3 * = 3/5 ⇔ 3/5 ÷ 2/3 = The quotient of this division represents the distance from the trailhead to the campsite. 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

a/b÷c/d=a/b*d/c

3/5 * 3/2

Multiply fractions

3 * 3/5 * 2

9/10

The trail to the campsite is 910 mile long.

Tearrik poured 34 gallon of water evenly into 6 bottles. The diagram illustrates the total amount of water and the unknown amount per bottle.

Dividing 34 by 6 gives how many gallons of water each bottle holds. 3/4 ÷ 6 This is a division of a fraction by a whole number. Let's rewrite the whole number as a fraction to calculate the quotient. 3/4 ÷ 6/1 Now we follow the same steps as before when when dividing two fractions.

3/4 ÷ 6/1

a/b÷c/d=a/b*d/c

3/4 * 1/6

Multiply fractions

3 * 1/4 * 6

3/24

Since the denominator and the numerator have a common factor of 3, let's simplify the fraction by dividing both by 3. 3/24 &= 3÷ 1/24÷ 3 &⇕ 3/24 &= 1/8 This means that Tearrik poured 18 gallon of water into each bottle.

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.

$quotient of random fractions$

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

15 + 1/5 ÷ 30+2/15

AddTerms

Add terms

16/5 ÷ 32/15

Now follow the usual steps to divide the two fractions. The division sign is changed to a multiplication sign and the second fraction is replaced with its reciprocal.

16/5 ÷ 32/15

a/b÷c/d=a/b*d/c

16/5 * 15/32

▼

Evaluate

Multiply fractions

16 * 15/5 * 32

SplitIntoFactors

Split into factors

16 * 5 * 3/5 * 16 * 2

Cancel out common factors

16 * 5 * 3/5 * 16 * 2

Simplify quotient

3/2

The quotient is 32, which can be rewritten as 1 12.

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.

What is the width of each piece?

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

Let's start by multiplying the length of a small wood piece by 6 to find the length of the plank of wood.

4/5 * 6

MoveRightFacToNum

a/c* b = a* b/c

4 * 6/5

24/5

The length of the plank of wood is 245 feet.

Now recall that the formula for the area of a rectangle is the rectangle's width times its length. Area = Width * Length We already know the area and the length of the plank of wood and we want to know its width. At this point of the process, it would be helpful to rearrange the formula to isolate the width to one side. Width = Area ÷ Length We can calculate the width of the plank of wood 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. Let's rewrite the mixed number as an improper fraction so we can perform the calculation.

1 35÷ 24/5

▼

Write mixed number as a fraction

MixedToFrac

a bc=a* c+b/c

1* 5 +3/5 ÷ 24/5

MultByOne

a * 1=a

5 +3/5 ÷ 24/5

AddFrac

Add fractions

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

a/b÷c/d=a/b*d/c

8/5 * 5/24

Multiply fractions

8 * 5/5 * 24

SplitIntoFactors

Split into factors

8 * 5/5 * 8 * 3

Cancel out common factors

8 * 5/5 * 8 * 3

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.

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.

Hint

What number times 1 112 is 1 56? Can we write the question as a division problem?

Solution

We want to compare the time it took to create the box and the time it took to paint it. Construction Time & & Painting Time 1 56 & & 1 112 We need 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 = Now we can find the answer by dividing the mixed numbers. First, let's convert the mixed numbers into improper fractions.

1 56÷ 1 112

▼

Write mixed number as a fraction

MixedToFrac

a bc=a* c+b/c

1* 6 +5/6 ÷ 1* 12+ 1/12

MultByOne

a * 1=a

6 +5/6 ÷ 12+1/12

AddTerms

Add terms

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

a/b÷c/d=a/b*d/c

11/6 * 12/13

Multiply fractions

11 * 12/6 * 13

SplitIntoFactors

Split into factors

11 * 6 * 2/6 * 13

Cancel out common factors

11 * 6 * 2/6 * 13

Simplify quotient

11* 2/13

22/13

We have been asked to give our answer as a mixed number, so let's convert it!

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 build the box than it did to paint it.

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.

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 Zero

Suppose 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