Download 4.5 Multiplying and Dividing Mixed Numbers Caution: is not the

Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
4.5 Multiplying and Dividing Mixed Numbers
Mixed Number
A mixed number is made up of two parts: an integer part and a fraction part
Mixed numbers are represented by writing an integer followed by a proper fraction
(with no symbol between).
2
3
3
is a mixed number since it is a “mix” of the integer 2 and the proper fraction .
4
4
4
9
9
is not considered a mixed number since the fraction part, is improper.
8
8
The Meaning of a Mixed Number
A mixed number is the sum of an integer and a proper fraction where both the signs of
the integer and proper fraction are the same.
2
Caution: 2
3
3
means 2  .
4
4
2
3
 3
means 2     .
4
 4
3
3
3
3
3
is not the same as 2   ! 2 is a mixed number while 2   means multiplication: 2   
4
4
4
4
4
Mixed numbers are another way to represent improper fractions. We can show this by using techniques of
section 4.4:
3
3
2  2
4
4
2 3
 
1 4
8 3
 
4 4
83
4
11

4

Use the meaning of a mixed number.
Rewrite the 2 with a denominator as 1.
Rewrite each fraction as an equivalent fraction with the LCD = 4 as the
Denominator.
Simplify.
3 11
So 2  .
4 4
1
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Math 40
Prealgebra
Similarly, 2
Section 4.5 – Multiplying and Dividing Mixed Numbers
3
 3
 2    
4
 4
2  3 

 
1  4 
8  3 

 
4  4 
8   3 

4
11
11
or  .

4
4
So 2
3
11
 .
4
4
From this process, we come up with a quicker way to convert a mixed number to an improper fraction.
Shortcut for Converting a Mixed Number to an Improper Fraction
1) If the mixed number is negative, hold the negative sign off to the side.
2) Multiply the integer part by the denominator of the fraction part. Then add the
numerator of the fraction part to this product.
3) Write the result of step 2 over the denominator of the fraction part.
4) If the original mixed number was negative, write the improper fraction with a
negative sign.
Example 1: Write the following mixed numbers as improper fractions.
2
5
a) 6
b) 3
3
8
Solution:
a) Multiply the integer part by the denominator of the fraction part. Then add the numerator.
2
 3
6
6  3  2  20
Write the result over the denominator of the fraction part.
So 6
20
3
2 20

.
3 3
2
2015 Campeau
Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
b) Since this is a negative mixed number, hold the negative sign off to the side.
Multiply the integer part by the denominator of the fraction part. Then add the numerator.

5
8
3
3  8  5  29
Write the result over the denominator of the fraction part. 
29
8
Don’t forget to write the negative sign!
5
29
So 3   .
8
8
You Try It 1: Write the following mixed numbers as improper fractions.
3
1
a) 5
b) 7
4
5
There will be times when it is necessary to convert an improper fraction to a mixed number. In these cases, we
simply use long division.
Converting an Improper Fraction to a Mixed Number
1) If the improper fraction is negative, hold the negative sign off to the side.
2) Use long division to divide the numerator by the denominator.
denominator numerator
3) The quotient is the whole number part for the mixed number, the remainder is the
numerator of the fraction part, and the denominator remains the same.
4) If the original improper fraction was negative, write the mixed number with a
negative sign.
3
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Example 2: Write the following improper fractions as mixed numbers.
15
39
a)
b) 
7
5
Solution:
a) Use long division: denominator numerator
2
7 15
14
1
So,
15
 2 remainder 1 . The quotient is the whole number part.
7
The remainder is the numerator of the fraction part.
1
The denominator remains the same.
2
7
b) Since this is a negative improper fraction, hold the negative sign off to the side.
39

5
Use long division: denominator numerator
7
5 39
35
4
So, 
39
 7 remainder 4 . The quotient is the whole number part.
5
The remainder is the numerator of the fraction part.
4
The denominator remains the same.
 7
5
Don’t forget to write the negative sign!
You Try It 2: Write the following improper fractions as mixed numbers.
38
43
a) 
b)
9
5
4
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Rounding Mixed Numbers to the Nearest Integer
Before we move on to multiplying and dividing mixed numbers, we would like to provide a nice way for you to
estimate your answer and use it at the end of the procedure to figure out if your answers are reasonable.
The key to rounding mixed numbers to the nearest whole number is to figure out whether the fraction part is
1
greater than or equal to .
2
In order to do this, it is helpful to construct a fraction that is equivalent to
1
having the same denominator as
2
the fraction in question.
1
k
are of the form
, where the numerator is half of the denominator
2
2k
(or the denominator is twice the numerator).
Recall that all fractions equivalent to
Example 3: Round the following mixed numbers to the nearest integer.
5
2
a) 1
b) 3
8
5
c) 9
2
3
Solution:
5
5
1
a) For 1 , when trying to determine if is greater than or equal to , we construct the fraction
8
8
2
4
.
8
5
4
Why? Half of 8 is 4! Now we only have to compare the with !
8
8
Since the denominators are the same, it is easier to see that 5  4 .
This means that
5 4
5 1
 or  .
8 8
8 2
Since the fraction part of the mixed number is greater than or equal to
1
, we round up to the
2
next integer.
5
So, 1  2 .
8
5
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
2
2
1
b) For 3 , when trying to determine if
is greater than or equal to , we construct the
5
5
2
2.5
fraction
.
5
2
2.5
Why? Half of 5 is 2.5! Now we only have to compare the
with
!
5
5
Since the denominators are the same, it is easier to see that 2  2.5 .
This means that
2 2.5
2 1
or  .

5
5
5 2
Since the fraction part of the mixed number is less than
1
, we round down an integer.
2
2
So, 3  3 .
5
2
is a negative mixed number. Hold the negative sign off to the side and attach it later.
3
2
1
1.5
When trying to determine if
is greater than or equal to , we construct the fraction
.
3
2
3
2
1.5
Why? Half of 3 is 1.5! Now we only have to compare the
with
!
3
3
c) 9
Since the denominators are the same, it is easier to see that 2  1.5 .
This means that
2 1.5
2 1

or  .
3 3
3 2
Since the fraction part of the mixed number is greater than or equal to
next integer. So, 9
1
, we round up to the
2
2
 10 .
3
2
Don’t forget the negative sign in your final answer! 9  10
3
You Try It 3: Round the following mixed numbers to the nearest integer.
5
3
a) 4
b) 8
9
7
c) 2
1
6
6
2015 Campeau
Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Once we get comfortable with rounding mixed numbers to the nearest integer and converting mixed numbers to
improper fractions (and vice versa), we are now ready to multiply and divide mixed numbers!
Multiplying and Dividing Mixed Numbers
1) Estimate the answer first.
 Round each mixed number to the nearest whole number.
 Perform the indicated operation to get an estimate of the answer.
2) Rewrite each mixed number in the problem as an improper fraction.
3) Multiply or divide using methods from 4.2 and 4.3.
4) Write the answer in lowest terms if possible.
5) If possible, convert the answer to a mixed number.
6) Make sure the answer is reasonable. Compare it to the estimate. It should be
relatively close.
Do I always have to convert my final answer to a mixed number?
If your answer is a proper fraction, it is impossible to convert it to a mixed number. Leave it alone!
If your answer is an improper fraction, you only have to conver it to a mixed number if:
 The directions call for it.
 The original problem started with any mixed numbers.
 Real life situations (also known as word problems).
2 3
Example 4: For 6  2  , first round each number to estimate the answer. Then find the exact answer.
3 4
Solution:
2
3
1 3
Estimate: 6  7 and 2  3 So, 6  2   7  3
3
4
3 4
 21
7
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
2 3
Actual: Convert each mixed number to an improper fraction. 6  2 
3 4
Multiply as usual.

20  11 
 
3  4

20 11
3 4

2  2  5 11
3 2  2

5 11
3

55
3
55
is improper.
3
Since the original problem started with mixed numbers, convert
55
to a mixed number.
3
55
1
1
 18 . Check! 18 is relatively close to our estimate of 21.
3
3
3
2 3
1
Therefore we are confident that 6  2   18
3 4
3
1 3
You Try It 4: For 5  1  , first round each number to estimate the answer. Then find the exact answer.
3 4
8
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
1  3
Example 5: For 2   3  , first round each number to estimate the answer. Then find the exact answer.
4  8
Solution:
Estimate: 2
1
3
1  3
 2 and 3  3 So, 2   3   2   3
4
8
4  8
2

3
1  3
Actual: Convert each mixed number to an improper fraction. 2   3 
4  8

Divide as usual.

9  27 
 
4  8 
9 8
 
4 27

9 8
4  27

33 2  2  2
2  2 333

2
3
2
is a proper fraction so we leave it as is..
3
Check! 
2
2
is exactly the same as our estimate of  .
3
3
1  3
2
Therefore we are confident that 2   3   
4  8
3
4  3
You Try It 5: For 4   5  , first round each number to estimate the answer. Then find the exact answer.
5  5
9
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Recall
Area of a Parallelogram
A  base  height
A  bh
height
base
Notice that we can create two equal-sized triangles if we draw a line through the diagonals of the parallelogram:
We can say that the area of 1 parallelogram is the same as the area of the two equal-sized triangles or more
importantly, the area of 1 of the equal-sized triangles is half of the area of the parallelogram. This is how
we determine a formula to calculate the area of a triangle!
Area of a Triangle
1
Area   base  height
2
1
A  bh
2
Remember to use square units when measuring area.
height
base
height
base
10
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Example 6: Find the area of the following triangles.
10
a)
1
yd
4
6 yd
9 yd
b)
11
7 ft
4
ft
5
8 ft
7 yd
9
1
ft
2
Solution:
a) The base is 10
1
yd and the height is 6 yd .
4
We do not need the 9 yd or 7 yd sides to find the area.
A
1
2

b

h
1 1 
A  10 yd   6 yd 
2 4 
Replace b with 10
1
yd and h with 6 yd .
4
Before we calculate the exact area, it is helpful to estimate the answer by first rounding any
mixed number to the nearest integer:
1 1 
1
A  10 yd   6 yd   10  6   30 yd 2
2 4 
2
(This way we know that our final answer should be relatively close to 30 yd 2 .)
Continuing with the actual calculation:
1 1 
A  10 yd   6 yd 
2 4 
A
1  41 yd   6 yd 



2  4  1 
Write 10
A
1 41 6  yd  yd
2  4 1
Simplify.
1
as an improper fraction.
4
Write 6 with 1 as the denominator.
11
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
A
1 41 2  3 2
yd
2  2  2 1
A
41  3 2
yd
22
A
123 2
yd
4
yd  yd  yd 2 (or “square yards”)
123
is improper!
4
Convert
A  30
123
to a mixed number for a better understanding of the measurement.
4
3 2
yd
4
Check! Is this close to our original estimate of 30 yd 2 ? Yes!
Now we can be confident that we have calculated the correct actual area.
The area of the triangle is 30
3 2
3
yd (or 30 square yards).
4
4
1
ft and the height is 7 ft .
2
1
We do not need the 8 ft or 11 ft sides to find the area.
4
1
A
 b  h
2
b) The base is 9
A
1 1 
 9 ft   7 ft 
2 2 
Replace b with 9
1
ft and h with 7 ft .
2
Before we calculate the exact area, it is helpful to estimate the answer by first rounding any
mixed number to the nearest integer:
1 1 
1
A   9 ft   7 ft   10  7   35 ft 2
2 2 
2
(This way we know that our final answer should be relatively close to 35 ft 2 .)
Continuing with the actual calculation:
A
1 1 
 9 ft   7 ft 
2 2 
12
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Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
A
1  19 ft   7 ft 



2  2  1 
Write 9
A
119  7  ft  ft
2  2 1
Simplify.
A
133 2
ft
4
1
as an improper fraction.
2
Write 7 with 1 as the denominator.
ft  ft  ft 2 (or “square feet”)
Convert
A  33
133
to a mixed number for a better understanding of the measurement.
4
1 2
ft
4
Check! Is this close to our original estimate of 35 ft 2 ? Yes!
Now we can be confident that we have calculated the correct actual area.
The area of the triangle is 33
1 2
1
ft (or 33 square feet).
4
4
You Try It 6: Find the area of the following triangles.
a)
8
1
5 m
2
6
3
m
4
5
m
16
b)
3
9 in
8
5 in
1
9 in
4
7 in
13
2015 Campeau
Math 40
Prealgebra
Section 4.5 – Multiplying and Dividing Mixed Numbers
Example 7: First, round any mixed numbers to estimate the answer. Then find the exact answer.
1
1
Capri used 2 packages of chocolate chips in her cookie recipe. Each package has 5 ounces of
2
2
chocolate chips. How many ounces of chips did she use in the recipe?
Solution:
First round each mixed number to the nearest integer: 2
1
1
 3 packages and 5  6 ounces
2
2
Reread the problem using the rounded numbers to get a better understanding of the problem so you
can select the correct operation to use:
Capri used 3 packages of chocolate chips in her cookie recipe. Each package has 6 ounces of
chocolate chips. How many ounces of chips did she use in the recipe?
3 packages with each having 6 ounces indicates multiplication: 3  6   18 ounces of chips
1  1  5  11 
Now calculate the actual answer: 2  5    
2 2 2 2 
55

4
3
3
Check! Is 13 relatively close to 18? Yes!
 13
4
4
3
State the final answer: Capri used 13 ounces of chocolate chips in the recipe.
4
You Try It 7: First, round any mixed numbers to estimate the answer. Then find the exact answer.
3
The directions for mixing a plant food say to use 1 ounces of chemical for each gallon of
4
1
water. How many ounces of chemical should be mixed with 5 gallons of water?
2
14
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