The Fractions Pack includes a huge collection of activity resources, teaching guides and display materials to make fractions a bit less scary for your class!
Download this popular teaching resource pack (as well as thousands of other resources) from http://teachr.co/fractionspack
The Fractions Pack includes a huge collection of activity resources, teaching guides and display materials to make fractions a bit less scary for your class!
Download this popular teaching resource pack (as well as thousands of other resources) from http://teachr.co/fractionspack
All About Fractions Powerpoint part 1 EDU 290charn1km
This powerpoint explains how to add, subtract, multiply, and reduce/simplify fractions. It also explains how to convert from an improper fraction to a mixed number and vice versa.
All About Fractions Powerpoint part 1 EDU 290charn1km
This powerpoint explains how to add, subtract, multiply, and reduce/simplify fractions. It also explains how to convert from an improper fraction to a mixed number and vice versa.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers.
p
q
Fractions
3. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers.
p
q
Fractions
3
6
4. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
p
q
Fractions
3
6
5. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
Fractions
3
6
6. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
3
6
Fractions
7. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3
6
Fractions
8. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3
6
Fractions
9. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions
10. Fractions are numbers of the form (or p/q) where
p, q 0 are whole numbers. Fractions are numbers that
measure parts of whole items.
Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions
3/6 of a pizza
11. Whole numbers can be viewed as fractions with denominator 1.
Fractions
12. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = .5
1
x
1
Fractions
13. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.5
1
x
1
0
x
Fractions
14. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
15. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
16. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0.
17. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
18. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
19. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
1
2
=
2
4
20. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
1
2
=
2
4
=
3
6
21. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
… are equivalent fractions.1
2
=
2
4
=
3
6
=
4
8
22. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
… are equivalent fractions.
The fraction with the smallest denominator of all the
equivalent fractions is called the reduced fraction.
1
2
=
2
4
=
3
6
=
4
8
23. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x 0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)
Fractions that represents the same quantity are called
equivalent fractions.
… are equivalent fractions.
The fraction with the smallest denominator of all the
equivalent fractions is called the reduced fraction.
1
2
=
2
4
=
3
6
=
4
8
is the reduced one in the above list.
1
2
24. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
a
b
a
b =
a / c
Fractions
b / c
25. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1,
a
b
a
b =
a / c
Fractions
b / c
26. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
=
a*c
b*c
a*c
b*c
1
Fractions
b / c
27. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
28. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
(Often we omit writing the 1’s after the cancellation.)
29. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
(Often we omit writing the 1’s after the cancellation.)
30. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
(Often we omit writing the 1’s after the cancellation.)
31. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
(Often we omit writing the 1’s after the cancellation.)
32. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
78/2
54/2
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
(Often we omit writing the 1’s after the cancellation.)
33. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
78/2
54/2
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
=
39
27
(Often we omit writing the 1’s after the cancellation.)
34. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
78/2
54/2
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
=
39/3
27/3
39
27
(Often we omit writing the 1’s after the cancellation.)
35. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
78/2
54/2
= 13
9 .
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
=
39/3
27/3
39
27
(Often we omit writing the 1’s after the cancellation.)
36. Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
In other words, a common factor of the numerator and the
denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54
78
54
=
78/2
54/2
= 13
9
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.
=
39/3
27/3
(or divide both by 6 in one step.)
39
27
(Often we omit writing the 1’s after the cancellation.)
37. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
38. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
A participant in a sum or a difference is called a term.
39. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
40. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
41. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
42. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
3
5
=
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
43. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
3
5
=
This is addition. Can’t cancel!
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
44. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
3
5
=
This is addition. Can’t cancel!
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
45. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
46. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!? 2 * 1
2 * 3
=
1
3
Yes
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
47. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
Improper Fractions and Mixed Numbers
2 * 1
2 * 3
=
1
3
Yes
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
48. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
A fraction whose numerator is the same or more than its
denominator (e.g. ) is said to be improper .
Improper Fractions and Mixed Numbers
3
2
2 * 1
2 * 3
=
1
3
Yes
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
49. Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
A fraction whose numerator is the same or more than its
denominator (e.g. ) is said to be improper .
We may put an improper fraction into mixed form by division.
Improper Fractions and Mixed Numbers
3
2
2 * 1
2 * 3
=
1
3
Yes
A participant in a sum or a difference is called a term.
The “2” in the expression “2 + 3” is a term (of the expression).
The “2” is in the expression “2 * 3” is called a factor.
Terms may not be cancelled. Only factors may be canceled.
50. 23
4
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
b. Convert into improper form.5 3
4
51. 23
4
23 4 = 5 with remainder 3.·
·
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
b. Convert into improper form.5 3
4
52. 23
4
23 4 = 5 with remainder 3.·
·
23
4
= 5 + 5 3
4 .
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
3
4
=
b. Convert into improper form.5 3
4
Hence,
53. 23
4
23 4 = 5 with remainder 3.·
·
23
4
= 5 + 5 3
4 .
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via a multiplication followed by an addition.
b. Convert into improper form.5 3
4
Hence,
54. 23
4
23 4 = 5 with remainder 3.·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via a multiplication followed by an addition.
b. Convert into improper form.5 3
4
Hence,
55. 23
4
23 4 = 5 with remainder 3.·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
23
4
=
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via a multiplication followed by an addition.
b. Convert into improper form.5 3
4
Hence,
56. 23
4
23 4 = 5 with remainder 3.·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
23
4
=
Improper Fractions and Mixed Numbers
Example B.
a. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via a multiplication followed by an addition.
b. Convert into improper form.5 3
4
Hence,
57. Multiplication and Division of Fractions
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
58. Multiplication and Division of Fractions
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
59. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
60. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
$108 $36$36$36
2
3
of
Divide into 3 piles
61. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
and we take 2 piles or 2 x 36 = $72.
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles
62. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
and we take 2 piles or 2 x 36 = $72.
This procedure is recorded as multiplication
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
* 108
2
3
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles
63. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
and we take 2 piles or 2 x 36 = $72.
This procedure is recorded as multiplication
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
* 108
2
3
36
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles
64. Multiplication and Division of Fractions
Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
and we take 2 piles or 2 x 36 = $72.
This procedure is recorded as multiplication
3/6 of a pizza
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.
* 108
2
3
36
= 2 * 36 = $72.
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles
65. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
Fractional multiplications of the form
Multiplication and Division of Fractions
* or
a
b
d *
66. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Fractional multiplications of the form
Multiplication and Division of Fractions
* or
a
b
d *
67. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18a.
Fractional multiplications of the form
11
16
48b.
Multiplication and Division of Fractions
*
*
*
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*
68. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18a.
Fractional multiplications of the form
6
11
16
48b.
Multiplication and Division of Fractions
*
*
*
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*
69. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
Multiplication and Division of Fractions
*
* *
*
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*
70. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
*
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*
71. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*
72. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
There are 60 minutes in one hour, 3
5
2
*
* 60 min.so there are
73. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
There are 60 minutes in one hour, 3
5
2
*
*
We split the mixed fraction to do the multiplication, i.e.
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5
74. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
There are 60 minutes in one hour, 3
5
2
*
*
We split the mixed fraction to do the multiplication, i.e.
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120
Distribute
75. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
There are 60 minutes in one hour, 3
5
2
*
*
We split the mixed fraction to do the multiplication, i.e.
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120 + 36
Distribute
12
76. a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
To calculate these, always divide or cancel first, then multiply.
Example C. Multiply by cancelling first.
2
3 18 = 2 6 = 12a.
Fractional multiplications of the form
6
11
16
48b.
3
Multiplication and Division of Fractions
*
* *
* = 3 * 11 = 33
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
There are 60 minutes in one hour, 3
5
2
*
*
We split the mixed fraction to do the multiplication, i.e.
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120 + 36 = 156,
Distribute
12
3
5
hours is 156 minutes.so 2
77. Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions
78. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
79. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
12
25
15
8
*a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
80. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
81. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
82. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
83. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
=
3*3
2*5
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
84. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
85. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
b.
8
9
7
8
*
10
11
9
10
**
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
86. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
87. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
88. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
89. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
90. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
91. c
d
=
a*c
b*d
a
b
*
Multiplication and Division of Fractions
Example D. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.
We can’t cancel like this for ± , i.e.
c
d = a c
b d
a
b ± ±
±
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
think of the case that: 1
2 = 2
4
1
2
+
92. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
Multiplication and Division of Fractions
93. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48
94. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
95. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
so there are 12 pieces of chocolate candies.
96. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is
so there are 12 pieces of chocolate candies.
97. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 12 pieces of chocolate candies.
98. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
so there are 12 pieces of chocolate candies.
99. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops.
so there are 12 pieces of chocolate candies.
100. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
so there are 12 pieces of chocolate candies.
101. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48 = 20/4
48/4
so there are 12 pieces of chocolate candies.
102. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48 = 20/4
48/4 = 5
12
so there are 12 pieces of chocolate candies.
103. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48 = 20/4
48/4 = 5
12
so there are 12 pieces of chocolate candies.
104. Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?
1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
3
4 * x.
Multiplication and Division of Fractions
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48 = 20/4
48/4 = 5
12
It translates into multiplication as
so there are 12 pieces of chocolate candies.
106. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
107. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
108. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
the reciprocal of is 3,1
3
109. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
110. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
111. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
112. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2* = 1,
113. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2* = 1, 5
1
5* = 1,
114. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2* = 1, 5
1
5* = 1, x
1
x* = 1,
115. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
116. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 ,*
1
2
117. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
118. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is,
119. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is, d
c
a
b
*
c
d =
a
b ÷
reciprocate
120. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
x
the reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is, d
c =
a*d
b*c
a
b
*
c
d =
a
b ÷
reciprocate
121. Example F. Divide the following fractions.
8
15
=
12
25
a. ÷
Reciprocal and Division of Fractions
122. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25
a. ÷
Reciprocal and Division of Fractions
123. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25 2
3
a. ÷
Reciprocal and Division of Fractions
124. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25 5
3
2
3
a. ÷
Reciprocal and Division of Fractions
125. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a. ÷
Reciprocal and Division of Fractions
126. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
÷
÷ =b.
Reciprocal and Division of Fractions
127. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
÷
÷ = *b.
Reciprocal and Division of Fractions
128. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
÷
÷ = *b.
Reciprocal and Division of Fractions
129. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
130. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
1
6
=5d. ÷
131. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1
= 30*
1
6
=5d. ÷ 5
132. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
6
1
= 30*
1
6
=5d. ÷ 5
133. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
In general, we convert mixes fractions into improper ones to
do multiplication or division.
6
1
= 30*
1
6
=5d. ÷ 5
134. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
Using division, the number of cakes can be made is
1
2
÷
11 =
In general, we convert mixes fractions into improper ones to
do multiplication or division.
7 4
6
1
= 30*
1
6
=5d. ÷ 5
135. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
1
2
÷
11 =
15
2
5
4
In general, we convert mixes fractions into improper ones to
do multiplication or division.
7 4
÷
6
1
= 30*
1
6
=5d. ÷ 5
Using division, the number of cakes can be made is
136. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
1
2
÷
11 =
15
2
5
4
3
= 6 cakes.
In general, we convert mixes fractions into improper ones to
do multiplication or division.
7 4
÷
15
2 5
4
2
*=
6
1
= 30*
1
6
=5d. ÷ 5
Using division, the number of cakes can be made is
137. Improper Fractions and Mixed Numbers
B. Convert the following improper fractions into mixed
numbers then convert the mixed numbers back to the
improper form.
9
2
11
3
9
4
13
5
37
12
86
11
121
17
1. 2. 3. 4. 5. 6. 7.
Exercise. A. Reduce the following fractions.
4
6 ,
8
12 ,
15
9 ,
24
18 ,
30
42 ,
54
36 ,
60
48 ,
72
108