13 fractions, multiplication and divisin of fractions
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Fractions
Fractions are numbers of the form (or p/q) where p, q 0 are whole numbers.
pq
Fractions
Fractions are numbers of the form (or p/q) where p, q 0 are whole numbers.
pq
Fractions
36
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.
pq
Fractions
36
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 ofthem, the fraction that represents this quantity is .
pq
36
Fractions
36
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 ofthem, the fraction that represents this quantity is .
pq
36
36
Fractions
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 ofthem, the fraction that represents this quantity is .
pq
36
The bottom number is the number of equal parts in the division and it is called the denominator.
36
Fractions
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 ofthem, the fraction that represents this quantity is .
pq
36
The bottom number is the number of equal parts in the division and it is called the denominator.
36
Fractions
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 ofthem, the fraction that represents this quantity is .
pq
36
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.
36
Fractions
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 ofthem, the fraction that represents this quantity is .
pq
36
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.
36
Fractions
3/6 of a pizza
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
How many slices should we cut the pizza into and how do we do this?
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
Cut the pizza into 8 pieces,
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
Cut the pizza into 8 pieces, take 5 of them.
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
5/8 of a pizza
Cut the pizza into 8 pieces, take 5 of them.
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Cut the pizza into 12 pieces,
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Cut the pizza into 12 pieces,
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Cut the pizza into 12 pieces, take 7 of them.
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Cut the pizza into 12 pieces, take 7 of them.
or
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
7/12 of a pizza
or
Cut the pizza into 12 pieces, take 7 of them.
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Note that or is the same as 1.88
1212
7/12 of a pizza
or
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
Fact: aa
Note that or is the same as 1.88
1212
= 1 (provided that a = 0.)
7/12 of a pizza
or
FractionsWe may talk about the fractional amount of a group of items.
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
100 ÷ 4 = 25 so each part is $25,
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25 so each part is $25,
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25 so each part is $25,3 parts make $75. So ¾ of $100 is $75.
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25 so each part is $25,3 parts make $75. So ¾ of $100 is $75.
712
Divide 72 people into 12 equal parts.
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25 so each part is $25,3 parts make $75. So ¾ of $100 is $75.
712
Divide 72 people into 12 equal parts.
72 ÷ 12 = 6 so each part consists of 6 people,
Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.To calculate such amounts, we always divide the group into parts indicated by the denominator, then retrieve the number of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them like the show very much. How many people is that?
34 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25 so each part is $25,3 parts make $75. So ¾ of $100 is $75.
712
Divide 72 people into 12 equal parts.
Take 7 parts. 72 ÷ 12 = 6 so each part consists of 6 people,7 parts make 42 people. So 7/12 of 92 people is 42 people.
Whole numbers can be viewed as fractions with denominator 1. Fractions
Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = .
51
x1
Fractions
Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x 0.
51
x1
0x
Fractions
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.
51
x1
0x
x0
Fractions
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.
51
x1
0x
x0
Fractions
The Ultimate No-No of Mathematics:
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.
51
x1
0x
x0
Fractions
The Ultimate No-No of Mathematics:The denominator (bottom) of a fraction can't be 0.
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.
51
x1
0x
x0
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.)
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.
51
x1
0x
x0
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.
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.
51
x1
0x
x0
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. 12 =
24
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.
51
x1
0x
x0
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. 12 =
24 =
36
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.
51
x1
0x
x0
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.12 =
24 =
36 =
48
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.
51
x1
0x
x0
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.
12 =
24 =
36 =
48
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.
51
x1
0x
x0
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.
12 =
24 =
36 =
48
is the reduced one in the above list.12
Factor Cancellation RuleGiven a fraction , then
that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.
ab
ab = a / c
Fractions
b / c
Factor Cancellation RuleGiven 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,
ab
ab = a / c
Fractions
b / c
Factor Cancellation RuleGiven 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.
ab
ab = a / c
=a*cb*c
a*cb*c
1
Fractions
b / c
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
(Often we omit writing the 1’s after the cancellation.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*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.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
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.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
=
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.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
= 78/254/2
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.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
= 78/254/2
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.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
= 78/254/2
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.
= 39/327/3
39
27
(Often we omit writing the 1’s after the cancellation.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
= 78/254/2
= 139 .
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.
= 39/327/3
39
27
(Often we omit writing the 1’s after the cancellation.)
Factor Cancellation RuleGiven 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.
ab
ab = a / c
ab .=a*c
b*c =a*cb*c
1
Fractions
b / c
Example B. Reduce the fraction . 7854
7854
= 78/254/2
= 139 .
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.
= 39/327/3
or divide both by 6 in one step.
39
27
(Often we omit writing the 1’s after the cancellation.)
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
FractionsOne 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.
FractionsOne 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).
FractionsOne 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.
FractionsOne 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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
35
=
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
35
=
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
35
=
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
= 13
35
=
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
= 13
35
=
This is addition. Can’t cancel!
!? 2 * 12 * 3 = 1
3Yes
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
= 13
35
=
This is addition. Can’t cancel!
!?
Improper Fractions and Mixed Numbers
2 * 12 * 3 = 1
3Yes
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
= 13
35
=
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 * 12 * 3 = 1
3Yes
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.
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
2 + 12 + 3
= 2 + 1 2 + 3
= 13
35
=
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 * 12 * 3 = 1
3Yes
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.
23 4
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
23 4
23 4 = 5 with remainder 3. ··
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 +
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
We may put a mixed number into improper fraction by doing the reverse via multiplication.
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
We may put a mixed number into improper fraction by doing the reverse via multiplication.
Example D. Put into improper form. 5 3 4
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
5 3 4
= 4*5 + 3 4
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
We may put a mixed number into improper fraction by doing the reverse via multiplication.
Example D. Put into improper form. 5 3 4
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
5 3 4
= 4*5 + 3 4
23 4=
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
We may put a mixed number into improper fraction by doing the reverse via multiplication.
Example D. Put into improper form. 5 3 4
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 + 5 3 4 .
5 3 4
= 4*5 + 3 4
23 4=
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
=
We may put a mixed number into improper fraction by doing the reverse via multiplication.
Example D. Put into improper form. 5 3 4
Rule for Multiplication of FractionsMultiplication and Division of Fractions
cd
=a*cb*d
ab
*
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
1225
158
*a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3=
3*32*5
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
b.89
78
*1011
910
**
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** =
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** =
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** =
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** =
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Each set of cancellation produces a “1”, which does not affect final the product.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** = =711
a.
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
cd
=a*cb*d
ab
*
Example E. Multiply by reducing first.
=15 * 12 8 * 25
1225
158
*2
3
5
3= =
910
3*32*5
7*8*9*10
8*9*10*11b.
89
78
*1011
910
** = =711
a.
Can't do this for addition and subtraction, i.e.cd
= a cb d
ab
±±±
Rule for Multiplication of FractionsTo multiply fractions, multiply the numerators and multiply the denominators, but always cancel as much as possible first then multiply.
Multiplication and Division of Fractions
ab d a
bd
d1
The fractional multiplications are important.or * *
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 a.
The fractional multiplications are important.or * *
*
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 a.
The fractional multiplications are important.
6
or * *
*
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 a.
The fractional multiplications are important.
6
or * *
* *
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
or * *
* *
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.
or * *
* *
*
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
*
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Example G. a. What is of $108?23
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Example G. a. What is of $108?23
* 108 23The statement translates into
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Example G. a. What is of $108?23
* 108 23
36The statement translates into
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
or * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Example G. a. What is of $108?23
* 108 = 2 * 36 23
36The statement translates into
Multiplication and Division of Fractions
ab d a
bd
d1
Example F. Multiply by cancelling first.23 18 = 2 6 = 12a.
The fractional multiplications are important.
6
1116
48
b.3
Multiplication and Division of Fractionsor * *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelledagainst d = .
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.
Example G. a. What is of $108?23
* 108 = 2 * 36 = 72 $.23
36The statement translates into
b. 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 lemon drops?
Multiplication and Division of Fractions
b. 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 lemon drops?
For chocolate, ¼ of 48 is 14
* 48
Multiplication and Division of Fractions
b. 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 lemon drops?
For chocolate, ¼ of 48 is 14
* 48 = 12,12
Multiplication and Division of Fractions
b. 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 lemon drops?
For chocolate, ¼ of 48 is 14
* 48 = 12,12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48
For chocolate, ¼ of 48 is 14
* 48 = 12,12
For caramel, 1/3 of 48 isso there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 48 = 12,12
For caramel, 1/3 of 48 is = 16, so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 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.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 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.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 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.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 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 = 20/448/4
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
For chocolate, ¼ of 48 is 14
* 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 = 20/448/4 = 5
12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
c. A class has x students, ¾ of them are girls, how many girls are there?
For chocolate, ¼ of 48 is 14
* 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 = 20/448/4 = 5
12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
c. A class has x students, ¾ of them are girls, how many girls are there?
34 * x.
For chocolate, ¼ of 48 is 14
* 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 = 20/448/4 = 5
12
It translates into multiplication as
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
b. 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 lemon drops?
13
* 48 16
c. A class has x students, ¾ of them are girls, how many girls are there?
34 * x.
For chocolate, ¼ of 48 is 14
* 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 = 20/448/4 = 5
12
It translates into multiplication as
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
So the reciprocal of is , 23
32
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5the reciprocal of is 3, 1
3
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32* = 1,
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32* = 1, 5 1
5* = 1,
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32* = 1, 5 1
5* = 1, x 1x* = 1,
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
For example, 10 ÷ 2 is the same as 10 , *12
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
For example, 10 ÷ 2 is the same as 10 , both yield 5. *12
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
For example, 10 ÷ 2 is the same as 10 , both yield 5. *12
Rule for Division of FractionsTo divide by a fraction x, restate it as multiplying by the reciprocal 1/x , that is,
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
For example, 10 ÷ 2 is the same as 10 , both yield 5. *12
Rule for Division of FractionsTo divide by a fraction x, restate it as multiplying by the reciprocal 1/x , that is, d
cab *
cd = a
b ÷reciprocate
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.
So the reciprocal of is , 23
32 the reciprocal of 5 is , 1
5and the reciprocal of x is . 1
xthe reciprocal of is 3, 13
23
32*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 15* = 1, x 1
x* = 1,1x
For example, 10 ÷ 2 is the same as 10 , both yield 5. *12
Rule for Division of FractionsTo divide by a fraction x, restate it as multiplying by the reciprocal 1/x , that is, d
c = a*db*c
ab *
cd = a
b ÷reciprocate
Example F. Divide the following fractions.
815
= 1225
a. ÷
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
1225
*8
15 =
1225
a. ÷
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
1225
*8
15 =
1225 2
3a. ÷
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
1225
*8
15 =
1225 5
3
2
3a. ÷
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a. ÷
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
÷
÷ =b.
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
÷
÷ = * b.
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
÷
÷ = * b.
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
165d. ÷
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61*1
6 = 5d. ÷ 5
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookiescan we make?
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookiescan we make?
We can make 34 ÷ 1
16
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookiescan we make?
We can make 34 ÷ 1
16 = 34 *
161
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookiescan we make?
We can make 34 ÷ 1
16 = 34 *
161
4
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
198 6
3
2
316
÷
÷ = * = b.
Reciprocal and Division of Fractions
61 = 30 *1
6 = 5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookiescan we make?
We can make 34 ÷ 1
16 = 34 *
161 = 3 * 4 = 12 cookies.
4
HW: Do the web homework "Multiplication of Fractions"
Multiplication and Division of Fractions
Remember to cancel first!
Multiplication and Division of FractionsExercise. B. 12. In a class of 48 people, 1/3 of them are boys, how many girls are there?13. In a class of 60 people, 3/4 of them are not boys, how many boys are there?14. In a class of 72 people, 5/6 of them are not girls, how many boys are there?15. In a class of 56 people, 3/7 of them are not boys, how many girls are there?16. In a class of 60 people, 1/3 of them are girls, how many are not girls?17. In a class of 60 people, 2/5 of them are not girls, how are not boys?18. In a class of 108 people, 5/9 of them are girls, how many are not boys?A mixed bag of candies has 72 pieces of colored candies, 1/8 of them are red, 1/3 of them are green, ½ of them are blue and the rest are yellow.19. How many green ones are there?20. How many are blue?21. How many are not yellow?20. How many are not blue and not green? 21. In a group of 108 people, 4/9 of them adults (aged 18 or over), 1/3 of them are teens (aged from 12 to 17) and the rest are children. Of the adults 2/3 are females, 3/4 of the teens are males and 1/2 of the children are girls. Complete the following table.22. How many females are there and what is the fraction of the females to entire group?23. How many are not male–adults and what is the fraction of them to entire group?
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.46 ,
812 ,
159 ,
2418 ,
3042 ,
5436 ,
6048 ,
72108
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