13 fractions, multiplication and divisin of fractions
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Transcript of 13 fractions, multiplication and divisin of fractions
![Page 1: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/1.jpg)
Fractions
![Page 2: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/2.jpg)
Fractions are numbers of the form (or p/q) where p, q 0 are whole numbers.
pq
Fractions
![Page 3: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/3.jpg)
Fractions are numbers of the form (or p/q) where p, q 0 are whole numbers.
pq
Fractions
36
![Page 4: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/4.jpg)
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
![Page 5: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/5.jpg)
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
![Page 6: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/6.jpg)
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
![Page 7: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/7.jpg)
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
![Page 8: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/8.jpg)
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
![Page 9: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/9.jpg)
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
![Page 10: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/10.jpg)
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
![Page 11: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/11.jpg)
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
![Page 12: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/12.jpg)
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?
![Page 13: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/13.jpg)
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
Cut the pizza into 8 pieces,
![Page 14: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/14.jpg)
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.
![Page 15: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/15.jpg)
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.
![Page 16: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/16.jpg)
For larger denominators we can use a pan–pizza for pictures. For example,
58
Fractions
712
5/8 of a pizza
![Page 17: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/17.jpg)
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,
![Page 18: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/18.jpg)
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,
![Page 19: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/19.jpg)
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.
![Page 20: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/20.jpg)
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
![Page 21: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/21.jpg)
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.
![Page 22: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/22.jpg)
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
![Page 23: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/23.jpg)
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
![Page 24: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/24.jpg)
FractionsWe may talk about the fractional amount of a group of items.
![Page 25: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/25.jpg)
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?
![Page 26: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/26.jpg)
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?
![Page 27: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/27.jpg)
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.
![Page 28: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/28.jpg)
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,
![Page 29: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/29.jpg)
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,
![Page 30: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/30.jpg)
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.
![Page 31: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/31.jpg)
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.
![Page 32: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/32.jpg)
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,
![Page 33: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/33.jpg)
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.
![Page 34: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/34.jpg)
Whole numbers can be viewed as fractions with denominator 1. Fractions
![Page 35: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/35.jpg)
Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = .
51
x1
Fractions
![Page 36: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/36.jpg)
Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x 0.
51
x1
0x
Fractions
![Page 37: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/37.jpg)
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
![Page 38: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/38.jpg)
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:
![Page 39: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/39.jpg)
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.
![Page 40: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/40.jpg)
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.)
![Page 41: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/41.jpg)
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.
![Page 42: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/42.jpg)
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
![Page 43: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/43.jpg)
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
![Page 44: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/44.jpg)
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
![Page 45: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/45.jpg)
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
![Page 46: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/46.jpg)
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
![Page 47: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/47.jpg)
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
![Page 48: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/48.jpg)
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
![Page 49: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/49.jpg)
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
![Page 50: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/50.jpg)
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
![Page 51: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/51.jpg)
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.)
![Page 52: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/52.jpg)
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.)
![Page 53: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/53.jpg)
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.)
![Page 54: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/54.jpg)
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.)
![Page 55: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/55.jpg)
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.)
![Page 56: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/56.jpg)
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.)
![Page 57: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/57.jpg)
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.)
![Page 58: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/58.jpg)
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.)
![Page 59: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/59.jpg)
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.)
![Page 60: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/60.jpg)
FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.
![Page 61: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/61.jpg)
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.
![Page 62: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/62.jpg)
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).
![Page 63: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/63.jpg)
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.
![Page 64: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/64.jpg)
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.
![Page 65: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/65.jpg)
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.
![Page 66: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/66.jpg)
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.
![Page 67: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/67.jpg)
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.
![Page 68: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/68.jpg)
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.
![Page 69: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/69.jpg)
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.
![Page 70: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/70.jpg)
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.
![Page 71: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/71.jpg)
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.
![Page 72: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/72.jpg)
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.
![Page 73: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/73.jpg)
23 4
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
![Page 74: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/74.jpg)
23 4
23 4 = 5 with remainder 3. ··
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
![Page 75: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/75.jpg)
23 4
23 4 = 5 with remainder 3. Hence, ··
23 4
= 5 +
Improper Fractions and Mixed NumbersExample C. Put into mixed form.
3 4
![Page 76: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/76.jpg)
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
=
![Page 77: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/77.jpg)
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.
![Page 78: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/78.jpg)
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
![Page 79: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/79.jpg)
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
![Page 80: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/80.jpg)
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
![Page 81: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/81.jpg)
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
![Page 82: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/82.jpg)
Rule for Multiplication of FractionsMultiplication and Division of Fractions
![Page 83: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/83.jpg)
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
![Page 84: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/84.jpg)
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
![Page 85: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/85.jpg)
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
![Page 86: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/86.jpg)
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
![Page 87: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/87.jpg)
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
![Page 88: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/88.jpg)
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
![Page 89: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/89.jpg)
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
![Page 90: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/90.jpg)
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
![Page 91: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/91.jpg)
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
![Page 92: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/92.jpg)
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
![Page 93: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/93.jpg)
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
![Page 94: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/94.jpg)
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
![Page 95: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/95.jpg)
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
![Page 96: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/96.jpg)
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
![Page 97: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/97.jpg)
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
![Page 98: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/98.jpg)
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
![Page 99: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/99.jpg)
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
![Page 100: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/100.jpg)
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
![Page 101: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/101.jpg)
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
![Page 102: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/102.jpg)
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
![Page 103: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/103.jpg)
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
![Page 104: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/104.jpg)
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
![Page 105: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/105.jpg)
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
![Page 106: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/106.jpg)
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
![Page 107: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/107.jpg)
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
![Page 108: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/108.jpg)
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
![Page 109: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/109.jpg)
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
![Page 110: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/110.jpg)
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
![Page 111: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/111.jpg)
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
![Page 112: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/112.jpg)
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
![Page 113: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/113.jpg)
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
![Page 114: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/114.jpg)
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
![Page 115: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/115.jpg)
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
![Page 116: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/116.jpg)
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
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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
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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
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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
![Page 120: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/120.jpg)
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
![Page 121: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/121.jpg)
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
![Page 122: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/122.jpg)
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
![Page 123: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/123.jpg)
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
![Page 124: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/124.jpg)
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
![Page 125: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/125.jpg)
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
![Page 126: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/126.jpg)
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
![Page 127: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/127.jpg)
The reciprocal (multiplicative inverse) of is . ab
ba
Reciprocal and Division of Fractions
So the reciprocal of is , 23
32
![Page 128: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/128.jpg)
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
![Page 129: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/129.jpg)
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
![Page 130: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/130.jpg)
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
![Page 131: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/131.jpg)
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
![Page 132: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/132.jpg)
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
![Page 133: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/133.jpg)
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,
![Page 134: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/134.jpg)
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,
![Page 135: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/135.jpg)
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,
![Page 136: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/136.jpg)
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
![Page 137: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/137.jpg)
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
![Page 138: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/138.jpg)
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
![Page 139: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/139.jpg)
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,
![Page 140: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/140.jpg)
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
![Page 141: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/141.jpg)
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
![Page 142: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/142.jpg)
Example F. Divide the following fractions.
815
= 1225
a. ÷
Reciprocal and Division of Fractions
![Page 143: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/143.jpg)
Example F. Divide the following fractions.
158
1225
*8
15 =
1225
a. ÷
Reciprocal and Division of Fractions
![Page 144: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/144.jpg)
Example F. Divide the following fractions.
158
1225
*8
15 =
1225 2
3a. ÷
Reciprocal and Division of Fractions
![Page 145: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/145.jpg)
Example F. Divide the following fractions.
158
1225
*8
15 =
1225 5
3
2
3a. ÷
Reciprocal and Division of Fractions
![Page 146: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/146.jpg)
Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a. ÷
Reciprocal and Division of Fractions
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Example F. Divide the following fractions.
158
= 1225
*8
15 =
1225 5
3
2
3 910
a.
698
÷
÷ =b.
Reciprocal and Division of Fractions
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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
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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
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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
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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. ÷
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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
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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
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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?
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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
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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
![Page 157: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/157.jpg)
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
![Page 158: 13 fractions, multiplication and divisin of fractions](https://reader035.fdocuments.net/reader035/viewer/2022081517/58f047e71a28abe3758b4601/html5/thumbnails/158.jpg)
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"
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Multiplication and Division of Fractions
Remember to cancel first!
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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?
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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