FRACTIONS MATHEMATICS Bombay Cambridge Gurukul FRACTIONS.
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Transcript of FRACTIONS MATHEMATICS Bombay Cambridge Gurukul FRACTIONS.
FRACTIONS
MATHEMATICS
Bombay Cambridge Gurukul
FRACTIONS
Standard IV
Standard III
Standard V
Choose the level
What are fractions?
Parts of a collection
RevisionMore about fractions……numerator and denominator
More about fractions…III
Back
How to read fractions?
Back
Equivalent fractions
Types of fractionsFraction as divisionMixed numbersComparison of fractionsAddition of like fractionsSubtraction of like fractions
IV
Addition of……unlike fractions, mixed numbersSubtraction of……unlike fractions, mixed numbersMultiplication of fractions
Reciprocal of a fractionDivision of fractions
V
Back
Reduced form of fractionsFactors and Multiples
Standard III
What are
fractions?
Look at the figure given below.
We can divide itinto 2 equal partsby drawing a line.
Shade only one part of the figure.
Each part is called one half of the whole.
It is a whole figure.
We write it as 12
12
12
Look at the figure given below.
We can divide itinto 2 equal partsby drawing a line.
Shade only one part of the figure.
Each part is called one half of the whole.
It is a whole figure.
We write it as 12
12
12
Back
How to read fractions?
How to read a fraction?
12
is read as 1 upon 2 or 1 by 2.
37
25
79
is read as
is read as
is read as
3 upon 7 or
2 upon 5 or
7 upon 9 or
3 by 7.
2 by 5.
7 by 9.
Back
More about fractions…
The following figures are divided into two equal parts.
When a whole is divided into two equal parts,
Two halves make a whole.
12
One half is written as
each part is called half of the whole.
12
12
12
12whole whole
Each figure is divided into two parts.Are both parts equal?
Yes Yes No Yes
No No Yes No
Which of the following figures are divided into two equal parts?
The following figures are divided into three equal parts.
When a whole is divided into three equal parts,
13
One third is written as
each part is called one third of the whole.
13
13
13
13
13
13
Each figure is divided into three parts.Are all the three parts equal ?
No Yes Yes No
Yes No No Yes
Which of the following figures are divided into three equal parts?
When a whole is divided into four equal parts,
14
One fourth is written as
each part is called one fourth of the whole.
The following figures are divided into four equal parts.
14
14
14
14
14
14
14
14
YesNo YesNo
Each figure is divided into four parts.Are all the four parts equal?
NoYes NoYes
Which of the following figures are divided into four equal parts?
Draw a line or lines to divide each of the
following shapes into:
two equal partsfour equal partsthree equal parts
Shade half (1/2) of each shapeShade one fourth (1/4) of each shapeShade one third (1/3) of each shape
12
12
13
13
131
4
14
14
14
12
12
13
13
13
14
14
14
14
Look at the figure given below:
23
The fraction for the shaded part is
It has 3 equal parts.2 parts are shaded.
It is read as two third.
34
The fraction for the shaded part is
It has 4 equal parts.3 parts are shaded.
It is read as three fourth.
Back
Match the following12
13
34
14
23
One fourth
One third
One half
Two third
Three fourth
Parts of a
collection
The box given below has 12 stars.They can be divided into 2 equal parts.
Each part has 6 stars.
6 6
To find the number of objects in one half of a collection, we
divide the total number of objects by 2.
They can be divided into 3 equal parts.
Each part has 4 stars.
4
4
4
The box given below has 12 stars.
To find the number of objects in one third of a collection, we
divide the total number of objects by 3.
They can be divided into 4 equal parts.
Each part has 3 stars.
3 33 3
The box given below has 12 stars.
To find the number of objects in one fourth of a collection, we
divide the total number of objects by 4.
Encircle one half(
43
6
One half of 12 is 6One fourth of 12 is 3One third of 12 is 4
Total number of insects shown below is 12.How many insects are there in 1
2of the collection?
=12 2 612)of each collection.
How many insects are there in 13
of the collection? How many insects are there in 14
of the collection?
Encircle one third( 13)of each collection.Encircle one fourth(14)of each collection.
=12 3 4=12 4 3
Colour one half of the collection.Colour one fourth of the collection.Colour one third of the collection.
Back
Revision
How many equal parts is each rod divided into?
2 equal parts
4 equal parts
3 equal parts
5 equal parts
What fraction do the colored portions in each of the following show?
25
34
14
23
Match the following fractions to the figures.59
28
15
67
26
46
59
28
15
67
26
46
HALF
QUARTER (ONE FOURTH)
THREE QUARTERS(THREE FOURTH)
ONE THIRD
WHOLE 1
12
13
34
14
TWO THIRD 23
Back
More about fractions…
…numerator and denominator
PARTS OF A WHOLE ARE CALLED FRACTIONS.
e.g.
12
Parts considered
Total number ofequal parts
NUMERATOR
DENOMINATOR
FRACTIONNUMERATOR
DENOMINATOR=
3838
Remember : Letter‘u’ is in the word ‘numerator’ and the word ‘ up’ .
Remember : Letter ‘d’ starts the word denominator’ and the word ‘down’ .
So, in the fraction , 3 is the numerator.38
So, in the fraction , 8 is the denominator.3 8
Write the numerator and denominator for each of the following fractions.
Fraction Numerator Denominator23
34
15
57
23
34
15
57
Write the fraction for the numerator and denominator given below.
Numerator Denominator Fraction
15
47
34
58
15
47
34
58
15
47
34
58
15
47
34
58
15
47
34
58
Write the fraction for the shaded part.
Numerator
Denominator
Fraction
Numerator
Denominator
Fraction
5
58
510
5
810
(shaded parts) (shaded parts)
(total parts)
(total parts)
The EndThe End
BOMBAY CAMBRIDGE GURUKULBack
Standard IV
Equivalent fractions
Is the shaded part in each pair of figures same?
Yes YesYes
Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that =
Yes
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =
Fractions which are equal in value to each other are calledequivalent fractions.
is equivalent to12
24
is equivalent to34
68
e.g.
Match the following equivalent fractions.
33
28
24
26
13
14
12
1
Back
Types of fractions
27
35
14
49
etc.
Fractions where the numerator is smaller than the denominatorare called
proper fractions.
e.g.
43
72
94
87
etc.
Fractions where the numerator is greater than the denominatorare calledimproper fractions.
e.g.
Fractions which have same denominator are calledlike fractions.
29
39
59
49
e.g. etc.
Fractions which have different denominators are calledunlike fractions.
23
34
59
45
e.g. etc.
Fractions which have numeral 1 as numerator are calledunit fractions.
13
14
19
15
e.g. etc.
Back
Fraction as
division
We can write each division sum as a fraction. 4
12=4 12
36
=3 6
15
=1 5
710
=7 10
We can write each fraction as a division sum.18 = 1 8
69 = 6 9
412 = 4 12
29 = 2 9
Back
Mixed numbers
Mixed numbers include a whole number and a fraction.
=+
(whole number) (fraction) (mixed number)+ =
Converting mixed numbers to improper fractions.
Convert 12
4 to a improper fraction.
Step 1 : Multiply the denominator 2 with whole number 4. =2 4 8
Step 2 : Add numerator 1 to 8 =1 8 9
Step 3 : Write 9 as the numerator of the improper fraction.
9
Step 4 : Write denominator 2 as the denominator of the improper fraction.
92
12
4 = 92(improper fraction)(mixed number)
Converting improper fractions to mixed numbers.
Convert 73
to a mixed number.
Step 1 : Divide 7 by 3.
Step 2 : Write the mixed number.
The quotient becomes the whole number.
Divisor : 3
Quotient: 2
Remainder : 1
The divisor becomes the denominator.
The remainder becomes the numerator.
73
13
2=(improper fraction) (mixed number)
**
2
*3
2
13
2
Improper fractions
Mixed numbers
Improper fractions
Mixed numbers
can be changed to
Converting improper fractions to mixed numbers.
can be changed to
Back
Comparison of fractions
like fractionsHow to compare like fractions ?
Look at the figures shown below.
Each figure is divided into 4 equal parts.
Which figure has more shaded parts?
The first figure (A) has more shaded parts.
(A) (B)
like fractionsHow to compare like fractions ?
Look at the figures shown below.Write the fraction for both figures.
26
46
Which fractions has more shaded area?
So, we can say that 46
26
>
46
like fractionsHow to compare like fractions ?
Look at the figures shown below.Write the fraction for both figures.
27
67
Which fraction has less shaded area?
So, we can say that 27
67
<
27
like fractions
If there are two like fractions, then the fraction with greater numerator is greater in value.
If there are two like fractions, then the fraction with smaller numerator is lesser in value.
How to compare like fractions ?
e.g. 47
37
e.g. 89
29
>
<
Compare the following using < , > or = .
45
37
29
46
15
67
29
36
>
<
=
>Back
Addition oflike fractions
Addition of like fractions
Two more parts of the circle are shaded.
In the circle given below only one part out of five is shaded.
The circle has three shaded parts.
Addition of like fractions
14
24
34
14
24
34
=+
=+
Addition of like fractions
26
36
56
=+
=+
26
36
56
Addition of like fractions
13
13
23
=+
+
13
13
23
When two or more like fractions are added, then only the numerators are added together.
The denominators are not added together.
Addition of like fractions
Addition of like fractions
The answer should be written in the reduced form of fractions.
+
+
4
4
Addition of like fractions
39
39
69=+
58
28
78=+
17
27
37=+
25
25
45=+
Back
=
=
=
=2 + 25
1 + 27
5 + 28
3 + 39
Subtraction oflike fractions
Subtraction of like fractions
In the figure given below, three parts out of five parts are shaded.
Two parts are taken away.
One part out of five is left.
Subtraction of like fractions
34
24
14
=-
Two parts are taken away.
In the figure given below, three parts out of four are shaded.
One part out of four is left.
Subtraction of like fractions
When two like fractions are subtracted, then the smaller numerator is subtracted from the bigger numerator.
The denominators are not
subtracted.
Subtraction of like fractions
The answer should be written in the reduced form of fractions.
612
412
212
=- 2
2 = 16
1416
216
1216
=-2
2 = 68
2
2 = 34
79
59
29
=-
68
18
58
=-
57
27
37
=-
36
26
16
=-
Subtraction of like fractions
=
=
=
= 3 - 26
5 - 27
6 - 18
7 - 59
The EndThe End
BOMBAY CAMBRIDGE GURUKULBack
Standard V
Reduced form of fractions
A fraction is said to be in the reduced form if its numerator and denominatorcannot be divided by a common number.
Reduced form of fractions
Look at the fraction given below.
We can divide the numerator and denominator both by 2.
68
22
68
=So,
Now we can not divide 3 and 4 both by any number.
So, we can say that 34is the reduced form of 6
8
2
234
=68
6 divided by 2 is 3.8 divided by 2 is 4.
=
Reduce the given fraction to its lowest form.
39
3
313
We can divide both, the numeratorand the denominator by 3.
The reduced form of 39
13
is
=1012
2
256
We can divide both, the numeratorand the denominator by 2.
The reduced form of 1012
56
is
1012
39
28
14
Circle the fractions which are in the reduced form.
39
56
57
412
35
918
38
614
49
812
Back
3
3
2
2
9
9
2
2
2
2
4
4
Factors and Multiples
A number that divides a given number completely (without leaving a remainder) is called its
factor.
e.g. 5 divides 20 exactly.
So, 5 is a factor of 20.
And 20 is a multiple of 5.
Is 20 exactly divisible by 3? No
Is 3 a factor of 20? NoIs 20 a multiple of 3? No
List the numbers that divide 15 exactly.
1 53 15
So, we can say that factors of 15 are 1, 3, 5 and 15.
List the numbers that divide 12 exactly.
1 2 3 4 6 12
So, we can say that factors of 12 are 1, 2, 3, 4, 6 and 12.
Every number has at least 2 factors :the number itself.
1
1
15
12
1 and
Which of the following are factors of 16?
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16
1
2 4 8 16
Is 4 a factor of 14 ? No
Is 6 a factor of 24 ?
Is 4 a factor of 32 ?
Is 3 a factor of 17 ?
Yes
Yes
No
Try the following….
Which of the following are multiples of 4 ?8, 10, 12, 14, 16, 18,
20, 22, 24, 26, 28, 30
8
12 16 20 24
Is 15 a multiple of 6 ? No
Yes
Yes
No
28
Is 28 a multiple of 7 ?
Is 24 a multiple of 8 ?
Is 21 a multiple of 9 ?
Try the following….
Common factors
The factors of 24 are :
1, 2, 3, 4, 6, 8, 12 and 24.
The factors of 30 are :
1, 2, 3, 5, 6, 10, 15 and 30.
Common factors of 24 and 30 are :
1,
Highest common factor (H.C.F.) of 24 and 30 is : 6
2,
3,
6
Common multiples
The multiples of 3 are :
3, 6, 9, 12, 15, 18, 21, 24 …
The multiples of 4 are :
4, 8, 12, 16, 20, 24, 28 ...
Common multiples of 3 and 4 are :
12 ,
Least common multiple (L.C.M.)of 3 and 4 is :
12
24 …
Back
Addition of……unlike fractions,
mixed numbers
When we add two unlike fractions(with different denominators),
we need to find the least common multiple ( L.C.M.)
of the two denominators.
Addition of unlike fractions
To change both fractions to like fractions, we find the L.C.M. of 2 and 6.
Multiples of 2 are :
Multiples of 6 are :
2, 4, 6, 8, 10, 12…
6, 12, 18, 24, 30…
Common multiples of 2 and 6 are : 6, 12…
Least common multiple (L.C.M.)of 2 and 6 is : 6
16
1 2
+
Now we can addThe denominator of both the fractions is the same as the L.C.M.
12
16
+ L.C.M. of 2 and 6 is 6.
________6
=
Divide the common denominator with the denominator of the first fraction.
Step 1 :
Step 2 :
Step 3 :
Multiply 3 with the numerator of the first fraction.Write 3 in place of the first numerator.
Step 4 : = 3 +
6
6 2 = 3
3 1 = 3
Divide the common denominator with the denominator of the second fraction.
Step 5 :
Step 6 :
Step 7:
Add the numeratorsStep 8 :
= 3 + 1 6
Multiply 1 with the numerator of the second fraction.
Write 1 in place of the second numerator.
= 46
12
16
+So,
= 46
6 6 = 1
1 = 11
Addition of unlike fractions
=
The denominators are different,so, we find the L.C.M. of 2 and 4.
L.C.M. of 2 and 4 is 4.
Then, numerators are added.
Addition of mixed numbers
Change the mixed number to an improper fraction.
Add both the fractions.
15
4 25
=
+
15
4
Back
4 5 + 1
5215
=
Step1:
Step 2:
25
+215
= 21 + 25
= 235
So, 15
4 25
+ = 235
(21 + 2) 5 5
Subtraction of……unlike fractions,
mixed numbers
When we subtract two unlike fractions (with different denominators),
we need to find the least common multiple ( L.C.M.)
of the two denominators.
Subtraction of unlike fractions
To change both fractions to like fractions, we find the L.C.M. of 3 and 6.
Multiples of 3 are :
Multiples of 6 are :
3, 6, 9, 12, 15, 18…
6, 12, 18, 24, 30…
Common multiples of 8 and 4 are : 6, 12…
Least common multiple (L.C.M.)of 8 and 4 is : 6
16
2 3
_
Now we can subtract
The denominator of both the fractions is the same as the L.C.M.
23
16
- L.C.M. of 3 and 6 is 6.
________6
=
Divide the common denominator with the denominator of the first fraction.
Step 1 :
Step 2 :
Step 3: Multiply 2 with the numerator of the first fraction.Write 4 in place of the first numerator.
Step 4 :
= 4 - 6
6 3 = 2
2 2 = 4
Divide the common denominator with the denominator of the second fraction.
Step 5 :
Step 6 :
Step 7:
Subtract the numeratorsStep 8 :
= 4 - 1 6
Multiply 1 with the numerator of the second fraction.
Write 1 in place of the second numerator.
= 36
23
16
-So,
= 36
1 1 = 1
6 6 = 1
Subtraction of unlike fractions
Then, numerators are subtracted.
The denominators are different,so, we find the L.C.M. of 2 and 4.
L.C.M. of 2 and 4 is 4.
Subtraction of mixed numbers
Change the mixed number to an improper fraction.
Subtract both the fractions.
=
27
3 27
-
27
3
Back
3 7 + 2
7237
=
Step 1:
Step 2:
27
-237
= 23 - 27
= 217
So, 27
3 27
- = 217
(23 - 2) 7 7
Multiplicationof fractions
How to multiply a fraction by a whole number ?58
4
We multiply only the numerator of the fraction with the whole number.
The denominator remains the same.
= 20 8
45 8
We should write the answer in the reduced form of fractions.20 8
22
= 10 4
22
= 52
So, 58
4 = 52
How to multiply a whole number by a fraction ?
69
We multiply the whole numberonly with the numerator of the fraction.
The denominator remains the same.
= 30 9
65 9
We should write the answer in the reduced form of fractions.
30 9
33
= 10 3
So, 69 =
10 3
5
5
How to multiply a fraction by a fraction ?
67
We multiply both the numerators.
= 1221
We should write the answer in the reduced form of fractions.
1221
33
= 47
So, =47
23
And we multiply both the denominators. 62 73
672
3 Back
Reciprocalof a fraction
How to write a reciprocal fraction ?
The numerator becomes the denominator.
And the denominator becomes the numerator.
79
Fraction Reciprocal fraction
97
The reciprocal of 17
is 71
The reciprocal of 7 or is 17
or 7
71
The reciprocal of a unit fraction is a whole number.
The reciprocal of a whole number is a unit fraction.
REMEMBER
Back
Divisionof fractions
How to divide a whole number by a fraction ?68
4
=
We change the division sign to multiplication.
4 Then we write the reciprocal of the second fraction.
4 86
Multiply the numerators.32 6Reduce the fraction to its lowest form. 3
2 6
16 3
2
2
So, 68
4 = 16 3
How to divide a fraction by a whole number ?45
4
=
We change the division sign to multiplication.
Then we write the reciprocal of the whole number.4
5Multiply the numerators.
Reduce the fraction to its lowest form. 420
2
2
So, =
45
14
420 2
10
2
2
= 15
45
4 15
How to divide a fraction by a fraction ?48
=
We change the division sign to multiplication.
Then we write the reciprocal of the second fraction.4
8Multiply the numerators and the denominators.
Reduce the fraction to its lowest form. 1216
2
2
So, =
48
32
12 16 6
8
2
2
= 34
48
34
23
23
The EndThe End
BOMBAY CAMBRIDGE GURUKUL