Download Unit 3 Introduction to Rational Number Class - VII - CBSE

CLASS
VII
CBSE-i
Introduction to
Rational
Numbers
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Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India
UNIT-3
CLASS
VII
UNIT-3
CBSE-i
Mathematics
Introduction to
Rational
Numbers
Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India
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ACKNOWLEDGEMENTS
Advisory
Conceptual Framework
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Contents
1.
Study Material
1
2.
Student's Support Material
34
C
SW 1: Warm Up Activity (W1)
35
• Recall Fractions
C
SW 2: Warm Up Activity (W2)
36
• Types of Fractions
C
SW 3: Warm Up Activity (W3)
37
• Arithmetical Operations on Fractions
C
SW 4: Pre Content Worksheet (P1)
39
• Need of Rational Numbers
C
SW 5: Pre Content Worksheet (P2)
40
• Another Number System
C
SW 6: Content Worksheet (C1)
41
• Defining Rational Numbers
C
SW 7: Content Worksheet (C2)
43
• Rational Numbers and Fractions
C
SW 8: Content Worksheet (C3)
45
• Standard form of Rational Numbers
C
SW 9: Content Worksheet (C4)
46
• Rational Numbers on Number Line
C
SW 10: Content Worksheet (C5)
51
• Comparison of Rational Numbers
C
SW 11: Content Worksheet (C6)
53
• Rational Numbers Between given Rational Numbers
C
SW 12: Content Worksheet (C7)
55
• Addition and Subtraction
C
SW 13: Content Worksheet (C8)
• Skill Drill 1
56
C
SW 14: Content Worksheet (C9)
58
• Skill Drill 2
C
SW 15: Content Worksheet (C10)
60
• Multiplication of Rational Numbers
C
SW 16: Content Worksheet (C11)
61
• Multiplication Skill Drill
C
SW 17: Content Worksheet (12)
64
• Division of Rational Numbers
C
SW 18: Content Worksheet (C13)
67
• Rational Numbers as Decimals
C
SW 19: Content Worksheet (C14)
69
• Real Life Problems
C
SW 20: Post Content Worksheet (PC1)
73
• Rational Numbers Drill
C
SW 21: Post Content Worksheet (PC2)
78
• Test Your Progress
7.
Suggested Videos/ Links/ PPT's
83
1
INTRODUCTION TO RATIONAL NUMBERS
Introduction
You have already learnt about natural numbers, whole numbers, integers and fractional
numbers (or fractions) along with fundamental operations on them.
Recall that we had extended the collection of whole numbers to integers to represent the
situations like ‘profit and loss’, ‘temperatures below and above oC’, ‘height below and
above sea level’ etc., by including negative numbers like.., 4, 3, 2, 1,… etc.
Recall that in daily life the quantities above cannot always be expressed in whole
numbers above and that is why, we had to introduce fractions’ such as , ,
,
, etc.,
In this unit, we shall further extend these numbers to new type of numbers called
rational numbers by introducing numbers such as
, ,
, etc., to handle
opposite situations corresponding to situations represented by fractions
, ,
, etc. We
shall also study fundamental operations on rational numbers and their use in our day
today life.
1. Fractions – A Review
Recall that fractions are numbers of the type ,
,
,
, etc, which can be written in
the form , where p and q are natural numbers. In a fraction , p is called the numerator
and q is called the denominator.
Proper fraction
A fraction is called a proper fraction if the numerator is less than the denominator.
The fractions , ,
,
, etc., are examples of proper fractions.
Improper fraction
A fraction is called an improper fraction if the numerator is greater than or equal to
the denominator. The fractions ,
,
,
, etc., are example of improper fractions.
Mixed fractions
A combination of a natural number and proper fraction is called a mixed fraction or
mixed number.
Examples of mixed fractions are
2
2 , 1 , 69
, 101
2 is same as
1
, etc.
or
is same as
(improper fraction)
or
(improper fraction) etc.
Thus, every mixed fraction can be converted into an improper fraction.
Conversely, every improper fraction (with numerator greater than denominator) can
be expressed as a mixed fraction. For example,
=2 ,
=2
,
=1
,
etc
Unit fraction
A fraction whose numerator is 1, is called a unit fraction.
For example, ,
,
,
, etc, are unit fractions.
Note that fractions such as , , etc, are not unit fractions.
Like fractions
Fractions having the same denominators are called like fractions.
For example,
, ,
,
, etc
are like fractions. Similarly,
,
,
,
,
are also like fractions.
Unlike fractions
Fractions having different denominators are called unlike fractions.
For example,
, ,
, , etc.
3
are unlike fractions.
Simplest or Lowest form of a fraction
A fraction in which numerator and denominator have no common factor other than
1 (i.e., numerator and denominator are coprime) is said to be in its simplest or
lowest form.
For example,
,
,
, , etc., are fraction in simplest form.
Equivalent fractions
Fractions having same simplest form are called equivalent fractions.
,
For example.
, , ,
,
are equivalent fractions because each of there can be expressed in the simplest form
as .
=
=
=
=
=
=
=
etc. Similarly, the fractions
,
,
,
4
are equivalent fractions as
=
=
=
=
=
=
,
etc, but
,
,
are not equivalent fractions as
=
=
=
= , which is not equal to .
but
Recall that we can obtain a fraction equivalent to a fraction
by multiplying (or
dividing) p and q by same non zero numbers.
Operations on fractions
(i)
Addition
Sum of two like fractions is a fraction whose numerator is the sum of numerators of
given fractions and denominator is the same as that of given fractions, i.e.,
+ =
For example, + =
+
=
=
=
For adding unlike fractions, we first convert them into like fraction and then add as
above.
+ =
=
=
+
+
=
=1
5
+
=
=
+
=
[Note that denominator
of each fraction has
been made 60 which is
LCM of 15 and 20]
+
=
This can be extended to addition of more than two fractions.
(ii)
Subtraction
Like fractions
=
[Subtracting numerators keeping denominator same]
=
Unlike fractions
=
[converting fractions into like fractions]
=
=
(iii)
Multiplication
Product of two (or more) fractions
=
For example,
×
(iv)
=
=
× ×
=
1 × ×
=
=1
=
× ×
=
=
=6
Division
To divide a fraction by another fraction, we multiply the first fraction by the
reciprocal of the second fraction.
6
For example,
=
of
×
=
=
=
3
=
=
=
× reciprocal
x reciprocal of
×
=
=
2. Rational Numbers
Need for further extention of numbers
Recall that sum and difference of two fractions is always a fraction. For example.
+
=
=
–
Again
What about
Fraction
=
–
=
Fraction
?
Is it a fraction? Think!!
If yes, find that fraction.
If no, what to do then?
Now, Suppose we have the equation
3x – 6 = 0
You can verify that 2 is a solution of this equation which is a whole number.
7
If we have the equation
3x + 6 = 0
You can verify that ‘ 2’ is a solution of this equation which is an integer but not a whole
number. That is
3x + 6 = 0
cannot be solved in whole numbers but can be solved in integers.
If we have the equation
3x -7 = 0
You can verify that is a solution of this equation, which is a fraction.
Thus, this equation has no solution in integers but can be solved in fractions.
If we have the equation
3x + 7 = 0
try to find a fraction which is its solution.
Can you find such fraction?
If yes, find the fraction.
If no, what to do?
In both the above situations, you are not able to find fractions providing answers.
To overcome this difficulty there is a need to extend the number system further by
including numbers like
,
,
,
, … etc.
You may call such numbers as ‘negative fractions’.
The extended collection so obtained is known as collection of rational numbers.
In general
The numbers which can be expressed in the form
q
, where p and q are integers and
0 are called rational numbers.
p is called its numerator and q is called its denominator.
8
,
Thus, the numbers
,
,
,
,
,
,
, etc., are all rational
numbers.
From the above definition, you can observe:
(a) Every whole number is a rational number because it can be written in the form
when p, q are integers and q
For example 0 =
,
0
, 1= , 2 = , 3 = , 4 = , etc
(b) Every integer is a rational number because it can be written in the form
q are integers and q
For example, -4 =
, where p,
0.
, -11 =
,3= ,0=
, etc.
(c) Every fractions is a rational number.
What about converser of (a), (b) and (c)? Clearly, their converses are not true. That is
(a’) Every rational number is not a whole number for example,
but it is not a whole number.
(b’) Every rational number is not an integer. For example,
also a rational number but none of these is an integer.
(c’) Every rational number is not a fraction. For example,
is a rational number
is a rational number,
is
is a rational number but not
is a rational number but not a fraction.
a fraction. Similarly,
Positive and Negative Rational Numbers
In a rational number , if both p and q are of the same sign i.e., both positive or both
negative, then it is called a positive rational number,
If p and q are of opposite signs, then rational number p/q is called a negative
rational number.
For example,
and
,
,
,
,
,
,
, etc., are positive rational numbers
, etc , are negative rational numbers
Rational Number 0 (i.e., ,
,
, etc., is neither positive nor negative.
9
Example 1:
,
,
Which of the following rational numbers are fractions?
,
,
,
,
,
, , -5 , ,
,
Solution: Only following rational numbers are fractions:
, , ,
3. Standard form of a Rational Number
A rational number is said to be in standard form if
(i)
Its denominator is positive, (q > 0)
(ii)
there is no common factor other than 1 of numerator and denominator, i.e.,
HCF of p, q = 1
For example rational numbers
The rational numbers
,
,
,
,
,
, etc., are in standard form.
, etc., are not in standard form as
in
, q is not positive although HCF of 3, 2 is 1.
in
, q > 0, but HCF of 8, 16 is 8 not 1
in
, q > 0 but HCF of 15, 30 is 15 not 1
Example 2: Convert the following rational numbers in standard form:
(i)
(ii)
(iii)
(iv)
10
Solution:
=
(i)
=
[Denominator has been made positive by multiplying
numerator and denominator by same number -1]
(ii)
In the rational number
, the denominator is positive but HCF of 16 and 20 is
4 (not 1)
So, we divide both numerator and denominator by 4.
=
(iii)
In
=
, which is in standard form.
, denominator is not positive. Also
HCF of 10 and 2, not 1.
So, we have
=
=
=
= ,
Which is in standard form
=
(iv)
=
=
=
,
which is in standard form.
4. Equivalent Rational Numbers
Two rational numbers are said to be equivalent rational numbers if they have the
same standard form.
For example
and
are equivalent rational numbers because
=
and
=
i.e., they have the same standard form.
Similarly,
,
,
,
,
are equivalent rational numbers because standard form of each of them is
11
.
Conversion of a rational number into equivalent rational number
Recall how to convert a fraction into an equivalent fraction. Likewise, a rational
number can be converted into an equivalent rational number by multiplying (or
dividing) both the numerator and denominator by the same non zero number.
For example,
=
So,
=
is equivalent to the rational number
=
Similarly,
So,
Also,
=
is also equivalent to
is equivalent to
=
Example 3 : Convert each of the following rational numbers into an equivalent rational
number.
(i)
(ii)
(iii)
Solution:
(i)
=
(ii)
=
(iii)
=
Example 4: Write
There can be infinitely many
equivalent rational numbers
to a given rational number
=
=
=
in an equivalent rational number form so that.
(i)
its numerator is 60
(ii)
its denominator is 15
12
Solution:
=
(i)
=
[Multiplying both numerator and denominator by 30
so that numerator becomes 60]
(ii)
=
=
[Multiplying both numerator and denominator by
( 3), so that denominator becomes -15]
(iii)
5. Operations on Rational Numbers
The operation of addition, subtraction, multiplication and division of rational
numbers are similar to that of integers and fractions which you have already
studied. We explain these through examples.
(i)
Addition
(a) Addition of rational numbers with same denominator.
Let two rational number be
+
and
=
Adding numerators
Keeping denominator same
=
Similarly
[Adding integers (-19) and 5]
=
[Dividing numerator and denominator by 2]
=
[Rational number in standard form]
+
=
Adding numerators
Keeping denominator same
=
In general, to add two rational numbers and
+ =
The above rule can be extended to more than two rational numbers.
13
Example 5: Add
and
Solution: We write
+
=
+
=
=
=
,
Example 6: Add
and
Solution: We write
=
Now,
+
+
=
+
+
=
=
=
(b) Addition of rational numbers with different denominators
Let two rational numbers be and
+
=
+
[Making the denominators same]
+
=
=
[Adding numerators and keeping denominator same]
=
Thus, +
=
14
Similarly, let us add
+
=
=
=
and
+
=
+
[Making the denominator = 15 = LCM of 15 and 3]
+
=
=
=
Again consider two rational numbers
+
=
+
,
[Making the denominators same i.e., 30 which is LCM
of 10 and 15]
+
=
=
=
You may also do it as follows:
+
=
=
+
+
=
=
=
You may note that it is always convenient to add two rational numbers with different
denominators, by converting their denominators into LCM of two denominators.
15
In general,
To add two (or more) rational numbers with different denominators, we first convert
them into equivalent rational numbers with common denominator equal to their LCM
and then add as done before for adding rational numbers with same denominator.
, and
Example 7: Add
+ +
Solution: =
+
=
=
+
+
[LCM of 3, 9 and 12 = 36]
+
=
=
=
(ii)
Subtraction of Rational Numbers
(a) Subtraction of rational numbers with same denominator
Let two other rational numbers be
– =
and
[Subtracting 2nd numerator from first numerator keeping
same denominator]
=
Let us go back to the problem of subtracting
We have
–
=
from
faced by us in (2) above.
[Subtracting 2nd numerator from first numerator
keeping the denominator same]
=
Similarly,
–
=
[ Subtracting 2nd numerator from first numerator,
keeping denominator same]
16
=
=
and
-
=
=
=
= -1
In general, if and are two rational numbers
– =
Example 8: Subtract
from
Solution:
–
=
=
=
Example 9: Subtract
Solution:
So,
–
=
from
(Making denominator positive)
=
=
=
=
(b) Subtraction of rational numbers with different denominators
Here again, we convert the given rational numbers into equivalent form with
same denominator, preferably their LCM. We explain it through examples.
Example 10: Subtract from .
Solution:
– =
–
[Making denominator same i.e., 15 = LCM of 5 and 3]
17
=
–
=
=
Example 11: Subtract
from
Solution:
=
[LCM of 24 and 36 is 72]
=
=
=
Suppose we add
Similarly,
and
+
=
+
=
. We have
=
=0
=0
In such case , we say that
additive inverse of
.
Similarly, additive inverse of
+
+
In general, if
then
is additive inverse of rational number
is
=
=
and that of
=
is
=0
=0
and are two rational numbers such that
is called the additive inverse of and vice versa.
=
In Example 11,
=
, as
+ additive inverse of
+
18
+ = + =0
and
is
=
+
=
=
Thus, to subtract
,
from
we can add the additive inverse of
to
.
In general,
– = + additive inverse of
+
=
(iii)
Multiplication of Rational Numbers
You already know that for two fractions
product of fractions =
For example,
=
=
=
=
=
[diving numerator and denominator by HCF of 450 and 180
i.e., 90]
=
The above rule of multiplication of two rational numbers can be extended to more than
two rational numbers.
Example 12: Multiply -2,
and
Solution: We have
19
=
=
=
(iv)
=
= 4
Division of Rational Numbers
Recall that to divide a fraction by another fraction, we multiply the first
fraction by the reciprocal of the second fraction.
We follow the same rule for dividing a rational number by a non-zero
rational number.
As
= 1 and
4 = 1., is the
For example, to divide
reciprocal of 4 and 4 is the reciprocal
of .
by
, we multiply
by the reciprocal of
=
Similarly, as
i.e.,
the reciprocal of
reciprocal of
=
=
=
= 1;
and
is reciprocal
of .
=
=
Reciprocal of
is
as
=1
Example 13: Divide
Solution:
=
reciprocal of
=
=
=
=
20
is
6. Representation of a Rational Number as a Decimal
Recall that a fraction can be represented as a decimal
For example, = 0.5
= 0.75
0.75
28
20
20
0
= 1.8
1.8
5
5
40
40
0
Similarly, a rational number can represented in the form of a decimal.
For example,
= -0.5
=
=
0.75
= 1.8
Thus, if a rational number is negative, we first find decimal representation of
corresponding fraction and then assign sign ( ) to the decimal.
Let us find decimal representation of rational number .
By long division,
21
= 0.3333…
0.333
9
10
9
10
9
10
9
1
Here the decimal representation is not ending as each time, we get remainder 1.
Also, the digit 3 in the decimal representation is repeating.
We say that decimal representation of rational number is
(i)
Non terminating and
(ii)
repeating
Clearly, decimal representation of
,
,
,
,
, are terminating and so is the case with
.
Let us find decimal representation of some more rational numbers.
Example14: Final decimal representation of
(i)
(ii)
(ii)
(vi)
(iii)
Solution:
(i)
= 0.625
Here, decimal representation is terminating
22
(iv)
0.625
48
20
16
40
40
0
(ii)
= 3.666
Here, decimal representation non terminating and repeating
3.666
9
20
18
20
18
20
18
20
18
2
Here, digit 6 is repeating we write
= 3.666 = 3.
23
(iii)
= 3.1666
Here, decimal representation is non terminating and repeating
3.1666
18
10
6
40
36
40
36
40
36
4
= -3.1
(iv)
=
0. 3555…. = -0.3
Here, decimal representations is non terminating and repeating
0.355
135
250
225
250
250
25
24
(v)
= 2.142857142857
Here, decimal representation is non-terminating and repeating. A block of 142857 is
repeating. We write
= -2.
2.142857 142857
14
10
7
30
28
20
14
60
56
40
35
50
49
10
7
30
28
2
(vi)
= 0.12
Decimal representation is terminating
0.12
25
50
50
0
Observe that the decimal representation of a rational number is either terminating or
non-terminating repeating.
7. Representation of Rational Numbers on the Number Line
Recall that integers can he represented on the number line and so are the fractions.
In the same way, rational number can also be represented on the number line. We
proceed as follows:
25
We draw a line and take a point 0 on it to represent the rational number 0. The
positive rational numbers will be on the right side of 0 and the negative rational
number will be on the left side of 0.
Fig.1
Recall that consecutive points representing number 1, 2, 3, …, 1, 2, 3, … are at unit
distance from each other.
To represent a rational number say,
on the number live, we divide the line segment
OA into two equal parts OM and MA.
The mid-point M represents the rational number .
To represent rational number
, we divide the line segment OA’ into two equal
parts,… ON and NA’.
The point N represents the rational number
.
Similarly, we can represent the rational number ,
, ,
by dividing the each unit
distance into two equal parts as shown in Fig.2.
Fig.2
For representing the rational number say , we divide each unit distance into four equal
parts and take the 5th point from O towards right as shown in Fig.3. Clearly, the rational
number
will be represented by the 5th point to the left of O.
26
Fig.3
,
In the same way, we can represent rational number,
,
,
,
,
, etc on the
number line.
Example 15: Represent the rational numbers
and
on the number line
Solution: Divide each unit distance into three equal parts. The second point.
from O to the right represent .
Fig.4
Redraw: Divide the line segment (0,1) into 3 equal parts. Similarly (0, (-1) into 3 equal
parts.
Note: Distance (0,1), (-1, -2), (-2, -3) or (0,1), (1, 2), (2, 3) all are equal.
Similarly, the second point from O to the left represents
In the same way, we can represent , , . . .,
In this manner, every rational number
,
.
, . . . on the number line.
can be represented on the number line by
dividing each unit distance into q equal parts and taking the pth point from O
toward right (for positive) and left (for negative) rational number.
8. Comparison of Rational Numbers
(a) Graphically: For comparing two rational number graphically, we represent them
on the number line.
27
The number on the right is greater than the other (See Fig. 5) and the number on
the left is smaller than the other.
Fig. 5
Redraw: Divide the line segment (0,1) into 3 equal parts. Similarly (0, (-1) into 3 equal
parts.
Note: Distance (0,1), (-1, -2), (-2, -3) or (0,1), (1, 2), (2, 3) all are equal.
<
Clearly,
>
,
<
,
>
>
and so on.
Algebraically: Let us compare two rational numbers say and
=
=
=
and
=
.
[Converting to equivalent rational numbers
having same denominator. HCF of 3 and 4 = 12]
As denominator of
and
is the same, so we compare their numerators 9 and 8.
>
Since, 9 > 8, so,
>
Or,
In fact, every positive rational number is greater than every negative rational number.
Now, let us compare
=
and
=
and
=
=
=
=
[Making denominators same and positive]
28
Denominators of
and
Here, 6 > 25
are the same. So, we compare their numerators 25 and 6.
(Why?)
>
So,
>
Hence,
Example 16: Which is smaller?
,
(i)
Solution: (i)
=
and
,
(ii)
=
=
[LCM of 11 and 7 = 77]
=
Here, denominators are the same, we compare numerators 55 and 42.
Clearly, 55 > 42.
>
So,
. Hence,
(ii)
and
=
=
is smaller.
=
=
Here, we compare numerators 55 and 42, because denominators are the same.
55 < 42
<
So,
Hence,
is Smaller.
To compare two rational numbers.
(i)
First make their denominator positive.
(ii)
If they are of opposite signs, positive rational number is always greater than
the negative rational number.
(iii)
If both are positive, compare them as fractions.
29
(iv)
If both are negative, compare their corresponding fractions negative (ignoring
the –sign). If one fraction is greater than the other, then the corresponding
rational number is less than the other.
(v)
Every positive rational number is greater than 0 and every negative rational
number is less than 0.
9. Finding Rational Numbers between any two Rational Numbers.
Take any two integers say 7 and 2. Recall that there are only eight integers 6, 5,
4, 3, 2, 1, 0, and 1 between these two integers.
Since 6, 5, 4, 3, 2, 1, 0 and 1 are rational numbers also, so we can say that
there are eight rational numbers between rational numbers 7 and 2.
Can you find some more rational numbers between the two given rational numbers.
7 can be written as
and 2 can be written as
,
So, rational numbers
,
, …,
,
.
, and
, are between
and
or
between 7 and 2.
How many number are these?
They are 89 such numbers.
Similarly 7 =
and
2=
Clearly, rational numbers
,
,…,
and
,
are between rational number 7 and 2.
In this way, we can say that there are infinitely many rational numbers between
any two rational numbers.
Example 16: Find three rational numbers between
Solution:
and
=
=
30
and .
,
Clearly, rational numbers
,
,…,
and
and . We can
are between
choose any three of them.
Alternate Method
Let us find:
=
=
=
Let us compare and
=
and
Further
So,
i.e.,
and
and
.
>
<
lies between
and .
In general, if
and
i.e.,
is always a rational number lying between
+
are two district rational numbers then, their average (or mean)
and
By this method
also, we can find infinitely many rational numbers, between two rational number.
Another rational number between
Similarly,
=
=
and will be
=
is another rational number between
=
and .
10. Applications of Rational Numbers in Problem solving.
Example 18: A person travels a distance of
km towards west and then a distance of
km towards east on the same road. Find the distance and direction of the person from
the starting point.
Solution: Let the person start from a point, say, O toward west.
Distance travelled towards west =
Distance travelled towards east =
km
km
31
Position of the person is given by the expression
+
=
=
+
=
=
So, the person is at a distance of
Example 19: As a part of a
cycles,
km from his original point in west direction.
km long marathon race,
km was to be covered by
km by boat and rest on foot. How much is the distance travelled on foot?
Solution: Distance to be covered by cycles and boat =
km
Distance to be covered on foot
=
km
=
km
=
km
=
km
=
km
=
=
km
km
Example 20: A field is km long and km wide. It is prosposed to sow wheat in part
of the field and grow vegetable in the remaining part. Find the area of the field in which
vegetables are to be grown?
Solution: Area of the field =
32
Part of the filed for sowing wheat =
=
So, the part of the field left for vegetables = 1
Area of the field for vegetable =
km
=
=
=
Example 21: The length of a ribbon is
packets each of length is
km.
metres. Its
metre. Find the number of gift packets.
Solution: Length of ribbon used for gift packets
= of
= x
=
metres
metres
metres.
Length of the piece used for one gift pack =
So, number gift packets =
=
part is used for wrapping gift
x
= 50
33
metres
34
Student’s Worksheet – 1
Recall Fractions
Warm Up W1
Name of the student _______________________
Date _____________
Activity – I recall
The tiling shown below is the basic unit that makes an entire floor pattern.
What fraction of the tiling is?
a)
BLUE-
b)
PINK-
c)
WHITE35
Student’s Worksheet – 2
Types of Fractions
Warm Up W2
Name of the student _______________________
Date _____________
Activity – Types of Fractions
Q1.
Rearrange the letters to form meaningful words in fractions
a) KIEL:
b) ROPPRE:
c) NIUT:
d) EMDXI:
e) RRPPEOMI:
f) INEUKL:
g) UEEVIQLTAN:
Q2.
Write a fraction for the shaded part in each case and answer the questions:
a)
d)
b)
e)
36
c)
(i)
The like fractions are:
(ii)
The unit fractions are:
(iii)
The unlike fractions are:
(iv)
Give two equivalent fractions of each:
Student’s Worksheet – 3
Arithmetical Operations on Fractions
Warm Up W3
Name of the student _______________________
Date _____________
Activity – Add or subtract
Weight=
4
kg
7
Weight=
3
kg
7
37
Weight=
2
kg
7
Now answer the following:a)
The total weight of the teddy and the car is:
b)
Whose weight is less, Rattle or Car, and by how much?
d)
How much more does the teddy weigh than the car?
e)
Together the weight of the car and the rattle is:
38
Student’s Worksheet - 4
Need of Rational Numbers
Pre Content Worksheet P1
Name of the student _______________________
Activity – I feel the need - I
Give 4 real life situations to express need of :
WHOLE NUMBERS
•Counting the number of RED tiles in a RUBRICS puzzle.
•
•
•
INTEGERS
•
•
•
•
FRACTIONS
•
•
•
•
Solve given fraction problem:
Is your answer a fraction? ___________________
So, we need more numbers which are
neither an integer nor a fractional number.
39
Date _____________
Student’s Worksheet - 5
Another Number System
Pre Content Worksheet P2
Name of the student ______________________
Date _____________
Activity – I feel the need - II
http://www.amathsdictionaryforkids.com/dictionary.html
Find out the definitions of given terms by following the given link, to understand and
appreciate the difference between the various number systems.
Whole numbers
Integers
Decimal numbers and Fractions
40
Rational numbers
Task 2: Write the given rational numbers in the given arrow boxes as per their type.
Negative Rational Numbers
Positive Rational Numbers
Student’s Worksheet -6
Defining Rational Number
Content Worksheet C1
Name of the student ______________________
Date _____________
Activity- Defining Dialogues
Mr. Int is a member of INTEGER CLUB and Mr. Rat is a member of RATIONALS
CLUB.
41
Given below are 4 sets of dialogues(in outer rectangles) and 4 options (in the
inner circle).
Read the dialogues and write the correct option in the dialogue box.
42
Student’s Worksheet – 7
Rational Numbers and Fractions
Content Worksheet C2
Name of the student ______________________
Date _____________
Activity 1 – I see I understand
Observe the given ‘Venn Diagram’ . . . .
Classify the given numbers as per your observations :
1. 56/3
a) fraction
b) rational
c) rational, whole, integer
d) fraction, rational, integer
2. -12/4
a) fraction
b) rational
c) rational, whole, integer
d) fraction, rational, integer
43
3. Classify 0.3527974
a) rational, whole
b) rational, integer
c) whole
d) decimal, rational
4. 1/3
a) rational
b) rational, integer
c) rational, whole, natural
d) rational, whole, natural, integer
5. √100
a) rational, whole, natural, integer
b) rational, integer
c) whole, integer
d) rational
6. 48 + √36
Its 54 when expressed
in standard form. .!
a) rational, whole, natural, integer
b) whole, integer
c) rational, whole, integer
d) none of these
7. √63
a) not rational
b) rational
c) rational, integer
d) rational, whole, integer, natural
8. 40
a) not rational
b) rational, whole, natural, integer
c) rational, whole, integer
d) rational, integer
44
9. |-12|
a) not rational
b) rational
c) rational, integer
d) rational, whole, integer, natural
10. 0
a) not rational
b) rational, whole, natural, integer
c) rational, whole, integer
d) rational, integer
Student’s Worksheet – 8
Standard form of Rational Numbers
Content Worksheet C3
Name of the student ______________________
Date _____________
Activity- Standard Forum
A rational number is said to be in its standard form if its numerator and denominator
have no common factor other than 1, and its denominator is a positive integer.
Which of the following rational numbers are not in the standard form? (Circle
them)
Filter out the circled rational and write their standard form in the space
provided.
45
Student’s Worksheet – 9
Rational Numbers on Number Line
Content Worksheet C4
Name of the student ______________________
Activity - Mirror Images
On the number line:
Positive rational numbers are represented to the right of 0.
Negative rational numbers are represented to the left of 0.
46
Date _____________
A)
Number Line 1:
Take 1
as 7/7.
Take
Given Shape
Rational number for given
Rational number for image of
shape
given shape
Repeat the same on Number Line 2 and 3 and fill the given tables.
47
Number Line 2:
Given Shape
Rational number for given
Rational number for image of given
shape
shape
48
Number Line 3:
Given Shape
Rational number for given
Rational number for image of
shape
given shape
B) Study the given number lines and write the rational number marked.
49
C) Draw the number lines and represent given rational numbers on it:
1)
2)
3)
4)
50
Student’s Worksheet – 10
Comparison of Rational Numbers
Content Worksheet C5
Name of the student ______________________
Date _____________
Activity - Mirror Distance
A)
Use the three number lines given in the previous worksheets and compare and
arrange their equivalent rational numbers as per their placement on number line
in the table given below.
Number
Shapes as per their
Rational number for
Rational number for
Line
position on
given shapes in
images of given shapes
number line(in
increasing order
in increasing order
increasing order)
1
2
3
51
Observe that rational numbers having like positive denominators can easily be
arranged and compared.
B)
Convert to rational numbers having like denominators and put =, <, or > :
1.
______
2.
______
3.
_______
4.
-ve numbers < +ve numbers
________
5.
_______
6.
_______
7.
_______
8.
_______
9.
_______ 0
10.
________
52
C)
Arrange in descending order:
1.
4.
2.
5.
3.
Student’s Worksheet -11
Rational Numbers Between given Rational Numbers
Content Worksheet C6
Name of the student ______________________
Date _____________
Activity- Search Is On. .
Little Winnie is working on ‘after numbers’. She went to Ms. Rational to clear her
doubts. Please help her search for more rational numbers that can be inserted between
two given rational numbers.
Winnie : Write 5 rational numbers inserted between 4/7 and 5/7.
_____________________________________________________
Ms. Rational : Rewrite them in standard form.
______________________________________________________
_
53
1. Find 5 rational numbers between given rational numbers:
a)
Make use of
the gap
between
the
numerators.
b)
c)
2. Find 5 rational numbers between given rational numbers by converting them to like
rational numbers:
1.
2.
3.
4.
54
Student’s Worksheet -12
Addition and Subtraction
Content Worksheet C7
Name of the student ______________________
Date _____________
Activity- Complete the circle
Use the cut outs having the question and answer texts running parallel with the top and
bottom edges of the segments. Place the answer text next to the question text to make a
complete circle and colour it.
55
Student’s Worksheet -13
Skill Drill 1
Content Worksheet C8
Name of the student ______________________
Date _____________
Activity- Solution Search
Step 1: Use the answers given in the box and write them in front of the
questions(without solving).
Step 2: Solve the questions and check the answers written in the step 1.
1.
2.
3.
4.
5.
56
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
57
Student’s Worksheet – 14
Skill Drill 2
Content Worksheet C9
Name of the student ______________________
Date _____________
Activity- Matching Cards
In the following cards activity use the given cards and find their correct order. For e.g.
Card 1 follows card 7 and one of the given cards shall follow card 1 and so on.
Card No.
7
1
Front
Back
44
62
35
I have 44
35
. Who has 5
7
less than it?
I have 6 2 . Who has 2 less than it?
5
5
Now try it out yourself.
Card No.
Front
1
19
2
½
3
31
4
3
5
43
6
11
7
44
35
Back
I have 19
35
.Who has 26
I have ½. Who has
4
35
less than it?
7 less than it?
8
I have 3 1 . Who has 1.25 less than it?
4
I have 3 . Who has double of 1 more?
2
7
2
4
I have 4 3 . Who has 3 less than it?
4
2
4
I have 1.25. Who has 3.5 more than it?
35
I have 44
58
35
. Who has 5
7
less than it?
8
19
7
I have 19 . Who has 2 more than it?
7
9
2
I have 2. Who has 3 1
10
13
11
4
12
6/7
I have 6/7. Who has 2/5 more than it?
13
11
14
I have 11 . Who has 1/14 more?
14
8
2
less than it?
I have 1 3 . Who has 1 less than it?
8
8
I have 4. Who has 5/8 less than it?
14
1
15
27
16
5
5
I have 1 . Who has 4 1 more?
5
5
8
I have 27 . Who has
8
23 more?
8
I have 5 . Who has 1.5 more?
7
7
59
Student’s Worksheet – 15
Multiplication of Rational Numbers
Content Worksheet C10
Name of the student ______________________
Date _____________
Activity - Catch the butterfly
Solve correctly and catch the butterfly in the net by colouring them in same the shade.
60
Student’s Worksheet – 16
Multiplication Skill Drill
Content Worksheet C11
Name of the student ______________________
Date _____________
Activity - Mission Multiplication
Solve the given multiplication problems that come on your way to ‘MISSION
MULTIPLICATION’.
1.
2.
3.
4.
5.
61
6.
7.
8.
9.
10.
11.
12.
62
13.
14.
15.
16.
17.
63
Student’s Worksheet – 17
Division of Rational Numbers
Content Worksheet C12
Name of the student ______________________
Activity I- Divide and join the correct jigsaw pieces with a line.
64
Date _____________
Activity II - Division Drill
1.
Evaluate:
i) 2
1
17
( 14)
51
ii)
13 65
19 42
iii)
55
21
iv)
1 1
5 6
1
4
15
3 16
8 33
1 1
6 15
65
2.
The product of two numbers is -28/27. If one of the numbers is -4/9, find the
other.
3.
By what number should we divide 27/16 so that the quotient is -15/8?
4.
A small scale company pays Rs.1056.86 per week for advertising in the local paper.
What is the total cost of advertising for the company for one year?
3
so that the quotient is 60?
4
5.
What number must be divided by 15
6.
What number must be multiplied by 15
7.
Sam buys a laptop for Rs.45000, and pays for it with a two year interest-free loan.
2
1
so that the product is 56 ?
3
2
If he makes equal monthly payments, how much are his monthly payments?
66
Student’s Worksheet – 18
Rational Numbers as Decimals
Content Worksheet C13
Name of the student ______________________
Date _____________
Activity: Convert rational numbers to decimals
Q1. Divide and write the decimal equivalent beside each of the following rational
numbers.
4
10
i)
ii)
2
3
iii)
11
12
iv)
1
4
v)
3
8
67
vi)
7
3
vii)
21
55
viii)
1
6
ix)
2
11
x)
5
4
Find a relation between a rational number and its decimal representation:
68
Q2.
Give five examples of rational numbers that fit between each of the following
sets of rational numbers.
a.
-0.56 and -0.65
b.
-5.76 and -5.77
c.
3.64 and 3.46
Student’s Worksheet – 19
Real Life Problems
Content Worksheet C14
Name of the student ______________________
Date _____________
Activity- Solve the real life problems
1.
The price of oranges is Rs. 65
3
per kilogram. What quantity of oranges can be
4
1
bought for Rs. 230 .
8
69
2.
1
The price of eleven cricket balls is Rs. 1432 .Find the price
5
of one ball.
3.
3
A car is moving at a speed of 72 km per hour. Find the
4
1
distance it will cover in 3 hours.
2
1
1
1
3
and by the sum of
and
.
4
5
10
7
4.
Divide the sum of
5.
By what number should we multiply
70
13
to get the product as -1?
17
6.
A farmer has a rectangular piece of land of dimensions
50000 30000
(in meters). He distributes it among his five
13
11
children. How much area will each child get?
7.
The product of two rational numbers is
. One of the rational numbers is −1/6.
What is the other rational number?
8.
A boy has a cardboard of length
10
5
m and breadth m .He
7
4
makes five pieces of it to form play cards. What is the area of
three such cards?
71
9.
Divide
10.
A
train
by the sum of
travels
a
and
distance
.
of
1000km400m
in
15
2
5
hours30minutes.Find the average speed of the train.
11.
By what number should we divide
so that the quotient is
72
?
Student’s Worksheet -20
Rational Numbers Drill
Post Content Worksheet PC1
Name of the student ______________________
1.
Date _____________
Draw the number lines and represent given rational numbers on it:
1)
2)
2.
The correct increasing order of the fractions
73
is _________________.
3.
The numbers
can be arranged in descending order as____________.
4.
Arrange in descending order:
1.
2.
3.
74
5.
Solve the following
a)
b)
6.
Which number should be subtracted from -1/5 to obtain -2/5?
7.
Simplify a)
8.
The product of two rational numbers is
b)
What is the other rational number?
75
. One of the rational numbers is −1/6.
9.
By what number should we multiply
10.
Divide the sum of
11.
Solve
12.
Solve
and
by
.
76
to get the product as
?
13.
Find
14.
Simplify
15.
Find
)
16.
Solve
77
Student’s Worksheet – 21
Test Your Progress
Post Content Worksheet PC2
Name of the student ______________________
Date _____________
Activity- Test your progress.
Q1.
Multiple choice questions: (5 X 1=5)
i)
If x =
1
1
and y =
,then
3
3
A) x + y =1
ii)
3
16
The value of
A)
iv)
-3
C) x – y = 1
D) x /y = 0
Which of the following rational numbers are in standard form?
A)
iii)
B) x + y
1
3
B)
26
91
C)
5
105
D)
52
38
C)
1
6
D)
1
6
9 24
is
16 81
B) -
1
3
The reciprocal of a negative rational number
A)
Is a positive rational number
B)
Doesnot exist rational number
C)
Is a negative rational number
D)
can be either positive or negative
78
v)
What number should be subtracted from
7
5
A)
Q2.
B)
12
5
C)
7
5
3
to get -3?
5
D)
13
5
TRUE/FALSE : (5 X 1 = 5)
i)
All rational numbers can be represented on a number line.
ii) We can insert finite number of rational numbers between -2 and -5.
iii) The product of two rational numbers is always a positive rational number.
Q3.
iv)
2 3
5 7
3 2
7 7
v)
3
lies to the right of 0 on a number line.
5
Fil in the blanks: (5 X 1 = 5)
3
5
7
,then x =………………..
x
i)
If
ii)
7
12
iii)
The denominator of a rational number cannot be …………….
5
4
5
4
…………
79
Q4.
iv)
……….. is not reciprocal of any number.
v)
The additive inverse of
23
is……………
13
Match the following: (5 X 1 = 5)
i)
3
4
2
3
88
33
ii) A rational number between
1
1
and
2
3
1
12
11
4
iii)
iv)
1
6
v) Standard form of
Q5.
3
1
3
Arrange in ascending order: (2 X 2 = 4)
5 2 3 7
i) , , ,
6 5 4 8
Q6.
88
33
ii)
3 7
1
, , 0,
4 5
2
Write four rational numbers between: (3 X 2 = 6)
i)-1 and 1
ii) -3 and -2
80
iii)
4
2
and
5
3
(Q7 to Q16 each carry three marks)
Q7.
Express in decimal form
i)
19
8
ii)
Q8.
Multiply:
Q9.
Divide:
5
3
37
3
16
by
.
4
69
77
5
by 4 .
36
18
4 11
13
and
Q10. Add ,
.
9 18
24
Q11. Evaluate: 1
3
4
2
Q12. Subtract the sum
Q13. Simplify:
5
8
64
.
147
5 11
from the sum
6 12
13 2
9 15
7 5
3 8
3 17
1
.
4
18
3 1
5 2
Q14. The product of two rational numbers is
81
5
3
.If one of them is
,find the other.
18
20
Q15. Represent on number line:
i)
5
6
ii)
11
3
1
1
Q16. How many pieces of rope of each 5 metres long can be cut from a rope of 77
6
2
metres long?
Q17 to Q21 each carry four marks)
Q17. Find two rational numbers whose sum is
1
and if greater is divided by the
2
1
smaller, the result is 1 .
2
Q18. A basket contains three types of fruits weighing 19
be pineapples, 3
1
1
kg in all. If 8 kg of these
3
9
1
kg be plums and the rest pears. What is the weight of the
6
pears in the basket?
Q19. Riya had Rs.300. She spent 1/3 of her money on notebooks and 1/4 of the
remainder on stationery items. How much money is left with her?
20.
Sandy wanted to go to his friend’s house from school. He walked
east and then from there he walked
5
km towards
4
12
km towards west to reach his friend’s
7
house. What is the distance between the school and his friend’s house?
21.
Aryan earns Rs.160000 per month. He spends 1/4 of his income on food; 3/10 of
the remainder on house rent and 5/21 of the remainder on the education of
children. How much money is still left with him?
82
Suggested Video Links
Name
Video Clip 1
Title/Link
Rational Numbers
http://www.authorstream.com/Presentation/aSGuest17486180490-rational-numbers-entertainment-ppt-powerpoint/
Video Clip 2
Rational numbers
http://www.wlcsd.org/webpages/mrjoseph/files/4CH3L1.ppt
Video Clip 3
Adding and Subtracting Rational Numbers
iadmin.bismarckschools.org/.../alg-2-ch-8-add-and-subtractrational-numbers.ppt.
Video Clip 4
Multiplying and Dividing Rational Numbers
teachers.henrico.k12.va.us/math/int10405/.../mult_div_fracNT2.ppt
Video Clip 5
Order Rational Numbers
http://www.youtube.com/watch?gl=SG&hl=enGB&v=KTR10AmibwM
Weblink1
Solve online problems on Rational Numbers
http://www.mathsisfun.com/algebra/rational-numbersoperations.html
Weblink 2
Comparing and Ordering Rational Numbers
http://www.math-play.com/Comparing-RationalNumbers/comparing-rational-numbers.html
83
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Delhi-110 092 India