Download Rational Numbers Grade 8

ID : ww-8-Rational-Numbers [1]
Grade 8
Rational Numbers
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Answer t he quest ions
(1)
Write a multiple of
-1
rational number ?
-3
(2)
Find the multiplicative inverse of the f ollowing.
A)
13
22
B)
7
14
(3)
If
p
=
q
(4)
(5)
pxm
, then what is the relation between m and n?
qxn
1
Is 0.003 the multiplicative inverse of 333
Consider the f ollowing statement
a
? Why or why not ?
3
+(
b
c
+
d
e
)=(
f
a
b
+
c
d
)+
e
. What property is
f
being described here?
(6) Solve the f ollowing and simplif y to lowest term:
A)
2
+
3
B)
-5
-2
+
6
+
3
(7)
4
+
2
-2
6
+
2
6
4
5
+
-6
6
Out of the three rational numbers,
8
-16
(8)
if
-7
X
=
X
,
-8
and
13
-8
, which is smaller ?
18
, then X is ________ a rational number
7
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ID : ww-8-Rational-Numbers [2]
Choose correct answer(s) f rom given choice
(9) Which of the f ollowing statements is f alse:
a. Every f raction is a rational number
b. Any rational number when multiplied by 8 is a
rational number
c. Every rational number is a f raction
d. Every negative number is a rational number
(10) Which of the f ollowing is a rational number(s) ?
a.
3
b.
-7
c.
2
9
d. All of these
-8
-1
(11) Which of the f ollowing statement is true ?
a.
7
>
-4
c.
7
-35
b.
20
>
-3
7
-35
<
-4
7
d.
-4
20
7
-35
=
-4
20
(12) Which of the statements below is f alse?
(13)
a. T here is no f raction that cannot be
represented as a rational number
b. A decimal number cannot be a rational
number
c. T here is no integer that cannot be
represented as a rational number
d. All negative numbers can be represented as
rational numbers
If
3
-9
(14)
=
-12
, then X = ___________
X
a. 36
b. -36
c. 27
d. None of these
T he average of the middle two rational numbers if
4
,
2
-9
15
,
3
5
,
-6
are arranged in
15
ascending order.
a.
7
b.
10
c.
-1
2
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1
10
d.
13
10
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ID : ww-8-Rational-Numbers [3]
(15) Which of the f ollowing is a negative integer and a rational number?
a.
b. -23
7
2
c. 0.65
d.
37
9
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ID : ww-8-Rational-Numbers [4]
Answers
(1)
4
(Answers may vary)
3
Step 1
Multiples of rational number means the multiplication of the rational number with some
given number or any number.
x1
For example, suppose
is the rational number.
x2
x1
multiple of
are
x2
2x1
x2
,
3x1
, etc.
x2
Step 2
T heref ore, one example of multiple of
-1
is
-3
(2)
A)
4
.
3
22
13
Step 1
Multiplicative Inverse: When we multiply a number by its "Multiplicative Inverse",
we get 1.
Mathematically,
n×
1
=1
n
Step 2
T he multiplicative inverse of
13
22
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is
22
13
, since
13
22
×
22
= 1.
13
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ID : ww-8-Rational-Numbers [5]
B)
14
7
Step 1
Multiplicative Inverse: When we multiply a number by its "Multiplicative Inverse",
we get 1.
Mathematically,
n×
1
=1
n
Step 2
T he multiplicative inverse of
7
14
(3)
is
14
, since
7
7
14
×
14
= 1.
7
m= n
Step 1
If you look at the question caref ully, you will notice that you have the relation
p
q
=
pxm
qxn
.
Step 2
Now
p
=
q
pxm
qxn
⇒p × (q × n) = q × (p × m)
⇒ (p × q) × n = (q × p) × m
⇒
n
=
m
⇒
n
q×p
=
m
⇒
n
p×q
p×q
p×q
=1
m
⇒n=m
or
⇒m= n
Step 3
T heref ore the relation between m and n is m = n .
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ID : ww-8-Rational-Numbers [6]
(4) Yes - T he product or f raction and its multiplicative inverse must be equal to 1.
Step 1
When we multiply a number by its Multiplicative Inverse we get 1.
For example, x ×
1
= 1 [It means
x
1
is a multiplicative inverse of x.]
x
Step 2
According to the question, we have to prove that 0.003 is the multiplicative inverse of
333
1
. T o prove this we have to multiply the given number to the given multiplicative
3
inverse and check if result is 1 or not.
Product,
= 0.003 × 333
=
3
1000
1
3
× 333
1
3
=1
It satisf ies the condition of product being equal to 1.
Step 3
T heref ore, 0.003 is the multiplicative inverse of 333
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1
.
3
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ID : ww-8-Rational-Numbers [7]
(5)
Associative property
Step 1
If you look at the question caref ully, you will notice that the f ollowing statement is given
a
+(
b
c
d
+
e
)=(
f
a
b
+
c
)+
d
e
.
f
Step 2
According to the statement sum of
a
+(
b
+
e
c
d
+
e
) is equal to the sum of (
f
a
b
+
c
)
d
.
f
Which is equal to the associative property.
Associative property is states that if you are adding or multiplying it does not matter
where you put the parenthesis.In other words, associative property states that you can add
or multiply regardless of how the numbers are grouped. By 'grouped' we mean 'how you use
parenthesis'.
For example : 2 + (3 + 5) = (2 + 3) + 5
10 = 10
Step 3
T heref ore the property of above statement is associative property.
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ID : ww-8-Rational-Numbers [8]
(6)
A)
2
3
Step 1
Let’s add the given numbers by taking the LCM of the denominators:
2
+
3
=
4
-2
+
6
2
+
2
6
(2 × 2) + (4 × 1) + (-2 × 3) + (2 × 1)
6
=
4 + 4 -6 + 2
6
=
4
6
Step 2
T o simplif y
4
into lowest term, we need to divide the numerator and
6
denominator by their HCF.
T he HCF of 4 and 6 is 2.
4
T heref ore,
4
=
6
2
=
6
2
3
2
Step 3
T hus, the simplif ied term is
2
.
3
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ID : ww-8-Rational-Numbers [9]
B)
-11
5
Step 1
Let’s add the given numbers by taking the LCM of the denominators:
-5
+
3
=
-2
+
6
4
+
-6
5
6
(-5 × 10) + (-2 × 5) + (4 × 6) + (-6 × 5)
30
=
-50 -10 + 24 -30
30
=
-66
30
Step 2
T o simplif y
-66
into lowest term, we need to divide the numerator and
30
denominator by their HCF.
T he HCF of -66 and 30 is 6.
-66
T heref ore,
-66
30
=
6
=
-11
5
30
6
Step 3
T hus, the simplif ied term is
-11
.
5
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(7)
-8
13
Step 1
Let's f irst ignore the negative sign of the numbers, and check the order of absolute values
of the f ractions.
We know that if numerator of f ractions are same, f raction with smaller denominator is
larger. T heref ore,
8
>
13
8
>
16
8
18
Step 2
We also know that f or negative numbers, number with higher absolute value is smaller,
-8
<
13
-8
<
16
-8
18
Or,
-8
<
13
8
<
-16
-8
18
Step 3
T heref ore,
-8
is smaller than other two given f ractions.
13
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ID : ww-8-Rational-Numbers [11]
(8)
not
Step 1
Rational numbers: A rational number is any number that can be expressed as the quotient
or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q
may be equal to 1, every integer is a rational number.
For example: reciprocal of
p
is
q
q
.
p
Step 2
If you look at the question caref ully, you will notice that you have
-7
X
Now,
-7
X
=
=
X
.
7
X
7
⇒ -7 × 7 = X × X
-49 = X2
X = √-49
√-49 is not a rational number.
Step 3
T heref ore X is not a rational number, if
-7
=
X
X
.
7
(9) c. Every rational number is a f raction
Step 1
A rational number is a number that can be written as a ratio.
For example a/b where a and b are whole numbers.
All prime numbers are 2,3,5,7,11,13,17,19,23, ......
All natural numbers are 0,1,2,3,4,5,6,7,8 ......
All integers are ..... -3, -2, -1, 0, 1, 2, 3 .....
Step 2
Now you can say that the statement 'Every rational number is a fraction' is f alse.
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ID : ww-8-Rational-Numbers [12]
(10) d. All of these
Step 1
Rational numbers: A rational number is any number that can be expressed as the quotient
or f raction
p
of two integers, p and q, with the denominator q not equal to zero.
q
Step 2
Now,
2
is the rational number with p = 2 and q = 9.
9
3
is the rational number with p = 3 and q = -7.
-7
-8
is the rational number with p = 3 and q = -1.
-1
Step 3
T heref ore, the correct option is the all of these.
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ID : ww-8-Rational-Numbers [13]
(11)
d.
7
=
-4
-35
20
Step 1
Let's look at all the options and see which one is correct.
Step 2
7
>
-3
7
, is the wrong statement, because f or negative numbers, number with higher
-4
absolute value is smaller.
Step 3
7
>
-4
-7
-35
20
=
4
-7
, is the wrong statement. Because the numbers are equal.
4
Step 4
7
<
-4
-7
-35
20
=
4
-7
, is the wrong statement. Because the numbers are equal.
4
Step 5
7
=
-4
-7
-35
20
=
4
-7
, is the right statement. Because the numbers are equal.
4
Step 6
T heref ore, the correct option is
7
-4
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=
-35
.
20
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ID : ww-8-Rational-Numbers [14]
(12) b. A decimal number cannot be a rational number
Step 1
A rational number is a number which can be written in the f orm
p
where p and q both are
q
integers and q is not equal to zero.
Step 2
All integers can be written in the f orm of p/1 (where p is any integer). Hence, all integers are
rational numbers. T heref ore, the statement "T here is no integer that cannot be
represented as a rational number" is true.
Step 3
All f ractions are rational numbers as they are represented in
p
f orm where p and q are
q
integers. T heref ore, the statement "T here is no fraction that cannot be represented
as a rational number" is true.
Step 4
All integers (-ve or +ve) can be written in the f orm p/1 where p is the integer. T heref ore, the
statement "All negative numbers can be represented as rational numbers" is true.
Step 5
A decimal number can be written in p/q f orm, if digits af ter decimal are either terminating or
repeating. T heref ore, the statement "A decimal number cannot be a rational number" is
false.
Step 6
T he f alse statement is "A decimal number cannot be a rational number" and hence, the
correct answer is b.
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ID : ww-8-Rational-Numbers [15]
(13) a. 36
Step 1
According to the question, we have to f ind the value of X in the given equation:
3
=
-12
-9
X
Step 2
Now,
3
=
-12
-9
X
⇒ 3 × X = -9 × -12
⇒ 3 X = 108
⇒X=
108
3
⇒ X = 36
Step 3
T heref ore the value of X is 36 .
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ID : ww-8-Rational-Numbers [16]
(14)
1
b.
10
Step 1
T o compare f ractions, f irst we need to make sure that all denominators are same, so we
can just compare the numerators of f ractions.
Step 2
T he LCM of the denominators 2, 15, 5 and 15 = 30
Step 3
Now, divide the LCM by the denominators and multiply the result with the numerator and
denominator as f ollowing:
4 × 15
-9 × 2
,
2 × 15
or
60
3×6
,
15 × 2
-18
,
30
30
,
5×6
18
,
30
,
-6 × 2
15 × 2
-12
30
Step 4
Let’s arrange the given numbers in ascending order, we get:
-18
,
30
-12
30
,
18
30
,
60
30
Step 5
-12
Now, the average of the middle two rational numbers =
+
30
18
30
2
=
6
30
=
×
1
2
1
10
Step 6
T hus, the average of the middle two rational numbers is
1
.
10
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(15) b. -23
Step 1
Rational Number :- A rational number is a real number that can be written as a simple
f raction.
f or example
7
is a rational number.
1
Integer :- An integer is a number that can be written without a f ractional component.
For example all positive and negative integers are {.... -3, -2, -1, 0, 1, 2, 3 ....}
Step 2
If you look at the all options caref ully, you will notice that -23 is a negative integer, it can be
written as
-23
and hence -23 is a rational number.
1
Step 3
Now you can say that -23 is a negative integer and a rational number.
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