Download Grade 8 Rational Numbers

ID : pk-8-Rational-Numbers [1]
Grade 8
Rational Numbers
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Answer t he quest ions
(1)
2
What is the multiplicative inverse of
?
23
(2)
If
p
=
q
(3)
(4)
pxm
, then what is the relation between m and n?
qxn
Write the rational number that are equal to its negative.
Consider the f ollowing statement
a
+(
b
c
+
d
e
)=(
f
a
b
+
c
d
)+
e
. What property is
f
being described here?
Choose correct answer(s) f rom given choice
(5)
If
8
=
-4
-32
, then X = ___________
X
a. 12
b. 16
c. -16
d. None of these
(6) How many rational numbers are there between any two non equal rational numbers ?
a. 30000
b. Inf inite
c. 0
d. 433
(7) Which of the statements below is f alse?
a. All negative numbers can be represented as
rational numbers
b. T here is no f raction that cannot be
represented as a rational number
c. T here is no integer that cannot be
d. A decimal number cannot be a rational
represented as a rational number
number
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ID : pk-8-Rational-Numbers [2]
(8)
What is the largest rational number that can be f ormed using 2 of the numbers below in the
p
f orm of
?
q
47, 163, 340, 127, 202, 48
202
a.
b.
47
48
47
c.
d.
202
(9)
p
If
340
47
=
q
m
, then which of the f ollowing is always true:
n
q
a.
340
=
p
m
b.
n
c. p x n = m x q
p
n
=
m
q
d. p x m = n x q
(10) If the sum of both the horizontal and the vertical rows are the same, f ind the missing rational
number.
4
31
2
6
31
31
2
31
a.
5
b.
31
c.
4
31
(11) -3
3
31
d.
2
31
is _________________
0
a. neither positive nor negative rational number b. positive rational number
c. negative rational number
d. either positive or negative rational number
Fill in t he blanks
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ID : pk-8-Rational-Numbers [3]
(12) What rational number results f rom the f ollowing:
A)
4 + 5 × 9 - 7 + 322 ÷ 91 =
B)
2 + 5 × 9 - 2 + 164 ÷ 81 =
(13)
T he rational number that are equal to their reciprocals are
&
.
(14) Fill in the blank to make the two rational numbers equivalent
A)
10
=
B)
70
16
C)
17
41
(15)
108
=
252
28
D)
=
18
T he average of the middle two rational numbers if
-3
15
,
-4
30
,
-4
15
,
-4
are arranged in
2
.
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288
31
492
ascending order is
=
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ID : pk-8-Rational-Numbers [4]
Answers
(1)
23
2
Step 1
If you look at the question caref ully, you will notice that you have
2
.
23
Step 2
Multiplicative inverse:
T he Multiplicative Inverse of a number is its reciprocal. T he product of a number and the
reciprocal of a number is 1. Multiplicative inverse of a number n is represented as
1
.
n
So, the multiplicative inverse of
2
is
23
23
.
2
Step 3
T heref ore the multiplicative inverse of
2
23
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is
23
.
2
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ID : pk-8-Rational-Numbers [5]
(2)
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 .
(3)
0
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.
Rational number zero(0) is the only rational number that is equal to its negative.
Step 2
T heref ore zero(0) is the only rational number that is equal to its negative.
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ID : pk-8-Rational-Numbers [6]
(4) Associative property
Step 1
If you look at the question caref ully, you will notice that the f ollowing statement is given
a
c
+(
b
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.
(5)
b. 16
Step 1
According to the question, we have to f ind the value of X in the given equation:
8
=
-32
-4
X
Step 2
Now,
8
=
-32
-4
X
⇒ 8 × X = -4 × -32
⇒ 8 X = 128
⇒X=
128
8
⇒ X = 16
Step 3
T heref ore the value of X is 16 .
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ID : pk-8-Rational-Numbers [7]
(6) b. Inf inite
Step 1
Rational Number: A rational number is a number that can be expressed as a f raction. A
rational number is said to have numerator and denominator .
Step 2
For example:
3
or 1.5 ,
2
5
or 2.5
2
rational numbers between these two rational numbers are 1.51, 1.52, 1.53, 1.54, 1.52, 1.6
and so on upto inf inite.
Step 3
T heref ore, Infinite rational numbers are there between any two non equal rational
numbers.
(7) d. 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 d.
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ID : pk-8-Rational-Numbers [8]
(8)
340
d.
47
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the value of
largest rational number that can be f ormed using 2 of the numbers below in the f orm of
p
q
47, 163, 340, 127, 202, 48.
Step 2
Rational number:
A rational number is a number that can be written as a ratio. T hat means it can be written as
a f raction, in which both the numerator (the number on top) and the denominator (the
number on the bottom) are whole numbers.
For largest rational number numerator (p) is the largest number and denominator (q) is the
smallest number.
Now, p = 340 and q = 47
Hence, the largest rational number is
340
.
47
Step 3
T heref ore, the largest rational number that can be f ormed using 2 of the numbers is
340
47
.
(9) c. p x n = m x q
Step 1
If you look at the question caref ully, you will notice that you have the relation
p
=
q
m
.
n
Step 2
You have
p
=
q
m
n
By cross multiply
⇒pxn= mxq
Step 3
T heref ore p x n = m x q is always true.
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ID : pk-8-Rational-Numbers [9]
(10)
4
c.
31
Step 1
Let the missing rational number be x.
Step 2
2
Sum of the numbers in the horizontal row =
+
31
8
=
6
+x
31
+x
31
Step 3
Sum of the numbers in the vertical row =
4
+
31
6
31
+
2
31
12
=
31
Step 4
It is given that the sum of the horizontal row is same as the sum of the vertical row.
T hat is,
2
31
⇒
6
+
+x=
31
8
31
+x=
31
⇒x=
+
6
31
+
2
31
12
31
12
31
⇒x=
4
-
8
31
4
31
Step 5
T heref ore, the missing rational number is
4
.
31
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ID : pk-8-Rational-Numbers [10]
(11) a. neither positive nor negative rational number
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.
-3
is not a rational number because denominator of the rational numbers can not be
0
zero(0).
Step 2
T heref ore
-3
is neither positive nor negative rational number .
0
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ID : pk-8-Rational-Numbers [11]
(12) A)
592
4 + 5 × 9 - 7 + 322 ÷ 91 =
13
Step 1
In order to solve this expression, we will have to f ollow the correct precedence
of operators i.e. Division, Multiplication, Addition and Subtraction (in that order).
Step 2
Now, 4 + 5 × 9 - 7 + 322 ÷ 91 can be solved as:
4 +5×9- 7 +
322
91
4 +5×9 - 7 +
46
13
= 4 + 45 - 7 +
46
13
= 49 - 7 +
46
13
= 42 +
46
13
=
(42 × 13) + 46
13
=
546 + 46
13
=
592
13
Step 3
T heref ore, the result of 4 + 5 × 9 - 7 + 46 ÷ 91 is
592
.
13
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ID : pk-8-Rational-Numbers [12]
B)
3809
2 + 5 × 9 - 2 + 164 ÷ 81 =
81
Step 1
In order to solve this expression, we will have to f ollow the correct precedence
of operators i.e. Division, Multiplication, Addition and Subtraction (in that order).
Step 2
Now, 2 + 5 × 9 - 2 + 164 ÷ 81 can be solved as:
2+5×9- 2+
164
81
= 2 + 45 - 2 +
164
81
= 47 - 2 +
164
81
164
= 45 +
81
=
(45 × 81) + 164
81
=
3645 + 164
81
=
3809
81
Step 3
T heref ore, the result of 2 + 5 × 9 - 2 + 164 ÷ 81 is
3809
.
81
(13)
1
-1
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.
Rational numbers 1 and -1 are the only numbers which are equal to their reciprocals.
Step 2
T heref ore rational numbers 1 and -1 are equal to their reciprocals.
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ID : pk-8-Rational-Numbers [13]
(14) A)
112
T o make the two rational numbers equivalent, f irst of all divide the given greatest
numerator/denominator by the smallest numerator/denominator and multiply the
numerator/denominator which is in the f ront of blank by the result.
Now the number which make the rational numbers
10
and
70
equivalent is
16
112.
B)
12
T o make the two rational numbers equivalent, f irst of all divide the given greatest
numerator/denominator by the smallest numerator/denominator and divide the
numerator/denominator which is in the f ront of blank by the result.
Now the number which make the rational numbers
and
108
equivalent is
252
28
12.
C)
204
T o make the two rational numbers equivalent, f irst of all divide the given greatest
numerator/denominator by the smallest numerator/denominator and multiply the
numerator/denominator which is in the f ront of blank by the result.
Now the number which make the rational numbers
17
and
41
equivalent is
492
204.
D)
496
T o make the two rational numbers equivalent, f irst of all divide the given greatest
numerator/denominator by the smallest numerator/denominator and multiply the
numerator/denominator which is in the f ront of blank by the result.
Now the number which make the rational numbers
18
and
288
equivalent is
31
496.
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ID : pk-8-Rational-Numbers [14]
(15)
-7
30
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 15, 30, 15 and 2 = 30
Step 3
Now, divide the LCM by the denominators and multiply the result with the numerator and
denominator as f ollowing:
-3 × 2
-4 × 1
,
15 × 2
-6
or
,
30 × 1
-4
,
30
,
30
-4 × 2
15 × 2
-8
,
,
-4 × 15
2 × 15
-60
30
30
Step 4
Let’s arrange the given numbers in ascending order, we get:
-60
,
30
-8
,
30
-6
30
,
-4
30
Step 5
-8
Now, the average of the middle two rational numbers =
+
30
-6
30
2
=
-14
30
=
×
1
2
-7
30
Step 6
T hus, the average of the middle two rational numbers is
-7
.
30
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