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Page No 58:
Question 1:
Identify the property in the following statements:
(i) 2 + (3 + 4) = (2 + 3) + 4; (ii) 2.8 = 8.2; (iii) 8. (6 + 5) = (8.6) + (8.5).
Answer:
(i) 2 + (3 + 4) = (2 + 3) + 4
This is the associative property under addition.
(ii) 2·8 = 8·2
This is the commutative property under multiplication.
(iii) 8· (6 + 5) = 8·6 + 8·5
This is the distributive property of multiplication over addition
Page No 58:
Question 2:
Find the additive inverses of the following integers:
6, 9, 123, −76, −85, 1000
Answer:
Here, 6 + (−6) = 6 − 6 = 0
Thus, − 6 is the additive inverse of 6.
9 + (−9) = 9 − 9 = 0
Thus, −9 is the additive inverse of 9.
123 + (−123) = 123 − 123 = 0
Thus, −123 is the additive inverse of 123.
(−76) + 76 = −76 + 76 = 0
Thus, 76 is the additive inverse of −76.
(−85) + 85 = −85 + 85 = 0
Thus, 85 is the additive inverse of −85.
1000 + (−1000) = 1000 − 1000 = 0
Thus, −1000 is the additive inverse of 1000.
Page No 58:
Question 3:
Find the integer m in the following:
(i) m + 6 = 8; (ii) m + 25 = 15; (iii) m − 40 = − 26; (iv) m + 28 = −49.
Answer:
(i)
(ii)
(iii)
(iv)
Page No 58:
Question 4:
Write the following in increasing order:
21, −8, −26, 85, 33, −333, −210, 0, 2011.
Answer:
The increasing order of the numbers 21, − 8, −26, 85, 33, −333, −210, 0 and 2011 is:
−333 < −210 < −26 < −8 < 0 < 21 < 33 < 85 <2011
Page No 58:
Question 5:
Write the following in decreasing order
85, 210, −58, 2011, −1024, 528, 364, −10000, 12
Answer:
The decreasing order of the numbers 85, 210, −58, 2011, −1024, 528, 364, −10000 and 12 is:
2011 > 528 > 364 > 210 > 85 > 12 > −58 > −1024 > −10000
Page No 60:
Question 1:
Write down ten rational numbers which are equivalent to 5/7 and the denominator not exceeding 80.
Answer:
Thus, the 10 equivalent fractions are
Page No 60:
Question 2:
Write down 15 rational numbers which are equivalent to 11/5 and the numerator not exceeding 180.
Answer:
Thus, the 10 equivalent fractions are
Page No 60:
Question 3:
Write down ten positive rational numbers such that the sum of the numerator and the denominator of each is 11. Write them in decreasing order.
Answer:
The numbers are
Page No 60:
Question 4:
Write down ten positive rational numbers such that numerator − denominator for each of them is −2. Write them in increasing order.
Answer:
Ten positive rational numbers are
Page No 60:
Question 5:
Is
Answer:
We know that a number in the form
Page No 60:
Question 6:
Earlier you have studied decimals 0.9, 0.8. Can you write these as rational numbers?
Answer:
The decimals 0.9 and 0.8 can be represented as rational numbers as:
Page No 70:
Question 1:
Name the property indicated in the following;
Answer:
(i) 315 + 115 = 430
The sum of two positive integers is a positive integer.
Therefore, 315 + 115 = 430 shows closure property of addition.
(ii)
The product of two rational numbers is a rational number.
Therefore,
(iii) 5 + 0 = 0 + 5 = 5
Here, 0 is the additive identity.
(iv)
Here, 1 is the multiplicative identity.
(v)
Here,
(vi)
Here,
Page No 70:
Question 2:
Check the commutative property of addition for the following pairs:
Page No 70:
Question 3:
Check the commutative property of multiplication for the following pairs:
Page No 70:
Question 4:
Check the distributive property for the following triples of rational numbers:
Page No 70:
Question 5:
Find the additive inverse of each of the following numbers:
Answer:
Hence, the additive inverse of the numbers
Page No 70:
Question 6:
Find the multiplicative inverse of each of the following numbers:
Answer:
Hence, the multiplicative inverse of the numbers
Page No 73:
Question 1:
Represent the following rational numbers on the number line:
Answer:
To represent the rational number
Now, we consider a line segment PQ from −8 to 0 and then divide PQ into 5 equal parts such that PA = AB = BC = CD = DQ. Thus, point D represents the rational
number
To represent the rational
number
Now, we consider a line segment PQ from 0 to 3 and then divide PQ
into 8 equal parts such that PA = AB = BC = CD = DE = EF = FG = GQ. Thus, point C represents the rational number
To represent the rational
number
Now, we consider a line segment PQ from 0 to 2 and then divide PQ
into 7 equal parts such that PA = AB = BC = CD = DE = EF = FQ. Thus, point B represents the rational number
To represent the rational
number
Now, we consider a line segment PQ from 0 to 12 and then divide PQ
into 5 equal parts such that PA = AB = BC = CD = DQ. Thus, point A represents the rational number
To represent the rational
number
Now, we consider a line segment PQ from 0 to 45 and then divide PQ
into 13 equal parts such that PA = AB = BC = CD = … = KJ = JQ. Thus, point A represents the rational number
Page No 73:
Question 2:
Write the following rational numbers in ascending order:
Answer:
The rational numbers
Since a proper fraction with a negative sign is greater than an improper fraction and a proper fraction with a positive sign is smaller than an improper fraction, we get:
Since
Page No 73:
Question 3:
Write 5 rational number between
Answer:
We have two rational numbers
We can write
Thus, we choose any 5 rational numbers between
Page No 74:
Question 4:
How many positive rational numbers less than 1 are there such that the sum of the numerator and denominator does not exceed 10?
Answer:
The positive rational numbers that are less than 1, such that the sum of the numerator and the denominator does not exceed 10, are:
Thus, there are 15 such rational numbers.
Page No 74:
Question 5:
Suppose m/n and p/q are two positive rational numbers. Where does
Answer:
In the first case, we assume
Then, we get:
From (1) and (2), we get:
In the second case, we assume
Then, we get:
From (1) and (2), we get:
In the third case, we assume
Then, we get:
From (1) and (2), we get:
Therefore, if
If
Page No 74:
Question 6:
How many rational numbers are there strictly between 0 and 1such that the denominator of the rational number is 80?
Answer:
The rational numbers that lie between 0 and 1, with 80 as their denominator, are:
Thus, there are 79 such rational numbers.
Page No 74:
Question 7:
How many rational numbers are there strictly between 0 and 1 with the property that the sum of the numerator and denominator is 70?
Answer:
The rational numbers that lie between 0 and 1, such that the sum of their denominator and numerator is 70, are
Thus, there are 34 such rational numbers.
Page No 75:
Question 1:
Fill in the blanks:
(a) The number 0 is not in the set of __________.
(b) The least number in the set of all whole numbers __________.
(c) The least number in the set of all even natural numbers is __________.
(d) The successor of 8 in the set of all natural numbers is ___________.
(e) The sum of two odd integers is ___________.
(f) The product of two odd integers is___________.
Answer:
(a) The number 0 is not in the set of natural numbers.
(b) The least number in the set of all whole numbers is 0.
(c) The least number in the set of all even natural numbers is 2.
(d) The successor of 8 in the set of all natural numbers is 9.
(e) The sum of two odd integers is even.
(f) The product of two odd integers is odd.
Page No 75:
Question 2:
State whether the following statements are true or false:
(a) The set of all even natural numbers is a infinite set.
(b) Every non-empty subset of
(c) Every integer can be identified with a rational number.
(d) For each rational number, one can find the next rational number.
(e)There is no largest rational number.
(f) Every integer is either even or odd.
(g) Between any two rational numbers, there is an integer.
Answer:
(a) The set of all even natural numbers is an infinite set.
Thus, the given statement is FALSE.
(b) Every non-empty subset of natural numbers
Thus, the given statement is FALSE.
(c) Let n be an integer. It can be written in the form of
a rational number as
Thus, the given statement is TRUE.
(d) For any rational number, we cannot find the next rational number.
Thus, the given statement is FALSE.
(e)There is no largest rational number.
Thus, the given statement is FALSE.
(f)The set of integers is a collection of all even and odd numbers.
Thus, the given statement is TRUE.
(g) Between any two rational numbers, there are infinite rational numbers.
Thus, the given statement is FALSE.
Page No 76:
Question 3:
Simplify:
Answer:
(i)
(ii)
Page No 76:
Question 5:
Define an operation * on the set of all relational
numbers
r * s = r + s − (r × s),for any two rational numbers r, s. Answer the following with justification:
(i) Is
(ii) Is * an associative operation on
(iii) Is * a commutative operation on
(iv) What is a * 1 for any a in
(v) Find two integers a ≠ 0 and b ≠ 0 such that a * b = 0.
Answer:
We are given an operation on the set of all rational numbers
(i)
The set of all rational numbers is closed under addition and multiplication.
Thus,
(ii)
Since
(iii)
Since
(iv)
a*1 = 1 for any a in
(v)
It is true for
Page No 76:
Question 6:
Find the multiplicative inverses of the following rational numbers:
Answer:
The multiplicative inverse of the rational number
The multiplicative inverse of the rational number
The multiplicative inverse of the rational
numbers
The multiplicative inverse of the rational number
The multiplicative inverse of the rational
number
Page No 76:
Question 7:
Write the following in increasing order:
Answer:
The LCM of the numerators 10, 20, 5, 40, 25, 10 is 200.
Thus, we get:
Therefore, we get the increasing order as:
Page No 76:
Question 8:
Write the following in decreasing order:
Answer:
We have the given fractions as:
Therefore, we get the decreasing order as:
Page No 76:
Question 9:
(a) What is the additive inverse of 0?
(b) What is the multiplicative inverse of 1?
(c) Which integers have multiplicative inverses?
Answer:
(a)The additive inverse of 0 is 0.
(b)The multiplicative inverse of 1 is 1.
(c) The integers 1 and −1 have multiplicative inverses.
Page No 77:
Question 4:
Suppose m is an integer such that m ≠ –1 and m ≠–2. Which is larger
Answer:
Reason:
Page No 77:
Question 10:
In the set of all rational numbers, give 2 examples each illustrating the following properties:
(i) associativity; (ii) commutativity; (iii) distributivity of multiplication over addition.
Answer:
(i) The examples for associative property for rational number are listed below.
For addition:
For multiplication:
(ii) The examples for commutative property for rational number are listed below.
For addition:
For multiplication:
(iii) The examples for distributivity of multiplication over addition for rational numbers are listed below.
Page No 77:
Question 11:
Simplify the following using distributive property;
Page No 77:
Question 12:
Simplify the following:
Answer:
(i)
(ii)
(iii)
(iv)
Page No 77:
Question 13:
Which is the property that is there in the set of all rationals but which is not in the set of all integers?
Answer:
The multiplicative inverse of a non-zero rational number is also a rational number. However, the multiplicative inverse of an integer except 1 and −1 is not an integer. Thus, every non-zero rational number is invertible, but only 1 and −1 are invertible integers.
Page No 77:
Question 14:
What is the
value of
Answer:
The value of
Page No 77:
Question 15:
Find the value of
Answer:
Page No 77:
Question 16:
Find all rational numbers each of which is equal to its reciprocal.
Answer:
The reciprocals of the rational numbers 1 and −1 are also 1 and −1 respectively, i.e., they are same as the number itself.
Thus, the required rational numbers are 1 and −1.
Page No 77:
Question 17:
A bus shuttles between two neigbouring towns every two hours. It starts from 8 AM in the morning and the last trip is at 6 PM. On one day the driver observed that the first trip had 30 passengers and each subsequent trip had one passenger less than the previous trip. How many passengers travelled on that day?
Answer:
Since the bus starts at 8 a.m. and the last trip is at 6 p.m. and it takes two hours for one trip, the total number of trips is 4.
Number of passenger in first trip = 30
∴Total number of passengers = 30 + 29 + 28 + 27 + 26 = 140
Page No 77:
Question 18:
How many rational numbers p/q are there between 0 and 1 for which q < p?
Answer:
Since
This rational number does not lie between 0 and 1.
Thus, there is no such rational number.
Page No 77:
Question 19:
Find all integers such
that
Answer:
At n = 0, we get:
At n = −1, we get:
At n = −2, we get:
At n = −3, we get:
At n = −4, we get:
At n = −5, we get:
Thus, the required integers are n = 0, −1, −3 and −4.
Page No 78:
Question 20:
By inserting parenthesis (that is brackets), you can get several values for 2 × 3 + 4 × 5. (For example ((2 × 3) + 4) × 5 is one way of inserting parenthesis). How many such values are there?
Answer:
We are
given
Inserting brackets, we get:
Page No 78:
Question 21:
Suppose p/q is a positive rational in its lowest form. Prove that
Answer:
We are given that
Since
Thus,
Therefore,
Page No 78:
Question 22:
Show that for each nautral number n, the
fraction
Answer:
Using Euclidean Algorithm, we have
21n + 4 = 1(14n + 3) + (7n + 1)
14n + 3 = 2(7n + 1) + 1
7n + 1 = 1(7n + 1) + 0
∴ GCD (7n + 1, 1) = 1
Also, GCD (21n + 4, 14n + 3) = GCD (14n + 3, 7n + 1) = GCD (7n + 1, 1) = 1
Since GCD (21n + 4, 14n + 3) =1, therefore, the fraction
Page No 78:
Question 23:
Find all integers n for which the number (n + 3) (n − 1) is also an integer.
Answer:
From the number (n + 3)(n − 1) , we observe that it is an integer for all integral values of n. Thus, the given number is an integer for all integers n.
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