Rational Numbers
Class 8th Mathematics AP Board Solution
- Name the property involved in the following examples
- 2 (3/5 + 1/2) = 2 (3/5) Name the property involved in the following examples…
- 3/7 x 1 = 3/7 = 1 x 3/7 Name the property involved in the following examples…
- (-2/5) x 1 = -2/5 Name the property involved in the following examples…
- 2/5 + 1/3 = 1/3 + 2/5 Name the property involved in the following examples…
- 5/2 x 3/7 = 15/14 Name the property involved in the following examples…
- 7a + (-7a) = 0 Name the property involved in the following examples…
- x x 1/x = 1 (x not equal 0) Name the property involved in the following…
- (2 x x) + (2 x 6) = 2 x (x+6) Name the property involved in the following…
- -3/5 Write the additive and the multiplicative inverses of the following.…
- 1 Write the additive and the multiplicative inverses of the following.…
- 0 Write the additive and the multiplicative inverses of the following.…
- 7/9 Write the additive and the multiplicative inverses of the following.…
- −1 Write the additive and the multiplicative inverses of the following.…
- Fill in the blanks.
- Fill in the blanks.
- 1 x 9/11 Fill in the blanks.
- Fill in the blanks.
- Multiply by the reciprocal of -5/14
- Which properties can be used in computing 2/5 x (5 x 7/6) + 1/3 x (3 x 4/11)…
- Verify the following (5/4 + -1/2) + -3/2 = -5/4 + (-1/2 + -3/2)
- Evaluate after rearrangement.
- 3/4 from 1/3 Subtract
- -32/13 from 2 Subtract
- −7 from -4/7 Subtract
- What numbers should be added to -5/8 so as to get -3/2 .
- The sum of two rational numbers is 8 If one of the numbers is -5/6 find the…
- Is subtraction associative in rational numbers? Explain with an example.…
- x = 2/15 Verify that - (-x) = x for
- x = -13/17 Verify that - (-x) = x for
- Write- (i) The set of numbers which do not have any additive identity (ii) The…
- Represent these numbers on the number line. (i) 9/7 (ii) -7/5
- Write five rational numbers which are smaller than 5/6 .
- Find 12 rational numbers between -1 and 2.
- Find a rational number between 2/3 and 3/4 . [Hint: First write the rational…
- Find ten rational numbers between - 3/4 and 5/6 .
- Express each of the following decimal in the p/q form. (i) 0.57 (ii) 0.176 (iii)…
- Express each of the following decimals in the rational form (p/q) .…
- Express each of the following decimals in the rational form (p/q) . 0. bar 57…
- Find (x + y) (x y) if
- Find (x + y) ÷ (x − y) if x = 1/4 , y = 3/2
- Divide the sum of - 13/5 and 12/7 by the product of - 13/7 and - 1/2 .…
- If of a number exceeds 1/7 of the same number by 36. Find the number.…
- Two pieces of lengths 2 3/5 m and 3 3/10 m are cut off from a rope 11 m long.…
- The cost of meters of cloth is 12 3/4 . Find the cost per metre.
- Find the area of a rectangular park which is 18 3/5 m long and 8 2/3 m broad.…
- What number should be divided by to get - 11/4 ?
- If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is…
- When the repeating decimal 0.363636 .... is written in simplest fractional form…
Exercise 1.1
Question 1.Name the property involved in the following examples
Answer: Role of zero/additive identity:
0 is the additive identity for rational numbers, i.e. if 'a' is any rational number then
a + 0 = a = 0 + a
Question 2.
Name the property involved in the following examples
Answer:
Distributive law
This property is know as the distributive law of multiplication over addition.
For all rational numbers a, b and c
a(b + c) = ab + ac
⇒
Question 3.
Name the property involved in the following examples
Answer:
Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 4.
Name the property involved in the following examples
Answer:
Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 5.
Name the property involved in the following examples
Answer:
Commutative law of addition
In commutative law of addition, a, b is rational number where
a + b = b + a
Question 6.
Name the property involved in the following examples
Answer:
Closure law in multiplication
We find that rational numbers are closed under multiplication. For any two rational numbers a and b, a
b is also rational number.
Question 7.
Name the property involved in the following examples
7a + (-7a) = 0
Answer:
(vii) Additive inverse law
Any two numbers whose sum is 0 are called the additive inverses of each other. In general if ‘a’ represents any rational number then a + (-a) = 0 and (-a ) + a = 0
Then a, (-a) are additive inverse of each other.
Question 8.
Name the property involved in the following examples
Answer:
(viii) Multiplicative inverse
We say that a rational number
is called the reciprocal or the multiplicative inverse of another rational number if 
Question 9.
Name the property involved in the following examples
Answer:
Distributive property
This property is known as distribution law of multiplication over addition. For all rational numbers a, b and c
a + b = b + a
Question 10.
Write the additive and the multiplicative inverses of the following.
Answer:
Additive inverse, 
Multiplicative inverse, 
Explanation:- When a number is added to its additive inverse, the result zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for the additive inverse:-
(add 
to both sides)
Solving for the multiplicative inverse:-
(dividing both sides by )
Question 11.
Write the additive and the multiplicative inverses of the following.
1
Answer:
Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
1 + x = 0
(add (-1) to both sides)
X = -1
Solving for the multiplicative inverse:-
1X = 1
(dividing both sides by 1)
X = 1
Question 12.
Write the additive and the multiplicative inverses of the following.
0
Answer:
Does not exist as the answer will be 0 itself.
Question 13.
Write the additive and the multiplicative inverses of the following.
Answer:
Additive inverse, 
Multiplication inverse, 
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
(add (
) to both sides)
Solving for the multiplicative inverse:-
(dividing both sides by 
)
Question 14.
Write the additive and the multiplicative inverses of the following.
−1
Answer:
Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
(-1) + x = 0
(add 1 to both sides)
X = 1
Solving for the multiplicative inverse:-
(-1)
X = 1
(dividing both sides by(-1))
X = 1
Question 15.
Fill in the blanks.
Answer:
let the blank be x
⇒ 
⇒ 
⇒ 
Question 16.
Fill in the blanks.
Answer:
let the blank be x
(add 
on both sides)
X = 0
Question 17.
Fill in the blanks.
Answer:
let the blank be x
Question 18.
Fill in the blanks.
Answer: We know, that rational numbers are associative over addition, i.e. if
are three rational number then, 
Therefore, on comparing we get
Question 19.
Multiply 
by the reciprocal of
Answer:
Reciprocal of 
is 
According to the given question
Question 20.
Which properties can be used in computing
Answer:
Multiplicative associative, multiplicative inverse, multiplicative identity, closure with addition are the properties used in computing.
Question 21.
Verify the following
Answer:
LHS:- 
RHS:- 
LHS = RHS
Hence verified
Question 22.
Evaluate
after rearrangement.
Answer:
Given, 
(rearranging the like fractions at one place)
(Since + 
- = -)
(L.C.M. of 53 = 15)
Question 23.
Subtract
from
Answer:
Given, 
(By taking L.C.M.)
Question 24.
Subtract
from 2
Answer:
Given, 
(Since - - = + )
(By taking L.C.M.)
Question 25.
Subtract
−7 from
Answer:
Given, 
(Since - - = + )
(By taking L.C.M.)
Question 26.
What numbers should be added to
so as to get
.
Answer:
let the unknown number be x
According to the given question,
(Add 
on both the sides)
(By taking L.C.M.)
Question 27.
The sum of two rational numbers is 8 If one of the numbers is 
find the other.
Answer:
let the unknown number be x
According to the given question,
(Add 
on both the sides)
(By taking L.C.M.)
Question 28.
Is subtraction associative in rational numbers? Explain with an example.
Answer:
Subtraction is not associative for rational numbers because when the numbers (say a,b,c) are subtracted by grouping any two at first and the other two at second [(a-b)-c and then a-(b-c)] the answer is not same. Thus subtraction is not associative in rational numbers.
Example:- let the 3 numbers be 5,8,9
Then, at first --
(9-5)-8 = -4
And in second - 9-(5-8) = 9-(-3) = 12
Since, first case is not equal to second case’s answer
Question 29.
Verify that – (–x) = x for
x = 
Answer:
According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
(Since - - = + )
LHS = RHS
Hence verified
Question 30.
Verify that – (–x) = x for
x = 
Answer:
According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
(Since - - = + )
LHS = RHS
Hence verified
Question 31.
Write-
(i) The set of numbers which do not have any additive identity
(ii) The rational number that does not have any reciprocal
(iii) The reciprocal of a negative rational number.
Answer:
(i) Natural numbers
(ii) 0(zero) is the rational number which does not have a reciprocal.
(iii) Is a negative rational number
The reciprocal of a negative number must itself be a negative number so that the number and its reciprocal multiply to 1.
Example:- 
Reciprocal 
Exercise 1.2
Question 1.Represent these numbers on the number line.
(i) 9/7 (ii) -7/5
Answer:
In a rational number, the number below the bar i.e. the denominator tells the number of equal parts into which the first unit has been divided. The numerator tells ‘how many’ of these parts are considered.
(i) Here means 9 markings of 
each on the right of zero and starting from 0. The 9th marking is.
(ii) Here 
means 7 markings of 
each on the left of zero and starting from 0. The 7th marking is.
Question 2.
Write five rational numbers which are smaller than
.
Answer:
Now we have to write 5 numbers which are less than
It is very simple.
Therefore
are 5 numbers which are less than
Question 3.
Find 12 rational numbers between -1 and 2.
Answer:
let us multiply and divide (-1) and 2 by 12
12 rational numbers are,
Question 4.
Find a rational number between
and
.
[Hint: First write the rational numbers with equal denominators.]
Answer:
make denominators same
Therefore 
lies between and 
Question 5.
Find ten rational numbers between
and 
.
Answer:
make denominators same
10 rational numbers are,
Exercise 1.3
Question 1.Express each of the following decimal in the
form.
(i) 0.57
(ii) 0.176
(iii) 1.00001
(iv) 25.125
Answer:
(i) 
(ii) 
(iii) 
(iv) 
Question 2.
Express each of the following decimals in the rational form 
.
Answer:
let x =
x = 0.99999….. -→(i)
Here the periodicity of the decimal is one
So, we multiply both sides of (i) by 10 and we get
10x = 9.999…-→(ii)
Subtract (i) from (ii)
10x = 9.999….
x = 0.999…
10x - x = 9.999... - 0.999...9x = 9.0
x = 1
Hence = 1
Question 3.
Express each of the following decimals in the rational form 
.
Answer:
let x =
X = 0.575757….. -→(i)
Here the periodicity of the decimal is two
So, we multiply both sides of (i) by 100 and we get
100x = 57.575757…-→(ii)
Subtract (i) from (ii)
100x = 57.5757….
X = 0.5757…
99x = 57.0
(divide by 99 on both sides)
Hence =
Question 4.
Find (x + y) ÷ (x − y) if
Answer:
According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
Question 5.
Find (x + y) ÷ (x − y) if
x = 
, y = 
Answer:
According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
Question 6.
Divide the sum of 
and 
by the product of
and
.
Answer:
To find the sum,
To find the sum,
According to the given question,
Question 7.
If 
of a number exceeds 
of the same number by 36. Find the number.
Answer:
let the number is x
Then 
of x 
According to the question,
9x = 3635
(multiply 35 on both the sides)
9X = 1260
(divide by 9 on both the sides)
X = 140
Question 8.
Two pieces of lengths 
m and 
m are cut off from a rope 11 m long. What is the length of the remaining rope?
Answer:
total length of the rope = 11m
Let the third part of rope be x
Length of first piece = 
m = 
m
Length of second piece = 
m = 
m
According to the question,
(By taking L.C.M.)
Question 9.
The cost of 
meters of cloth is
. Find the cost per metre.
Answer:
cost of cloth = 
= 
Length of cloth = 
= 
Cost per meter = 
(divide cost of cloth by length of cloth)
= 1.66 is the cost per meter
Question 10.
Find the area of a rectangular park which is
m long and 
m broad.
Answer:
length = 
m = 
Breadth = 
m = 
Area = length breadth
= 
Question 11.
What number should
be divided by to get
?
Answer:
let the number which be divided by be x
According to the question
(Multiply by x on both the sides)
Question 12.
If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is the length of the cloth required for each trouser?
Answer:
number of trousers = 36
Length of cloth = 64m
Length required for each trouser = 
m
Question 13.
When the repeating decimal 0.363636 .... is written in simplest fractional form
, find the sum p + q.
Answer:
x = 0.363636…(i)
Periodicity = 2
So,
100x = 36.363636….(ii)
From (i) and (ii)
100x = 36.363636….
X = 0.3636363…
_____________________
99x = 36
X = 
Then p = 4 and q = 11
p + q = 4 + 11 = 15