Rational Numbers

For a fraction,

Where, n ≠ 0

(i) We have to express as a rational number with denominator 20.

In order to make the denominator 20, multiply 5 by 4.

Therefore,

(ii) We have to express as a rational number with denominator -30.

In order to make the denominator -30, multiply 5 by -6.

Therefore,

(iii) We have to express as a rational number with denominator 35.

In order to make the denominator 35, multiply 5 by 7.

Therefore,

(iv) We have to express as a rational number with denominator -40.

In order to make the denominator 20, multiply 5 by -8.

Therefore,

For a fraction,

Where, n ≠ 0 and n divides both a and b

(i) We have to express as a rational number with denominator 7.

In order to make the denominator 7, divide 98 by 14.

Therefore,

For a fraction,

Where, n ≠ 0 and n divides both a and b

We have to express as a rational number with denominator 5.

In order to make the denominator 5, divide 60 by 12.

Therefore,

A rational number is in standard or simplest or lowest form when-

1. Numerator and denominator have only 1 as its highest common factor.

2. Denominator is a positive integer.

(i) The HCF of 12 and 30 is 6

Therefore,

(ii) The HCF of 49 and 14 is 7

Therefore,

(iii) The HCF of 24 and 64 is 8

Therefore,

In order, to make the denominator positive, multiply both numerator and denominator by -1

(iv) The HCF of 36 and 63 is 9

Therefore,

In order, to make the denominator positive, multiply both numerator and denominator by -1

(i) is a positive number and all positive numbers are greater than 0.

Therefore,

(ii) is a negative number and all negative numbers are less than 0.

Therefore,

(iii) Both and have the same denominator 4.

Therefore, we can directly compare both the numbers.

Since, 1 > -3

Therefore,

(iv) Both and have the same denominator 7.

Therefore, we can directly compare both the numbers.

Since, -4 > -5

Therefore,

(v) and have different denominators.

Therefore, we take LCM of 3 and 4 that is 12.

Now,

And,

Since, 9 > 8

Therefore,

Hence,

(vi) We can write

and have different denominators.

Therefore, we take LCM of 1 and 2 that is 2.

Now,

And,

Since, -1 > -2

Therefore,

Hence,

(i) and have different denominators.

Therefore, we take LCM of 3 and 7 that is 21.

Now,

And,

Since, -24 > -28

Therefore,

Hence,

(ii)

and have different denominators.

Therefore, we take LCM of 9 and 8 that is 72.

Now,

And,

Since, -45 > -56

Therefore,

Hence,

(iii)

and have different denominators.

Therefore, we take LCM of 3 and 5 that is 15.

Now,

And,

Since, -5 > -12

Therefore,

Hence,

(iv)

And,

and have different denominators.

Therefore, we take LCM of 13 and 12 that is 156.

Now,

And,

Since, -91 > -108

Therefore,

Hence,

(v)

and have different denominators.

Therefore, we take LCM of 10 and 5 that is 10.

Now,

And,

Since, -7 > -8

Therefore,

Hence,

(vi)

We can write

and have different denominators.

Therefore, we take LCM of 1 and 5 that is 5.

Now,

And,

Since, -12 > -15

Therefore,

Hence,

(i) Clearly,

(ii)

and have different denominators.

Therefore, we take LCM of 13 and 91 that is 91.

Now,

And,

Clearly,

Hence,

(iii) We can write

and have different denominators.

Therefore, we take LCM of 1 and 5 that is 5.

Now,

And,

Since, -10 > -13

Therefore,

Hence,

(iv)

and have different denominators.

Therefore, we take LCM of 3 and 8 that is 24.

Now,

And,

Since, -16 < -15

Therefore,

Hence,

(v)

is a positive number and all positive numbers are greater than 0.

Therefore,

Hence,

(vi) and have different denominators.

Therefore, we take LCM of 9 and 10 that is 90.

Now,

And,

Since, -80 > -81

Therefore,

Hence,

(i)

And,

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 9, 12, 18 and 3 = 36

Clearly,

-24 < -16 < -15 < -14

Therefore,

Hence,

(ii)

And,

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 4, 12, 16 and 24 = 48

Clearly,

-36 < -21 < -20 < -18

Therefore,

Hence,

(iii)

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 5, 10, 15 and 20 = 60

Clearly,

-44 < -42 < -39 < -36

Therefore,

Hence,

(iv)

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 7, 14, 28 and 42 = 84

Clearly,

-54 < -48 < -46 < -39

Therefore,

Hence,

(i)

And,

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 1, 6and 3 = 6

Clearly,

2 > -12 > -13 > -16

Therefore,

Hence,

(ii)

And,

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 10, 15, 20 and 30 = 60

Clearly,

-18>-28>-33>-34

Therefore,

Hence,

(iii)

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 6, 12, 18 and 24 = 72

Clearly,

-42>-52>-60>-69

Therefore,

Hence,

(iv)

Since, the denominators of all the numbers are different therefore we will take LCM of the denominators.

LCM of 11, 22, 33 and 44 = 132

Clearly,

-92>-114>-117>-120

Therefore,

Hence,

(i) Every whole number a can be represented as

Therefore, every whole number is a rational number.

(ii) Every integer a can be represented as

Therefore, every integer is a rational number.

(iii) 0 can be represented as

Therefore, 0 is a whole number and a rational number.

(i) is greater than 0 and less than 1.

Therefore, it lies between 0 and 1

(ii) is greater than 0 and less than 1.

Therefore, it lies between 0 and 1

(iii)

is greater than 1 and less than 2.

Therefore, it lies between 1 and 2

(iv)

is greater than 2 and less than 3.

Therefore, it lies between 2 and 3.

(v)

is greater than 3 and less than 4.

Therefore, it lies between 3 and 4.

(vi)

is greater than 5 and less than 6.

Therefore, it lies between 5 and 6.

(vii)

is greater than 4 and less than 5.

Therefore, it lies between 4 and 5.

(viii) The number line representation of 8 is

(i) is greater than -1 and less than 0.

Therefore, it lies between -1 and 0

(ii) is greater than -1 and less than 0.

Therefore, it lies between -1 and 0

(iii)

is greater than -2 and less than -1`.

Therefore, it lies between -2 and -1

(iv)

is greater than -8 and less than -7`.

Therefore, it lies between -8 and -7

(v)

is greater than -5 and less than -4.

Therefore, it lies between -5 and -4

(vi)

is greater than -3 and less than -2.

Therefore, it lies between -3 and -2

(vii) The number line representation of -3 is

(viii)

is greater than -3 and less than -2.

Therefore, it lies between -3 and -2

(i) True

is a negative number.

All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line.

Hence, lies to the left of 0 on the number line.

(iii) False

is a negative number.

All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line.

Hence, lies to the left of 0 on the number line.

(iii)True

is a positive number.

All positive numbers are greater than 0 and therefore, lie to the right of 0 on the number line.

Hence, lies to the right of 0 on the number line.

is a negative number.

All negative numbers are less than 0 and therefore, lie to the left of 0 on the number line.

Hence, lies to the left of 0 on the number line.

Therefore, the rational numbers, and are on opposite sides of 0 on the number line.

(iv) False

is a positive number.

All positive numbers are greater than 0 and therefore, lie to the right of 0 on the number line.

Hence, lies to the right of 0 on the number line.

(i)

(ii)

(iii)

To convert it into lowest terms, divide both numerator and denominator by common divisor of both 6 and 8 that is, 2

(iv)

To convert it into lowest terms, divide both numerator and denominator by common divisor of both 6 and 3 that is, 3.

=2

(v)

To convert it into lowest terms, divide both numerator and denominator by common divisor of both 4 and 6 that is, 2.

(vi)

To convert it into lowest terms, divide both numerator and denominator by common divisor of both 18 and 15 that is, 3.

(i) Since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 4 and 5 = 20

And

Now,

(ii) Since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 8 and 12 = 24

And

Now,

(iii) Since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 9 and 6 = 18

And

Now,

(iv) Since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 16 and 24 = 48

And

Now,

(v) Since, the denominators of given rational numbers are negative therefore, we will make them positive.

Now, since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 18 and 27 = 54

And

Now,

(vi) Since, the denominators of given rational numbers are negative therefore, we will make them positive.

And,

Now, since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 12 and 15 = 60

And

Now,

(vii) We can write -1 as .

Now, since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 1 and 4 = 4

And

Now,

(viii) We can write 2 as .

Now, since, the denominators of given rational numbers are different therefore, we take their LCM.

LCM of 1 and 4 = 4

And

Now,

(ix)

On adding, any number to 0 we get the same number.

Therefore,

(i)LCM of 5 and 7 = 35

And,

Similarly,

LCM of 7 and 5 = 35

And,

i.e., LHS = RHS

Hence,

Verified

(ii)LCM of 13 and 8 = 104

And,

Similarly,

LCM of 8 and 13 = 104

And,

i.e., LHS = RHS

Hence,

Verified

(iii) 3 can be written as

LCM of 1 and 12 = 12

And,

Similarly,

LCM of 1 and 12 = 12

And,

i.e., LHS = RHS

Hence,

Verified

(iv) Since, the denominators are negative we will make them positive.

And,

LCM of 7 and 35 = 35

And,

Similarly,

LCM of 7 and 5 = 35

And,

i.e., LHS = RHS

Hence,

Verified

(i)

RHS = LHS

Verified

(ii)

RHS = LHS

Verified

(iii)

RHS = LHS

Verified

(i)

By Commutative property, i.e., a+b=b+a

Therefore,

(ii)

By Commutative property, i.e., a+b=b+a

Therefore,

(iii)

By Associative property, i.e., (a+b)+c=a+(b+c)

Therefore,

(iv)

By Associative property, i.e., (a+b)+c=a+(b+c)

Therefore,

(v)

By Associative property, i.e., (a+b)+c=a+(b+c)

Therefore,

(vi) 0,0

0 is the additive identity that is, if we add 0 to any number the result will be the number itself.

a+0=0+a

Also, By Commutative property, i.e., a+b=b+a

We get,

Additive inverse of a number is the number such that,

Therefore,

(i) Additive inverse of is

(ii) Additive inverse of is

(iii) Additive inverse of -18 is 18

(iv) Additive inverse of is

(v)

Therefore, Additive inverse of is

(vi)

Additive inverse of is

(vii) Additive inverse of is

(viii) Additive inverse of is

(ix)

Therefore, Additive inverse of is

(x)

Additive inverse of is

(i)

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 3 and 4 = 12

And,

Therefore,

(ii)

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 6 and 3 = 6

And,

Therefore,

(iii)

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 9 and 5 = 45

And,

Therefore,

(iv)

We can write,

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 1 and 7 =7

And,

Therefore,

(v)

We can write,

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 1 and 11 = 11

And,

Therefore,

(vi)

(vii)

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 13 and 5 = 65

And,

Therefore,

(viii)

We can write,

Since the denominators of both the numbers are different therefore, we will take their LCM

LCM 0f 1 and 7 = 7

And,

Therefore,

Rearrangement property says that, the numbers in an addition expression may be arranged and grouped in any order.

Therefore,

(i)

We arrange the numbers with same denominators together,

Now, we take LCM of 3 and 5=15

And,

Therefore,

(ii)

We arrange the numbers,

LCM of 3 and 6 =6

And,

LCM of 4 and 8 =8

And,

Now,

Now, we take LCM of 6 and 8=24

And,

Therefore,

In lowest terms,

(iii)

We arrange the numbers,

LCM of 20 and 10 =20

And,

LCM of 14 and 7 =14

And,

Now,

Now, we take LCM of 20 and 14=140

And,

Therefore,

(iv)

We arrange the numbers,

LCM of 4 and 9 =18

And,

Now,

In lowest terms,

Now, we take LCM of 1 and 18=18

And,

Therefore,

Sum of two rational numbers = -2

One number =

Let the other rational number = x

Now,

According to question,

Therefore, the other rational number is

Sum of two rational numbers =

One number =

Let the other rational number = x

Now,

According to question,

In lowest terms,

Therefore, the other rational number is

Let the number = x

Now,

According to question,

Therefore, should be added to so as to get

Let the number = x

Now,

According to question,

Therefore, should be added to so as to get

Let the number = x

Now,

According to question,

In lowest terms,

Therefore, should be subtracted from so as to get

(i) A Additive inverse of a number is the number such that,

0 is the rational number that is its own additive inverse

(ii) Let there be 2 rational numbers, and where, b≠0 and d≠0

LCM of b and d = bd

Where, bd ≠ 0

Therefore, is a rational number

Hence,

Yes, the difference of two rational numbers a rational number

(iii) Yes, addition is commutative on rational numbers

Let there be 2 rational numbers, and where, b≠0 and d≠0

Then,

(iv) Yes, addition is associative on rational numbers

Let there be 3 rational numbers, , and where, b≠0 , d≠0 and f≠0

Then,

(v) No, subtraction is not commutative on rational numbers

Let there be 2 rational numbers, and where, b≠0 and d≠0

Then,

(vi) No, addition is not associative on rational numbers

Let there be 3 rational numbers, , and where, b≠0 , d≠0 and f≠0

Then,

(vii) Negative of a negative rational number is the number itself without the negative sign.

(i)

(ii)

(iii)

In lowest terms,

(iv)

In lowest terms,

(v)

In lowest terms,

Further,

(vi)

In lowest terms,

(vii)

In lowest terms,

(viii)

In lowest terms,

Further,

(ix)

In lowest terms,

(x)

In lowest terms,

Further,

(xi)

In lowest terms,

Further,

(xii)

In lowest terms,

(i)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(ii)

LHS=RHS

Verified

(iii)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(iv)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(i)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(ii)

In lowest terms,

Further,

In lowest terms,

Further,

LHS=RHS

Verified

(iii)

In lowest terms,

Further,

In lowest terms,

Further,

LHS=RHS

Verified

(i)

By Commutative Property, i.e, a × b = b × a

(ii)

By Commutative Property, i.e, a × b = b × a

(iii)

By Associative Property, i.e, (a × b) × c = a × (b × c)

(iv)

By Associative Property, i.e, (a × b) × c = a × (b × c)

A multiplicative inverse for a number x, is a number which when multiplied by x yields the multiplicative identity, 1

The multiplicative inverse of a rational number is.

Therefore,

(i) The multiplicative inverse of =.

(ii) The multiplicative inverse of =.

In standard form,

(iii) The multiplicative inverse of =.

In standard form,

.

(iv) The multiplicative inverse of =.

(v) The multiplicative inverse of =.

(vi) The multiplicative inverse of =.

In standard form,

(vii) The multiplicative inverse of -1 =-1.

(viii) The multiplicative inverse of is undefined.

Since, is undefined.

(ix) The multiplicative inverse of =.

(x) The multiplicative inverse of =.

In standard form,

Let there be a rational number then =

Therefore,

(i)

(ii)

(iii)

(iv)

(i)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(ii)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(iii)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(iv)

In lowest terms,

In lowest terms,

LHS=RHS

Verified

(i) Commutative law i.e., a b = b a

(ii) Associative law i.e., a(bc) = (ab)c

(iii) Distributive law i.e., a(b + c) = ab + ac

(iv) Property of multiplicative identity i.e., a× 1=1× a

(v) Property of multiplicative inverse i.e., =1

(vi) Multiplicative property of 0 i.e., a× 0=0

(i) 1

(ii) No

(iii) 1 and -1

(iv) Not

(v)

(vi) a

(vii) Positive

(viii) Negative

(i)

In lowest terms,

(ii)

(iii)

In lowest terms,

(iv)

In lowest terms,

(v)

In lowest terms,

(vi)

In lowest terms,

Further,

(i)

Since, RHS = LHS

Therefore, True

(ii)

Since, RHS ≠ LHS

Therefore, False

(iii)

Since, RHS ≠ LHS

Therefore, False

(iv)

Since, RHS ≠ LHS

Therefore, False

(i)

RHS ≠ LHS

Hence, False

(ii)

RHS ≠ LHS

Hence, False

(iii)

RHS ≠ LHS

Hence, False

Product of two rational numbers = -9

One rational number = -12

Let the other rational number = x

Now,

According to the question,

-12 × x = -9

Hence, the other rational number is

Product of two rational numbers =

One rational number =

Let the other rational number = x

Now,

According to the question,

Hence, the other rational number is

Let x be multiplied by to get

It can be written as,

Hence, it should be multiplied by is

Let x be multiplied by to get

It can be written as,

Hence, it should be multiplied by is

Let be divided by x to get

It can be written as,

Hence, it should be multiplied by is

Sum of and -

Product of and -

Now,

According to the question,

According to the question,

(i)

Therefore,

(ii)

Therefore,

(iii)

Therefore,

(iv)

Therefore,

(i) No rational numbers are not always closed under division,

Since, which is not a rational number

(ii) No rational numbers are not always commutative under division,

Let and be two rational numbers.

And

Therefore,

Hence, rational numbers are not always commutative under division

(iii) No rational numbers are not always associative under division,

Let , and be two rational numbers.

And

Therefore,

Hence, rational numbers are not always associative under division.

(iv) No we cannot divide 1 by 0.

Since, which is not defined.

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between and

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between 2 and 3

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between and

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between -3 and -2

Now if we find a rational number between and -2 it will also be between -3 and -2 since lies between -3 and -2

Therefore, to find rational number y (let) between and -2

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between 4 and 5

Now if we find a rational number between and it will also be between 4 and 5 since lies between 4 and 5

Therefore, to find rational number y (let) between and

Now if we find a rational number between and 5it will also be between 4 and 5 since lies between 4 and 5

Therefore, to find rational number z (let) between and 5

To find a rational number x between two rational numbers and , we use

Therefore, to find rational number x (let) between and

Now if we find a rational number between and it will also be between and since lies between and

Therefore, to find rational number y (let) between and

Now if we find a rational number between andit will also be between and since lies between and

Therefore, to find rational number z (let) betweenand

We can write as (Since,

And,

We can write as (Since,

Now clearly, rational numbers between them are,

Any 10 rational numbers are,

We can write as (Since,

And,

We can write 2 as (Since,

Now clearly any 12 rational numbers between -1 and 2 are,

Length of rope = 11 m

Length of first piece cut = 2

Length of second piece cut = 3

Total length cut = Length of first piece cut + Length of second piece cut

Length of remaining rope = Length of rope - Total length cut

Hence, Length of remaining rope =

Weight of drum full of rice = kg

Weight of empty drum = kg

Weight of rice Weight of drum full of rice - Weight of empty drum

Hence, Weight of rice =

Weight of basket with three types of fruits = kg

Weight of apples = kg

Weight of oranges = kg

Weight of pears = Weight of basket with three types of fruits – (Weight of apples + Weight of oranges )

Hence, Weight of pears

Total Earnings = Rs 160

Spend on tea and snacks = Rs

Spend on food = Rs

Spend on repairs = Rs

Total Expenditure = Spend on tea and snacks + Spend on food + Spend on repairs

Savings= Total Earnings - Total Expenditure

Hence, Savings =

Cost of cloth per meter =

Total meters =

Cost of total cloth = Cost of cloth per meter × Total meters

Therefore, total cost

Speed of car =

Total hours =

Total Distance = Speed of car × Total hours

Therefore, Total Distance

Length of park =

Breadth of park =

Area of park = Length of park × Breadth of park

Hence, Area of park

Side of plot =

Area of plot = Side of plot × Side of plot

Hence, Area of plot

Cost of one litre petrol =

Cost of 34 litre petrol = 34 × Cost of one litre petrol

Cost of 34 litre petrol

Distance covered in one hour = 1020 km

Distance covered in hours = × Distance covered in one hour

Distance covered in hours

Cost of of cloth = Rs

Cost of of cloth = Cost of of cloth

Cost of of cloth

Length of cord = 71

No of pieces = 26

Length of each piece = Length of cord No of pieces

Length of each piece

Area of room =

Breadth of room =

Length of room = Area of room Breadth of room

Length of room

Product of two fractions = 9

First fraction = 9

Second fraction = Product of two fractions ÷ First fraction

Second fraction

Fraction of boys =

Fraction of girls =1- =

Number of girls= 240

Number of girls = Total students ×

⇒ 240 = Total students ×

⇒ Total students = 240 ÷

Total students =640

Number of boys = Total students - Number of girls

=640 – 240 = 400

Number of boys= 400

Fraction read =

Fraction left =

Pages left = 40

Pages left = Total pages

40= Total pages

⇒ Total pages =

Total pages = 180

Total money = Rs 300

Fraction spent on notebooks =

Amount spent on notebooks = = Rs 100

Amount left = Rs 300 – Rs 100 =Rs 200

Fraction spent on stationary =

Amount spent on stationary = = Rs 50

Money left = Rs 300 – Rs 150 = Rs 150

Total earnings = Rs 32000

Amount spend on food =

Amount left = Rs 32000 - Rs 8000 = Rs 24000

Amount spend on house rent =

Amount left = Rs 24000- Rs 7200 = Rs 16800

Amount spend on education =

Amount left = Rs 16800 - = Rs 12800

Money left = Rs 12800

Let the number be x

of x =

of x =

According to the question,

The number is 140

Fraction of spectators covered =

Fraction left =

Number of spectators in open = 15000

According to the question,

Number of spectators in open = Total number of spectators ×

LCM of 12 and 16 = 48

And,

Let the number added be x.

Then,

LCM of 3, 5, 15, 20

Let the number added be x.

Then,

Let the number added be x.

Then,

Let the number subtracted be x.

Then,

We know,
For any real number a≠0,

So,

Let the other number be x.

Then,

Let the other number be x.

Then,

Let the number subtracted be x.

Then,

Let the other number be x.

Then,

is not in standard form since 12 and 26 have a common divisor 2.

is not in standard form since its denominator is negative.

Therefore, only and are in standard forms as their numerator and denominator have no common divisor and their denominators are positive.

Additive inverse of a number is the number such that,

Therefore,

Additive inverse of is

Reciprocal of

Rational number between and

Let be a negative rational number

Then, its reciprocal will be which is also a negative rational number.

Hence, the reciprocal of a negative rational number is a negative rational number

Additive inverse of a number is the number such that,

Therefore,

(i)

Additive inverse of is

(ii) Additive inverse of is

Sum of two rational numbers = -4

First number =

Second number = Sum of two rational numbers - First number

Second number

Let the number added be x

Then,

Let the number subtracted be x

Then,

Multiplicative inverse of a rational number

Therefore,

(i) Negative inverse of

(ii) Negative inverse of

Let the other number be x

Then,

(i)

(ii)

(iii)

(i) Commutative law of multiplication i.e., a b = b a

(ii) 1 as multiplicative identity i.e., a × 1 = b × 1

(iii) Associative law of multiplication i.e., a(bc) = (ab)c

(iv) Multiplicative property of 0 i.e., a× 0=0

(v) Distributive law of multiplication over addition i.e., a(b + c) = ab + ac

Rational number between and

Now,

Rational number between and

Let the number added be x

Then,

Let the number added be x

Then,

Let the other number be x

Then,

Reciprocal of

Rational number between and

(i)

(ii)