Rationals Numbers

Division by 0 is not defined i.e. you cannot divide any number by 0.

Addition of 0 is defined. p + 0 = p;

Subtraction of 0 is defined p-0 = p

Multiplication by 0 is defined p×0 = 0.

A rational number is positive if

1.both numerator and denominator are greater than 0, i.e. positive.

2.both numerator and denominator are less than 0 , i.e. negative.

In option (C) both numerator and denominator are negative.

Option(A) :

Both numerator and denominator is positive,

∴ number is positive.

Option(B): Both numerator and denominator is negative,

∴ number is positive.

Option(C): Both numerator and denominator is positive,

∴ number is positive.

Option(D): Numerator is positive but denominator is negative

∴ number is negative

Standard form is the simplest form.

If there is any common factor other than 1, we will divide numerator and denominator both by that factor to convert it into standard form .

1 =

To find reciprocal of fraction,

Interchange its numerator and denominator .

Recipracal of 1 = = 1

2 =

∴ Reciprocal of 2 =

Reciprocal of = 2

Reciprocal of 0 = not defined

To find reciprocal of fraction,

Interchange its numerator and denominator .

∴ Reciprocal of = 2

Common factor of 4 and 5 is only 1.

Common factor of 4 and 5 is only 1

∴ is in standard form

Common factor of 5 and 4 is only 1

∴ is in standard form

Common factor of 16 and 25 is only 1

∴ is in standard form

Hence, is equivalent to

We can find any number of rational numbers between two rational numbers by making their denominator and increasing the numerator of the smaller number.

For example, we have to find rational numbers between = and

So, the rational numbers between and will be:

Similarly, we can find any number of rational numbers between

And

Standard form is such that 1 is the only common factor of numerator and denominator.

And denominator should be positive.

Standard form is such that 1 is the only common factor of numerator and denominator.

Since, HCF is the greatest number that is a factor of both numerator and denominator , if we will divide both by HCF, there will be no common factor except 1.

Positive number>zero>negative number.

negative

A rational number is negative if either of numerator and denominator is negative but not both.

Positive rational number.

Example, -1 is a negative rational number.

HCF of 8 and 36 is 4.

Now, dividing numerator and denominator by 4

To convert into standard form ,

Divide numerator and denominator by HCF and make denominator positive.

HCF of 18, 24 = 6

Dividing numerator and denominator by 6

Multiplying -1 to both numerator and denominator (to make denominator positive)

left

Numerator is negative and denominator is positive.

∴ is a negative rational number.

On a number line,

Negative numbers are to the left side of zero(0).

Right

Both numerator and denominator are positive.

∴ is a positive number.

On a number line,

Positive numbers are to the right side of zero(0).

less

is a negative rational number.

is a positive rational number.

Negative number<zero(0)<positive number.

less

is a negative number.

Negative number<zero(0)<positive number.

Negative

If either numerator or denominator is negative, but not both,

The rational number is negative.

equivalent

Two rational numbers are said to be equivalent if they are equal.

HCF of 3 and 5 is 1.

∴ is in standard form.

HCF of 27 and 45 is 9.

Dividing numerator and denominator of by 9

∴

Hence, both are equivalent rational numbers.

Additive inverse of a number is the number adding which, the result is zero.

To find additive inverse,

Reverse the sign of the rational number.

LCM of 5 and 6 is 30.

Multiplying numerator and denominator of by 6.

We get,

Multiplying numerator and denominator of by 5.

We get,

∴

LCM of 35 and 30 = 5

Dividing numerator and denominator by 5

HCF of 6, 12 = 6

Dividing numerator and denominator by 6

HCF of 15, 15 = 15

Dividing numerator and denominator by 15

-36

To make denominator 42,

We have to multiply numerator and denominator by 42÷7 = 6

Multiplying numerator and denominator by 6

∴

12

To make numerator 6,

We have to multiply numerator and denominator by 6÷1 = 6

Multiplying numerator and denominator by 6

∴

-1

LCM of 9, 9 = 9

Dividing Numerator and denominator by 9

<

In

Numerator is positive, Denominator is negative

∴ It is negative rational number.

In

Both numerator and denominator are positive

∴ it is positive rational number .

Negative number < Positive Number

>

In

Numerator is negative, Denominator is positive

∴ It is negative rational number.

In

Both numerator and denominator are positive

∴ it is positive rational number .

Positive number < Negative Number

<

LCM of 6 and 4 = 12

To make denominator 12,

Multiplying numerator and denominator of by 2

Multiplying numerator and denominator of by 3

10<24

∴

∴

<

To compare 2 negative numbers ,

First compare them without negative sign

And then reverse the sign of greater than/less than

Comparing and

9>4

∴

Reversing the sign

.

=

LCM of 8, 2 = 8

To make denominator of , 8

Multiply numerator and denominator by 8÷2 = 4

8 = 8

∴

∴

Zero

For Reciprocal, numerator should be made denominator and denominator should be made numerator.

Division by zero is not defined

∴ zero cannot be in denominator

Hence, reciprocal of zero does not exist.

1

1 =

Interchanging numerator by denominator ,

Reciprocal = 1.

For division by a rational number,

Multiply by its reciprocal.

Interchanging numerator and denominator

Reciprocal of

∴

Multiplying numerator and denominator by -1

0

Division of zero by any number is zero.

0

Multiplying any number by 0 gives 0.

Multiplying a rational number by its reciprocal gives 1

Numerator of -2

Denominator of 5

Interchanging numerator and denominator

Reciprocal of

In standard form

HCF of numerator and Denominator should be 1

And

Denominator should be positive.

b÷m

A fraction is not changed if both numerator and denominator is

1. Multiplied by same non zero number.

2. Divided by same non zero number.

positive, negative

If both numerator and denominator are positive,

Rational number is positive.

If either of numerator or denominator is negative but not both,

Rational number is negative.

Standard

If two numbers are equal, they will be reduced to same fraction after dividing numerator and denominator by their HCF.

Zero

Division by zero is not defined.

True

Every natural number N can be represented int the form of as

True

Zero can be represented in the form as or etc.

True

Every Integer I can be represented in the form as

∴ Every integer is a rational number

Every rational number cannot be an integer, as it can have irremovable fractional part . Example :

False

Every Integer I can be represented in the form as

∴ Every integer is a rational number

While representing the integer in the form

Its value will not be changed .

i.e. if integer is less than 0

corresponding rational number will also be less than 0

∴ every negative integer is a negative rational number.

True

A fraction is not changed if both numerator and denominator is

1. Multiplied by same non zero number.

2. Divided by same non zero number.

True

A fraction is not changed if both numerator and denominator is

1. Multiplied by same non zero number.

2. Divided by same non zero number.

True

It is true that, in a rational number of the form, , where q ≠ 0.

False

A rational number is not changed if both numerator and denominator is

1. Multiplied by same non zero number.

2. Divided by same non zero number.

True

To add two rational numbers,

We will make their denominators same.

Then, we will add numerator

So, the result is again in the form

Hence, Sum of two rational numbers is always a rational number

True

All decimal numbers are also rational numbers because every decimal number can be expressed in the form of , where p and q are integers and q ≠ 0

For example,

1. .88 is a decimal number, and it can be expressed as .

Converting it into its standard form,

⇒ =

⇒ , which is a rational number

Since 22 and 25 both are integers and q ≠ 0

False

Let’s take 1 and 0 as two numbers,

Both of these are rational numbers but their division is not defined.

⇒ = not defined

Also, is not a rational number as q = 0.

Hence, The quotient of two rationals is not always a rational number.

True

Every fraction can be expressed in the form , where p and q are integers and q ≠ 0

For, example:

one – fourth part

Since both 1 and 4 are integers and 4 ≠ 0

Hence, is a rational number.

Note: All integers and fractions are rational numbers.

False

Let’s take (first rational) and (second rational)

As given both the rationals have different numerators.

Since, 15 and 18 have 3 in common hence,

⇒ =

⇒ (second rational) = (first rational )

Hence, two rational with different numerators can be equal.

True,

8 can be written as , where ‘m’ is any integer.

For example:

True

First rational number =

4 = 2 × 2

6 = 2 × 3

H.C.F. = 2

So, to convert into its standard form, divide the numerator and denominator by their H.C.F.

⇒ =

⇒

Now, consider the second rational number,

=

= First rational number

Hence, is equivalent to .

False

is a negative rational number, and all negative numbers lie left of zero on the number line.

True

The rational number have ‘-‘ common in both numerator and denominator so,

⇒ =

⇒ A positive rational number.

Hence, it lies right of zero on the number line

is a negative rational number.

So, it lies left of zero on the number line.

⇒ and lies on the opposite side of zero on the number line.

False

is a rational number but it is not a whole number because,

The whole number is natural number including 0.

0,1,2,3,…….

False

A rational number can be negative for example,

-1,-8,……

Which are less than zero.

(i) (c)

⇒

⇒

= 1

Note: To divide one rational number by other non zero rational number, We multiply the first rational number by reciprocal of the other.

(ii) (e)

⇒

⇒

Note: To divide one rational number by other non-zero rational number, We multiply the first rational number by reciprocal of the other.

(iii) (a)

⇒

⇒

Reciprocal of (-1) is itself (-1).

(iv) (b)

⇒

= -1

Note: To divide one rational number by other non-zero rational number, We multiply the first rational number by reciprocal of the other.

(v) (d)

⇒

⇒

Multiplying by -1 in both numerator and denominator we get,

⇒

⇒

Now, for

Multiplying by -1 in both numerator and denominator we get,

⇒

⇒

Now, for

Multiplying by -1 in both numerator and denominator we get,

⇒

⇒

(i) as a rational number with denominator = 36

Multiplying by 9 in both numerator and denominator we get,

⇒

⇒

(ii) as a rational number with denominator = -80

Multiplying by -20 in both numerator and denominator we get,

⇒

⇒

(i) Considering the rational number =

-60 = 2 × 2 × 3 × 5 × -1

72 = 2 × 2 × 2 × 3 × 3

H.C.F. = 12

To convert into it’s lowest form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

(ii) considering the rational number =

91 = 7 × 13

-364 = 2 × 2 × 7 × 13 × -1

H.C.F. = 91

To convert into it’s lowest form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

=

A. Considering the rational number =

-12 = 2 × 2 × 3 × -1

72 = 2 × 2 × 2 × 3 × 3

H.C.F. = 12

To convert into it’s standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

B. Considering the rational number =

14 = 2 × 7

-49 = 7 × 7 × -1

H.C.F. = 7

To convert into it’s standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

C. Considering the rational number =

-15 = 3 × 5 × -1

35 = 5 × 7

H.C.F. = 5

To convert into it’s standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

D. Considering the rational number =

299 = 13 × 23

161 = 7 × 23

H.C.F. = 23

To convert into standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

Yes

Considering the rational number

-8 = 2 × 2 × 2 × -1

28 = 2 × 2 × 7

H.C.F. = 4

To convert into standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

Now, considering the rational number =

32 = 2 × 2 × 2 × 2 × 2

-112 = 2 × 2 × 2 × 2 × 7 × -1

H.C.F. = 16

To convert into standard form,

Divide the numerator and denominator by H.C.F.

⇒

⇒

=

Since, and have same standard form.

Hence, they are equivalent.

For comparing the rational numbers we should make the denominator equal of all rationals.

We have to compare so,

L.C.M. of 10, 8 , 3 , 4 , 5

= 120

⇒

⇒

As we know,

-84 < -80 < -75 < -72 < -30

So, order of ascending will be:

⇒

Since,

, we will divide the 0-1 in 8 parts, and will mark

Since,

, we will divide the (-3) - (-2) into three parts.

Since,

, we will divide (-4) – (-3) into six parts.

According to the question,

Using cross multiplication we get,

⇒ -5 × 28 = 7 × x

⇒ 7 × x = -5 × 28

⇒ x =

⇒ x = -5 × 4

⇒ x = -20

Hence, the value of x = -20.

i. NOTE: If the numerator and denominator of a rational number is multiplied or divided by a non – zero integer, we get a rational number which is said to be equivalent of given number.

First number equivalent to

⇒

⇒

Second number equivalent to

⇒

⇒

Third number equivalent to

⇒

⇒

ii. NOTE: If the numerator and denominator of a rational number is multiplied or divided by a non – zero integer, we get a rational number which is said to be equivalent of given number.

First number equivalent to

⇒

⇒

Second number equivalent to

⇒

⇒

Third number equivalent to

⇒

⇒

(i) As we can see the pattern is,

So, the next three rational numbers are:

(ii) As we can see the pattern is,

So, the next three rational numbers are:

A rational number between and

⇒

⇒

⇒

Second rational number between and

⇒

⇒

⇒

Third rational number between and

⇒

⇒

⇒

Fourth rational number between and

⇒

⇒ ()

⇒

i.

⇒

⇒

⇒

ii.

⇒

⇒

⇒ = 1.

i.

⇒

⇒

⇒

ii.

⇒

⇒

⇒ =

i.

Product of two rational numbers =

⇒

⇒

⇒

ii.

Product of two rational numbers =

⇒

⇒

⇒

i. Product of two rational numbers =

⇒

⇒

⇒

⇒

⇒ =

⇒ 13

ii. Product of two rational numbers =

⇒

⇒

⇒ =

i. To divide one rational by other non – zero rational number, We multiply the first rational number by the reciprocal of the other.

⇒

⇒

⇒

ii. To divide one rational by other non – zero rational number, We multiply the first rational number by the reciprocal of the other.

⇒

⇒ -2

(i) L.C.M. of 4 and 8 = 8

⇒ ,

⇒

Hence, is greater than .

(ii) Simplifying the above fractions we get,

=

L.C.M. of 7 and 9 = 63

⇒

⇒

Since, first number is ‘negative’ so,

⇒ is greater than .

Numerator is less than ‘-7 × 11’

⇒ Numerator is less than ‘-77’

⇒ Numerator = -78, -79,………

Denominator is greater than ’12 + 4’

⇒ Denominator is greater than ‘16’

⇒ Denominator = 17, 18,……

So, Rational number =

x + y = ()

⇒

⇒

⇒

x – y = ()

⇒

⇒

⇒ =

x × y = ()

⇒

x ÷ y = ()

⇒

⇒ =

i. () + ()

⇒ + 3

⇒

ii. ()

⇒

iii.

⇒

⇒

iv. () – ()

⇒ - ()

⇒ -4 +

⇒

⇒

I.

⇒

⇒

II.

⇒

⇒

III.

⇒

⇒

IV.

⇒

⇒

⇒

⇒

V.

⇒

⇒

VI.

⇒

⇒

VII.

⇒

⇒

VIII.

⇒

⇒

A.

B. =

C.

D.

E.

F.

P = m × t

q = n × t

⇒

⇒ =

(a) If p × s < r × q

By cross multiplication,

Since, and are two rational numbers

⇒

(b) Using cross multiplication we get,

⇒ p × s = q × r

(c) If p × s > r × q

Since, and are two rational numbers

⇒

(a) Numerator = 5 – 39

⇒ Numerator = -34

Denominator = 54 – 6

⇒ Denominator = 48

⇒ Rational number =

(b) Numerator = -4 × 6

Numerator = -24

Denominator = 8 ÷ 2

Denominator = 4

⇒ Rational number =

=

(c) Numerator =

⇒ Numerator = -5

Denominator = 35 – 18

⇒ Denominator = 17

⇒ Rational number =

(d) Numerator = 25 + 15

⇒ Numerator = 40

Denominator =

⇒ Rational number =

⇒ Rational number =

A. 35% =

⇒

B. 1.2 =

⇒

C. =

⇒

D. 240 ÷ (-840)

⇒ =

⇒

E. 115 ÷ 207

⇒

A.

⇒ [ Adding additive inverse to

= 0

B.

⇒

⇒

⇒ =

C.

⇒

⇒

⇒

⇒

D.

⇒

⇒

⇒

(a) Let’s the rational number is p.

⇒ x + p = y

⇒ p = y – x

⇒ p =

⇒ p =

⇒ p =

⇒ p =

(b) Let’s the rational number is p.

⇒ y – p = z

⇒ p = y – z

⇒ p =

⇒ p =

⇒ p =

(c) ⇒ z + p = x

⇒ p = x – z

⇒ p =

⇒ p =

⇒ p =

⇒ p =

⇒ p × y = x

⇒ p =

⇒ p = =

⇒ p =

(e) ⇒ The reciprocal of x + y =

⇒

⇒ =

(f) ⇒ The reciprocals of x and y are =

⇒ The sum of reciprocals of x and y are =

⇒

⇒ =

⇒ =

⇒

(g) () ×

⇒ ×

⇒

⇒

(h) (- ) +

⇒ () +

⇒

⇒

⇒

(i)

⇒ + ()

⇒ + ()

⇒

⇒

(j)

⇒ ÷ ()

⇒ ÷

⇒

⇒

(k)

⇒ - ()

⇒

⇒ =

⇒

Nearest natural number to is 1.

⇒

⇒ p = 1 +

⇒ p =

Nearest integer to is = -1.

⇒

⇒ p =

⇒ p =

Nearest integer to is = -1.

⇒

⇒ Using cross multiplication

⇒ p = =

Greatest negative integer = -1.

⇒ p ÷ = -1

⇒ p × 2 = -1

⇒ p =

Length of one piece = = m

Let’s number of such pieces = p

⇒ p × = 68 m

Using cross multiplication

⇒ p =

⇒ p = 16

Hence, the number of pieces = 16.

Let’s length of cloth required for each shirt = p m

⇒ 12 × p = 27

⇒ p =

⇒ p = m

i. First rational number =

⇒

⇒

Second rational number = =

Third equivalent rational number = =

ii. First rational number =

= = -5

Second rational number =

Third rational number =

(a) -114

⇒ Integer, Rational number

(b)

⇒ Fraction, Rational number

(c)

⇒ Natural number, Whole number, Integer, Rational number

(d) =

Rational number

(e)

⇒ Fraction, Rational number

(f) 0

⇒ Whole number, Integer, Rational number

For the largest value of .

Let’s a = 50 and b = 1;

⇒ =

⇒

For the largest value of .

Let’s a = 50 and b = 49;

⇒ =

⇒

Students studying English, Maths or both are = 150

Student studying English are = = 93

Now, students studying Maths are = = 102

Using the concept of sets,

Students studying both = Students studying Maths + Students Studying English – Students studying both English, Maths or both

⇒ Students studying both = 102 + 93 – 150

⇒ Students studying both = 195 – 150

⇒ Students studying both = 45

Let’s volume of the body = V

So, Exposed volume =

Now, the submerged volume of the body =

=

⇒ Ratio of submerged volume to it’s exposed volume

= = = 7:2

As a Rational number =

(D)

A. = 1 B. = 1

C. =1 but in option D. = -1.

(C)

All are standard form of

Except, which is =

(B)

All Rationals except have even numerator and odd denominator.

(A)

The Standard form of above rationals are

Since, All rationals have denominator = 5, except which have 7 as a denominator.

Considering

-25 = 5 × 5 × -1

-30 = 2 × 3 × 5 × -1

H.C.F. = 5

To get standard form,

Dividing by 5 in both numerator and denominator

⇒ =

ERROR: Chhaya did not divide numerator by 5 and denominator by -5.