Rational Numbers

Note: Rational number is any number that can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

(i) We have to find five rational numbers that lie between the numbers -1 and 0.
We need 5 rational number.
So multiply a number by 6/6

Five rational numbers between -1 and 0 are:

(ii) Here, we have to find five rational numbers that lie between the numbers -2 and -1.

Therefore,

Five rational numbers between -2 and -1 are:

(iii) Here, we have to find five rational numbers that lie between the numbers

and

Now make the denominator same,

And, =

Thus, we write five rational numbers between and as:

(iv) Here, we have to find five rational numbers that lie between the numbers

and

So,

And, =

Therefore, five rational numbers between and are:

Note: You can divide and multiply with any number to make the denominator of two rational numbers same.

The parts of the given question are solved below:

(a) We have,

,

We can observe that,

The numerator is a multiple of 3

And the denominator is a multiple of 5

Hence,

The next four rational numbers in this pattern are as follows:

… =

(i) Here, we have,

,

We can observe that,

The numerator and denominator are a multiple of increasing natural numbers.

Hence,

The next four rational numbers in this pattern are as follows:

, …

, …

(ii) Here, we have,

,

We can observe that,

The numerator and denominator are a multiple of increasing natural numbers.

Hence,

The next four rational numbers in this pattern are as follows:

, …

, …

(iii) Here, we have,

,

We can observe that,

The numerator and denominator are a multiple of increasing natural numbers.

Hence,

The next four rational numbers in this pattern are as follows:

, …

, …

(i) We can find the four equivalent fractions of as follows:

Thus, the required equivalent fractions are:

(ii) We can find the four equivalent fractions of as follows:

Thus, the required equivalent fractions are:

(iii) We can find the four equivalent fractions of as follows:

Thus, the required equivalent fractions are:

The parts of the given question are solved below:

(i) We can observe that the fraction represents 3 parts out of 4.

Hence,

Each space between two integers will be divided into four equal parts.

Therefore,

can be represented on the number line as:

(ii) We can observe that the fraction represents 5 parts out of 8 equal parts.

Hence,

Each space between two integers will be divided into eight equal parts.

Therefore,

can be represented on the number line as:

(iii) We can observe that the fraction represents 1 full part and 3 parts out of 4.

Hence,

Each space between two integers will be divided into four equal parts.

Therefore,

can be represented on the number line as:

(iv) We can observe that the fraction represents 7 parts out of 8.

Hence,

Each space between two integers will be divided into eight equal parts.

Therefore,

can be represented on the number line as:

Here,

Distance between U and T is 1 unit

And,

It is divided into three equal parts.

That are:

TR = RS = SU =

As the given points are in the Left side of -1.

Thus,

R =

=

=

S =

=

=

In the same way,

AB = 1 unit

And,

It is divided into three equal parts.

P =

=

=

Q=

=

=

The parts of the given question are solved below:

(i) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

And,

Clearly,

We can see that,

Hence,

It does not represent the same rational number.

(ii) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

And,

Clearly,

We can see that,

Hence,

It represents the same rational number.

(iii) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

Clearly,

We can see that,

Hence,

It represents the same rational number.

(iv) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

Clearly,

We can see that,

Hence,

It represents the same rational number.

(v) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

And,

Clearly,

We can see that,

Hence,

It can be concluded that,

It represent the same rational number.

(vi) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

And,

Clearly,

We can see that,

Hence,

It can be concluded that,

It does not represent the same rational number.

(vii) We have,

Now,

We have to check if the two given are the same rational number.

Thus,

And,

Clearly,

We can see that,

Hence,

It does not represent the same rational number.

The parts of the given question are solved below:

(i) We can simplify the given fraction as follows:

Hence, is the simplest form of the given fraction.

(ii) We can simplify the given fraction as follows:

Hence, is the simplest form of the given fraction.

(iii) We can simplify the given fraction as follows:

Hence, is the simplest form of the given fraction.

(iv) We can simplify the given fraction as follows:

Hence, is the simplest form of the given fraction.

In this question we have to compare two given fraction i.e. to find whether the given two fractions are equal or one is greater or less than the other.

Concept: It is easier to compare fractions with like denominators. So, we convert all the fractions into like denominators and the fraction which has greater denominator will be greater than the other fraction.

The parts of the given question are solved below:

(i) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

14 is greater than -15

Hence,

<

(ii) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

-25 is greater than -28

Hence,

<

(iii) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

-14 is equal to -14

Hence,

=

(iv) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

-32 is greater than -35

Hence,

>

(v) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

-3 is greater than --4

Hence,

<

(vi) We have,

Now, the two fractions already have like denominators, so we just need to check which is the fraction in which the numerator is larger.

Note: The fraction is considered negative if it has a negative sign either in numerator or in denominator.

-5 is equal to -5

Hence,

=

(vii) Here,

We have,

As you know, that Zero is greater than any negative rational number.

Hence,

0 >

In this question we have to compare two given fraction i.e. to find whether the given two fractions are equal or one is greater or less than the other.

Concept: It is easier to compare fractions with like denominators. So, we convert all the fractions into like denominators and the fraction which has greater denominator will be greater than the other fraction.

The parts of the given question are solved below:

(i) We have,

Writing the fractions in the form of like denominators we get,

Now,

And,

Since,

15 is greater than 4

Hence,

is greater than

(ii) We have,

Writing the fractions in the form of like denominators we get,

And,

Since,

-5 is greater than -8

Hence,

is greater than

(iii) We have,

Now, Writing the fractions in the form of like denominators we get,

And,

Since,

-8 is greater than -9

Hence,

is greater than

(iv) Here,

We have,

Now, in the question the fractions given already have like denominators.

But, one of the given fractions is positive and the other fraction is negative.

Since, positive is always greater than Negative

Hence,

is greater than

(v) Here,

We have,

First convert the mixed fraction into improper fraction. An improper fraction is the one whose numerator is bigger than the denominator. On doing so, we get,

And,

Since,

-115 is greater than -133

Hence,

is greater than

The parts of the given question are solved below:

(i) Here,

We have,

Now,

We have to arrange the given fractions in ascending order,

Since,

-3 < -2 < -1

Therefore,

It can be concluded that,

(ii) We have,

Now, we have to arrange the given fractions in ascending order,

Converting them into like fractions:

Since,

-12 < -3 < -2

Therefore,

It can be concluded that,

(iii) We have,

No, we have to arrange the given fractions in ascending order,

Now,

Converting them into like fractions:

Since,

-42< -21< -12

Therefore,

It can be concluded that,

The parts of this question are solved below:

(i) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(ii) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(iii) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(iv) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(v) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(vi) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(vii) In order to find the sum of the given fractions

We will follow the following steps:

Therefore,

(i) Here,

We have:

-

We can solve the above given fractions as follows:

-

=

=

Hence,

The solution for the given fraction is

(ii) Here,

We have:

– ()

We can solve the above given fractions as follows:

– ()

=

=

Hence,

The solution for the given fraction is

(iii) Here,

We have:

– ()

We can solve the above given fractions as follows:

– ()

=

=

Hence,

The solution for the given fraction is

(iv) Here,

We have:

– ()

We can solve the above given fractions as follows:

– ()

=

=

Hence,

The solution for the given fraction is

(v) Here,

We have:

- 2 – 6

= - 6

We can solve the above given fractions as follows:

– 6

=

=

Hence,

The solution for the given fraction is

(i) Here,

We have:

× ()

We can solve the above given fractions as follows:

× ()

=

=

Hence,

The solution for the given fraction is

(ii) Here,

We have:

× -9

We can solve the above given fractions as follows:

× - 9

=

=

Hence,

The solution for the given fraction is

(iii) Here,

We have:

× ()

We can solve the above given fractions as follows:

× ()

=

=

Hence,

The solution for the given fraction is

(iv) Here,

We have:

× ()

We can solve the above given fractions as follows:

× ()

=

=

Hence,

The solution for the given fraction is

(v) Here,

We have:

× ()

We can solve the above given fractions as follows:

× ()

=

=

Hence,

The solution for the given fraction is

(vi) Here,

We have:

× ()

We can solve the above given fractions as follows:

× ()

=

= 1

Hence,

The solution for the given fraction is 1

(i) Here

We have:

(-4)

Or we can write it as: -4 × ()

Now, we can solve the above given fractions as follows:

-4 × ()

=

=

Hence,

The solution for the given fraction is -6.

(ii) Here

We have:

2

Or we can write it as: - ×

Now, we can solve the above-given fractions as follows:

×

=

=

Hence,

The solution for the given fraction is

(iii) Here

We have:

(-3)

Or we can write it as: - ×

Now, we can solve the above given fractions as follows:

- × (- )

=

=

Hence,

The solution for the given fraction is

(iv) Here

We have:

Or we can write it as: - × ()

Now, we can solve the above-given fractions as follows:

- × ()

=

=

=

Hence,

The solution for the given fraction is

(v) Here

We have:

(-)

Or we can write it as: - × 7

Now, we can solve the above given fractions as follows:

- × 7

=

=

Hence,

The solution for the given fraction is

(vi) Here

We have:

(-)

Or we can write it as: - × ()

Now, we can solve the above given fractions as follows:

- × ()

=

=

=

Hence,

The solution for the given fraction is

(vii) Here

We have:

()

× ()

Now, we can solve the above given fractions as follows:

× ()

=

=

=

Hence,

The solution for the given fraction is