Mathematics – Class 7 – Chapter 9 – Rational Numbers – Exercise 9.1 – NCERT Exercise Solution

2. Write four more rational numbers in each of the following patterns:

(i) -3/5, -6/10, -9/15, -12/20, …..

Solution:

In the above question, we may observe that the numerator and denominator are the multiples of 3 and 5 respectively.

Then, the given numbers are like,

= (-3 × 1)/ (5 × 1), (-3 × 2)/ (5 × 2), (-3 × 3)/ (5 × 3), (-3 × 4)/ (5 × 4)

Hence, next four rational numbers in this pattern are,

= (-3 × 5)/ (5 × 5), (-3 × 6)/ (5 × 6), (-3 × 7)/ (5 × 7), (-3 × 8)/ (5 × 8)

= -15/25, -18/30, -21/35, -24/40 ….

(ii) -1/4, -2/8, -3/12, …..

Solution:

In the above question, we may observe that the numerator and denominator are the multiples of 1 and 4 respectively.

Then, the given numbers are like,

= (-1 × 1)/ (4 × 1), (-1 × 2)/ (4 × 2), (-1 × 3)/ (1 × 3)

Hence, next four rational numbers in this pattern are,

= (-1 × 4)/ (4 × 4), (-1 × 5)/ (4 × 5), (-1 × 6)/ (4 × 6), (-1 × 7)/ (4 × 7)

= -4/16, -5/20, -6/24, -7/28 ….

(iii) -1/6, 2/-12, 3/-18, 4/-24 …..

Solution:

In the above question, we may observe that the numerator and denominator are the multiples of 1 and 6 respectively.

Then, the given numbers are like,

= (-1 × 1)/ (6 × 1), (1 × 2)/ (-6 × 2), (1 × 3)/ (-6 × 3), (1 × 4)/ (-6 × 4)

Hence, next four rational numbers in this pattern are,

= (1 × 5)/ (-6 × 5), (1 × 6)/ (-6 × 6), (1 × 7)/ (-6 × 7), (1 × 8)/ (-6 × 8)

= 5/-30, 6/-36, 7/-42, 8/-48 ….

(iv) -2/3,2/-3 4/-6, 6/-9 …..

Solution:

In the above question, we may observe that the numerator and denominator are the multiples of 2 and 3 respectively.

= (-2 × 1)/ (3 × 1), (2 × 1)/ (-3 × 1), (2 × 2)/ (-3 × 2), (2 × 3)/ (-3 × 3)

Hence, next four rational numbers in this pattern are,

= (2 × 4)/ (-3 × 4), (2 × 5)/ (-3 × 5), (2 × 6)/ (-3 × 6), (2 × 7)/ (-3 × 7)

= 8/-12, 10/-15, 12/-18, 14/-21 ….

3. Give four rational numbers equivalent to:

(i) -2/7

Solution:

The four rational numbers equivalent to -2/7 are,

= (-2 × 2)/ (7 × 2), (-2 × 3)/ (7 × 3), (-2 × 4)/ (7 × 4), (-2 × 5)/ (7× 5)

= -4/14, -6/21, -8/28, -10/35

(ii) 5/-3

Solution:

The four rational numbers equivalent to 5/-3 are,

= (5 × 2)/ (-3 × 2), (5 × 3)/ (-3 × 3), (5 × 4)/ (-3 × 4), (5 × 5)/ (-3× 5)

= 10/-6, 15/-9, 20/-12, 25/-15

(iii) 4/9

Solution:

The four rational numbers equivalent to 5/-3 are,

= (4 × 2)/ (9 × 2), (4 × 3)/ (9 × 3), (4 × 4)/ (9 × 4), (4 × 5)/ (9× 5)

= 8/18, 12/27, 16/36, 20/45

4. Draw the number line and represent the following rational numbers on it:

(i) ¾

Solution:

We know that 3/4 is greater than 0 and less than 1.

So, it lies between 0 and 1.

(ii) -5/8

Solution:

We know that -5/8 is less than 0 and greater than -1.

So, lies between 0 and -1.

(iii) -7/4

Solution:

We can write the given number as:

Hence, lies between -1 and -2. It can be represented on the number line as,

(iv) 7/8

Solution:

We know that 7/8 is greater than 0 and less than 1.

Hence, it lies between 0 and 1. It can be represented on the number line as,

5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Solution:

According to the figure, we may say that,

The distance between A and B = 1 unit

It is divided into 3 equal parts = AP = PQ = QB = 1/3

P = 2 + (1/3)

= (6 + 1)/ 3

= 7/3

Q = 2 + (2/3)

= (6 + 2)/ 3

= 8/3

Similarly,

The distance between U and T = 1 unit

It is divided into 3 equal parts = TR = RS = SU = 1/3

R = – 1 – (1/3)

= (- 3 – 1)/ 3

= – 4/3

S = – 1 – (2/3)

= – 3 – 2)/ 3

= – 5/3

6. Which of the following pairs represent the same rational number?

(i) (-7/21) and (3/9)

Solution:

According to the question we have to check the given pair represents the same rational number.

Then,

-7/21 = 3/9

-1/3 = 1/3

So,  -1/3 ≠ 1/3

Hence,  -7/21 ≠ 3/9

Hence, the given pair do not represent the same rational number.

(ii) (-16/20) and (20/-25)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

-16/20 = 20/-25

-4/5 = 4/-5

So, -4/5 = -4/5

Hence, -16/20 = 20/-25

Hence, the given pair represent the same rational number.

(iii) (-2/-3) and (2/3)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

-2/-3 = 2/3

2/3= 2/3

So, 2/3 = 2/3

Hence, -2/-3 = 2/3

Hence, the given pair represent the same rational number.

(iv) (-3/5) and (-12/20)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

-3/5 = – 12/20

-3/5 = -3/5

So, -3/5 = -3/5

Hence, -3/5= -12/20

Hence, the given pair represent the same rational number.

(v) (8/-5) and (-24/15)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

8/-5 = -24/15

8/-5 = -8/5

 -8/5 = -8/5

 8/-5 = -24/15

Hence, the given pair is represent the same rational number.

(vi) (1/3) and (-1/9)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

1/3 = -1/9

 1/3 ≠ -1/9

 1/3 ≠ -1/9

Hence, the given pair do not represent the same rational number.

(vii) (-5/-9) and (5/-9)

Solution:

According to the question we have to check the given pair represents the same rational number.

Now,

-5/-9 = 5/-9

5/9 ≠ -5/9

-5/-9 ≠ 5/-9

Hence, the given pair do not represent the same rational number.

7. Rewrite the following rational numbers in the simplest form:

(i) -8/6

Solution:

Dividing both numerator and denominator by 2

= -4/3

(ii) 25/45

Solution:

Dividing both numerator and denominator by 5

= 5/9

(iii) -44/72

Solution:

Dividing both numerator and denominator by 4

= -11/18

(iv) -8/10

Solution:

Dividing both numerator and denominator by 2

 -4/5

8. Fill in the boxes with the correct symbol out of >, <, and =.

(i) -5/7 [ ] 2/3

Solution:

First, we make the numbers with same denominator:

LCM of 7 and 3 = 21

 (-5/7) = [(-5 × 3)/ (7 × 3)] = (-15/21)

And (2/3) = [(2 × 7)/ (3 × 7)] = (14/21)

Now,

-15 < 14

So, (-15/21) < (14/21)

Hence, -5/7 [<] 2/3

(ii) -4/5 [ ] -5/7

Solution:

First, we make the numbers with same denominator:

LCM of 5 and 7 = 35

 (-4/5) = [(-4 × 7)/ (5 × 7)] = (-28/35)

And (-5/7) = [(-5 × 5)/ (7 × 5)] = (-25/35)

Now,

-28 < -25

So, (-28/35) < (- 25/35)

Hence, -4/5 [<] -5/7

(iii) -7/8 [ ] 14/-16

Solution:

14/-16 can be simplified further,

Dividing both numerator and denominator by 2

= 7/-8

So, (-7/8) = (-7/8)

Hence, -7/8 [=] 14/-16

(iv) -8/5 [ ] -7/4

Solution:

First, we make the numbers with same denominator:

LCM of 5 and 4 = 20

(-8/5) = [(-8 × 4)/ (5 × 4)] = (-32/20)

And (-7/4) = [(-7 × 5)/ (4 × 5)] = (-35/20)

Now,

-32 > – 35

So, (-32/20) > (- 35/20)

Hence, -8/5 [>] -7/4

(v) 1/-3 [ ] -1/4

Solution:

First, we make the numbers with same denominator:

LCM of 3 and 4 = 12

(-1/3) = [(-1 × 4)/ (3 × 4)] = (-4/12)

And (-1/4) = [(-1 × 3)/ (4 × 3)] = (-3/12)

Now,

-4 < – 3

So, (-4/12) < (- 3/12)

Hence, 1/-3 [<] -1/4

(vi) 5/-11 [ ] -5/11

Solution:

Since, (-5/11) = (-5/11)

Hence, 5/-11 [=] -5/11

(vii) 0 [ ] -7/6

Solution:

We know that every negative rational number is less than 0.

So,

= 0 [>] -7/6

9. Which is greater in each of the following:

(i) 2/3, 5/2

Solution:

First, we make the numbers with same denominator:

LCM of 3 and 2 = 6

(2/3) = [(2 × 2)/ (3 × 2)] = (4/6)

And (5/2) = [(5 × 3)/ (2 × 3)] = (15/6)

Now,

4 < 15

So, (4/6) < (15/6)

2/3 < 5/2

Hence, 5/2 is greater.

(ii) -5/6, -4/3

Solution:

First, we make the numbers with same denominator:

LCM of 6 and 3 = 6

 (-5/6) = [(-5 × 1)/ (6 × 1)] = (-5/6)

And (-4/3) = [(-4 × 2)/ (3 × 2)] = (-12/6)

Now,

-5 > -12

So, (-5/6) > (- 12/6)

 -5/6 > -12/6

Hence, – 5/6 is greater.

(iii) -3/4, 2/-3

Solution:

First, we make the numbers with same denominator:

LCM of 4 and 3 = 12

(-3/4) = [(-3 × 3)/ (4 × 3)] = (-9/12)

And (-2/3) = [(-2 × 4)/ (3 × 4)] = (-8/12)

Now,

-9 < -8

So, (-9/12) < (- 8/12)

-3/4 < 2/-3

Hence, 2/-3 is greater.

(iv) -¼, ¼

Solution:

The given fraction is like friction,

So, -¼ < ¼

Hence ¼ is greater,

(v)

Then,

The LCM of the denominators 7 and 5 = 35

 (-23/7) = [(-23 × 5)/ (7 × 5)] = (-115/35)

And (-19/5) = [(-19 × 7)/ (5 × 7)] = (-133/35)

Now,

-115 > -133

So, (-115/35) > (- 133/35)

10. Write the following rational numbers in ascending order:

(i) -3/5, -2/5, -1/5

Solution:

(-3/5) < (-2/5) < (-1/5)

(ii) -1/3, -2/9, -4/3

Solution:

First we convert the given rational numbers into like fraction we have to find LCM,

LCM of 3, 9, and 3 = 9

Now,

(-1/3) = [(-1 × 3)/ (3 × 9)] = (-3/9)

(-2/9) = [(-2 × 1)/ (9 × 1)] = (-2/9)

(-4/3) = [(-4 × 3)/ (3 × 3)] = (-12/9)

So,

(-12/9) < (-3/9) < (-2/9)

Hence,

(-4/3) < (-1/3) < (-2/9)

(iii) -3/7, -3/2, -3/4

Solution:

First we convert the given rational numbers into like fraction, we have to find LCM,

LCM of 7, 2, and 4 = 28

Now,

(-3/7) = [(-3 × 4)/ (7 × 4)] = (-12/28)

(-3/2) = [(-3 × 14)/ (2 × 14)] = (-42/28)

(-3/4) = [(-3 × 7)/ (4 × 7)] = (-21/28)

Clearly,

(-42/28) < (-21/28) < (-12/28)

Hence,

(-3/2) < (-3/4) < (-3/7)