NCERT Solutions Class 7th Maths Chapter – 8 Rational Numbers Exercise – 8.1

1. List five rational numbers between-

(i) -1 and 0
Solution – Converting each of rational numbers as a denominator 5 + 1 = 6, we have

-1 = -1 × 6/6 = -6/6 and 0 × 6/6 = 0/6
So, -6/6 < -5/6, -4/6, -3/6, -2/6, -1/6, < 0/6
Or -1 < -5/6, -2/3, -1/2, -1/3, -1/6 < 0/6

Hence, the required five rational numbers between -1 and 0 are -5/6, -2/3, -1/2, -1/3 and – 1/6

(ii) -2 and -1
Solution – Converting each of rational numbers as a denominator 5 + 1 = 6, we have

-2 = -2 × 6/6 = -12/6,
-1 = -1 × 6/6 = -6/6
So, -12/6 < -11/6 < -10/6 < -9/6 < -8/6 < -7/6 < -6/6
Or -2 < -11/6 < -5/3 < -3/2 < -4/3 < -7/6 < -1

Hence, the required rational numbers are
-11/6, -5/3, -3/2, -4/3 and -7/6.

(iii) -4/5 and -2/3
Solution – Converting each of the rational numbers as a denominator 5 × 3 = 15, we have (∴ LCM of 5 and 3 = 15)

-4/5 = – 4 × 3/5 × 3 = – 12/15,
– 2/3 = -2 × 5/3 × 5 = -10/15

Since there is only one integer i.e. -11 between -12 and -10, we have to find equivalent rational numbers.

-12/15 = -12 × 3/15 × 3 = -36/45,
-10/15 = -10 × 3/15 × 3 = -30/45
∴ -36/45 < -35/45 < -34/45 – 33/45 < -32/45 < -31/45 < -30/45
or -4/5 < -7/9 < -34/45 < -11/15 < -32/45 < -31/45 < -2/3

Hence, the required rational numbers are
– 7/9, -34/45, -11/15, -32/45, -31/45

(iv) – 1/2 and 2/3
Solution – Converting each of the rational numbers in their equivalent rational numbers, we have
1/2 = 1 × 18/2 × 18 = 18/36
2/3 = 2 × 12/3 × 12 = 24/36
∴ 18/36 < 19/36 < 20/36 < 21/36 < 22/36 < 23/36 < 24/36
or 1/2 <19/36 < 5/9 < 7/12 < 11/18 < 23/36 < 2/3

Hence, the required rational numbers are
19/36, 5/9, 7/12, 11/18 and 23/36.

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

(i) -3/5, -6/15, -9/15, -12/20, …..
Solution – Given pattern is-
-3/5 = -3 × 1/5 × 1
-6/10 = -3 × 2/5 × 2
-9/15 = -3 × 3/5 × 3
-12/20 = -3 × 4/5 × 4

Proceeding with the same pattern, we have
-3 × 5/5 × 5 = -15/25
-3 × 6/5 × 6 = -18/30
-3 × 7/5 × 7 = -21/35
-3 × 8/5 × 8 = -24/40

Hence, the required rational numbers are
-15/25, -18/30, -21/35 and -24/40

(ii) -1/4, -2/8, -3/12, …..
Solution – Given pattern is-
-1/4 = -1 × 1/4 × 1
-2/8 = -1 × 2/4 × 2
-3/12 = -1 × 3/4 × 3

Proceeding with the same pattern, we have
-1 × 4/4 × 4 = -4/16
-1 × 5/4 × 5 = -5/20
-1 × 6/4 × 6 = -6/24
-1 × 7/4 × 7 = -7/28

Hence, the required rational numbers are
-4/16, -5/20, -6/24 and -7/28

(iii) -1/6, 2/-12, 3/-18, 4/-24, …..
Solution – Given pattern is-
– 1/6 = – 1 × 1/6 × 1
2/-12 = -2/12 = -1 × 2/6 × 2
3/-18 = -3/18 = -1 × 3/6 × 3
4/-24 = -4/24= -1 × 4/6 × 4

Proceeding with the same pattern, we have
-1 × 5/6 × 5 = 5/-30
-1 × 6/6 × 6 = 6/-36
-1 × 7/6 × 7 = 7/-42
-1 × 8/6 × 8 = 8/-48

Hence, the required rational numbers are
5/-30, 6/-36, 7/-42 and 8/-48

(iv) -2/3, 2/-3, 4/-6, 6/-9, …..
Solution – Given pattern is-
-2/3 = – 2 × 1/3 × 1
2/-3 = 2/-3 = 2 × 1/-3 × 1
4/-6 = -4/6 = – 2 × 2/3 × 2
6/-9 = -6/9 = – 2 × 3/3 × 3

Proceeding with the same pattern, we have
-2 × 4/3 × 4 = -8/12 or 8/-12
-2 × 5/3 × 5 = -10/15 or 10/-15
-2 × 6/3 × 6 = -12/18 or 12/-18
-2 × 7/3 × 7 = -14/21 or 14/-21

Hence, the required rational numbers are
8/-12, 10/-15, 12/-18, 14/-21

3. Give four rational numbers equivalent to-

(i) -2/7
Solution – -2/7 = -2 × 2/7 × 2 = -4/14
-2/7 = -2 × 3/7 × 3 = -16/21
-2/7 = -2 × 4/7 × 4 = -8/28
-2/7 = -2 × 5/7 × 5 = -10/35

Hence, the required equivalent rational numbers are
-4/14, -6/21, -8/28, -10/35

(ii) 5/-3
Solution – 5/-3 = 5 × 2/-3 × 2 = 10/-6
5/-3 = 5 × 3/-3 × 3 = 15/-9
5/-3 = 5 × 4/-3 × 4 = 20/-12
5/-3 = 5 × 5/-3 × 5 = 25/-15

Hence, the required equivalent rational numbers are
10/-6, 15/-9, 20/-12, 25/-15

(iii) 4/9
Solution – 4/9 = 4 × 2/9 × 2 = 8/18
4/9 = 4 × 3/9 × 3 = 12/27
4/9 = 4 × 4/9 × 4 = 16/36
4/9 = 4 × 5/9 × 5 = 20/45

Hence, the required equivalent rational numbers are
8/18, 12/27, 16/36, 20/45

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

(i) 3/4
Solution –

Hare, A represents 3/4.

(ii) -5/8
Solution –

Hare, A represents -5/8.

(iii) -7/4
Solution –

Hare, A represents -7/4.

(iv) 7/8
Solution –

Hare, A represents 7/8.

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 – Rational numbers represented by P, Q, R and S.
7/3, 8/3, -4/3 and -5/3 respectlvely.

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

(i) -7/21 and 3/9
Solution –
⇒ -7 × 9/21 × 3 and 3 × 21/9 × 21
⇒ – 63/189 and 63/189
Since -63 ≠ 63, So – 7/21 and 3/9 do not represent the same rational numbers.

(ii) -16/20 and 20/-25
Solution –
⇒ -16 × -25/20 × -25 and 20 × 20/-25 × 20
⇒ 400/-500 and 400/-500
Since 400 = 400, So – 16/20 and 20/-25 represent the same rational numbers.

(iii) -2/-3 and 2/3
Solution – Here, we have the same numerator and denominator. So -2/-3 and 2/3 represent the same rational numbers.

(iv) -3/5 and -12/20
Solution –
⇒ -3 × 20/5 × 20 and -12 × 5/20 × 5
⇒ -60/100 and -60/100
Hance, -60 = -60, so – 3/5 and 12/20 represent the same rational numbers.

(v) 8/-5 and -24/15
Solution –
⇒ 8 × 15/-5 × 15 and -24 × -5/15 × -5
⇒ 120/-75 and 120/-75
Here, 120 = 120, so 8/-5 and -24/15 represent the same rational numbers.

(vi) 1/3 and -1/9
Solution –
⇒ 1 × 9/3 × 9 and -1 × 3/9 × 3
⇒ 9/27 and -3/27
Here, 9 ≠ -3, so 1/3 and – 1/9 do not represent the same rational numbers.

(vii) -5/-9 and 5/-9
Solution – Here, denominators are same and -5 ≠ 5, so -5/-9 and 5/-9 do not represent the same rational numbers.

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

(i) -8/6
Solution – -8/6 = -8 ÷ 2/6 ÷ 2 = -4/3
{∴ HCF of 8 and 6 = 2}

(ii) 25/45
Solution – 25/45 = 25 ÷ 5/45 ÷ 5 = 5/9
{∴ HCF of 25 and 45 = 5}

(iii) -44/72
Solution – -44/72 = -44 ÷ 4/72 ÷ 4 = -11/18
{∴ HCF of 44 and 72 = 4}

(iv) -8/10
Solution – -8/10 = -8 ÷ 2/10 ÷ 2 = -4/5
{∴ HCF of -8 and 10 = 2}

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

(i) -5/7 ▭ 2/3
Solution –

(ii) -4/5 ▭ -5/7
Solution –

(iii) -7/8 ▭ 14/-16
Solution –

(iv) -8/5 ▭ -7/4
Solution –

(v) 1/-3 ▭ -1/4
Solution –

(vi) 5/-11 ▭ -5/11
Solution –

(vii) 0 ▭ -7/6
Solution –

9. Which is greater in each of the following-

(i) 2/3, 5/2
Solution –
⇒ 2 × 2/3 × 2, 5 × 3/2 × 3 ⇒ 4/6, 15/6
Since, 15/6 > 4/6, So 5/2 > 2/3.

(ii) -5/6, -4/3
Solution –
⇒ -5 × 3/6 × 3, -4 × 6/3 × 6
⇒ -15/18, -24/18
Since, -15/18 > -24/18, So -5/6 > -4/3

(iii) -3/4, 2/-3
Solution –
⇒ -3 × -3/4 × -3, 2 × 4/-3 × 4
⇒ 9/-12, 8/-12
Since 9/-12 < 8/-12 So, 2/-3 > -3/4

(iv) -1/4, 1/4
Solution –
⇒ 1/4 > -1/4 [∴ Each positive number is greater than its negative]

(v) -3 2/7, -3 4/5
Solution –
⇒ – 23/7, – 19/5 = -23 × 5/7 × 5, -19 × 7/5 × 7
⇒ -115/35, -133/35
Since -115/35 > -133/35 So, -3 2/7 > -3 4/5

10. Write the following rational numbers in ascending order-

(i) -3/5, -2/5, -1/5
Solution – Here, denominators are same.
∴ -3 < -2 < -1
Hence, the required ascending order is
-3/5 < -2/5 < -1/5

(ii) -1/3, -2/9, -4/3
Solution – LCM of 3,9 and 3 = 9
1 × 3/3 × 3, -2 × 1/9 × 1, -4 × 3/3 × 3
⇒ 3/9, -2/9, -12/9
Since -12/9 < -2/9 < 3/9
Hence, the required ascending order is
-4/3 < -2/9 < 1/3

(iii) -3/7, -3/2, -3/4
Solution – LCM of 7, 2 and 4 = 28
-3 × 4/7 × 4, -3 × 14/2 × 14, -3 × 7/4 × 7
⇒ -12/28, -42/28, -21/28
Since, -42/28 < -21/28 < -12/28
Hence, the required ascending order is
-3/2 < -3/4 < -3/7