Class 7 - Maths - Rational Numbers

Exercise 9.1

Question 1:

List five rational numbers between:

(i) -1 and 0               (ii) -2 and -1        (iii) -4/5 and -2/3            (iv) -1/2 and 2/3

(i) -1 and 0

Let us write -1 and 0 as rational numbers with denominator 6.

=> -1 = -6/6 and 0 = 0/6

So, -6/6 < -5/6 < -4/6 < -3/6 < -2/6 < -1/6 < 0/6

=> -1 < -5/6 < -2/3 < -1/2 < -1/3 < -1/6 < 0

Hence, the five rational numbers between -1 and 0 are:

-5/6 < -2/3 < -1/2 < -1/3 < -1/6

(ii) -2 and -1

Let us write -2 and -1 as rational numbers with denominator 6.

=> -2 = -12/6 and -1 = -6/6

So, -12/6 < -11/6 < -10/6 < -9/6 < -8/6 < -7/6 < -6/6

=> -2 < -11/6 < -5/3 < -3/2 < -4/3 < -7/6 < -1

Hence, the five rational numbers between -2 and -1 are:

-2 < -11/6 < -5/3 < -3/2 < -4/3 < -7/6 < -1

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

Let us write -4/3 and -2/3 as rational numbers with the same denominator.

=> -4/5 = -36/45 and -2/3 = -30/45

=> -36/45 < -35/45 < -34/45 < -33/45 < -32/45 < -31/45 < -30/45

=> -4/5 < -7/9 < -34/45 < -11/15 < -32/45 < -31/45 < -2/3

Hence, the five rational numbers between -4/5 and -2/3 are:

-4/5 < -7/9 < -34/45 < -11/15 < -32/45 < -31/45 < -2/3

(iv) -1/2 and 2/3

Let us write -1/2 and 2/3 as rational numbers with the same denominators

=> -1/2 = -3/6 and 2/3 = 4/5

=> -3/6 < -2/6 < -1/6 < 0/6 < 1/6 < 2/6 < 3/6 < 4/6

=> -1/2 < -1/3 < -1/6 < 0 < 1/6 < 1/3 < 1/2 < 2/3

Hence, the five rational numbers between -1/2 and 2/3 are:

-1/3 < -1/6 < 0 < 1/6 < 1/3

Question 2:

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

(i) -3/5, -6/10, -9/15, -12/20, …………….

(ii) -1/4, -2/8, -3/12, ……………

(iii) -1/6, 2/-12, 3/-18, 4/-24, …………….

(iv) -2/3, 2/-3, 4/-6, 6/-9, ………………

(i) -3/5, -6/10, -9/15, -12/20, ……….

= (-3 * 1)/(5 * 1), (-3 * 2)/(5 * 2), (-3 * 3)/(5 * 3), (-3 * 4)/(5 * 4), …………

Therefore, the next four rational numbers of 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, ……………

= (-1 * 1)/(4 * 1), (-1 * 2)/(4 * 2), (-1 * 3)/(4 * 3), ………….

Therefore, the next four rational numbers of 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, ……………

= (-1 * 1)/(6 * 1), (1 * 2)/(-6 * 2), (1 * 3)/(-6 * 3), (1 * 4)/(-6 * 4), …………..

Therefore, the next four rational numbers of 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/-42

(iv) -2/3, 2/-3, 4/-6, 6/-9, ………….

=> (-2 * 1)/(3 * 1), (2 * 1)/(-3 * 1), (2 * 1)/(-3 * 1), (2 * 1)/(-3 * 1), ………..

Therefore, the next four rational numbers of 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

Question 3:

Give four rational numbers equivalent to:

(i) -2/7                    (ii)5/-3                   (iii) 4/9

(i) -2/7

(-2 *2)/(7 * 2) = -4/14, (-2 *3)/(7 * 3) = -6/21, (-2 *4)/(7 * 4) = -8/28, (-2 *5)/(7 * 5) = -10/35

Therefore, four rational numbers are: -4/14, -6/21, -8/28, -10/35

(ii)5/-3

(5 * 2)/(-3 * 2) = 10/-6, (5 * 3)/(-3 * 3) = 15/-9, (5 * 4)/(-3 * 4) = 20/-12, (5 * 5)/(-3 * 5) = 25/-15

Therefore, four rational numbers are: 10/-6, 15/-9, 20/-12, 25/-15

(iii) 4/9

(4 * 2)/(9 * 2) = 8/18, (4 * 3)/(9 * 3) = 12/27, (4 * 4)/(9 * 4) = 16/36, (4 * 5)/(9 * 5) = 20/35

Therefore, four rational numbers are: 8/18, 12/27, 16/36, 20/45

Question 4:

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

(i) 3/4                  (ii) -5/8                    (iii) -7/4                     (iv) 7/8

(i) 3/4

(ii) -5/8

(iii) -7/4

(iv) 7/8

Question 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.

Each part which is between the two numbers is divided into 3 parts.

Therefore, A = 6/3, P = 7/3, Q = 8/3 and B = 9/3

Similarly, T = -3/3, R = -4/3, S = -5/3 and U = -6/3

Thus, the rational numbers represented P, Q, R and S are 7/3, 8/3 -4/3 and -5/3 respectively.

Question 6:

Which of the following pairs represent the same rational numbers:

(i) -7/21 and 3/9          (ii) -16/20 and 20/-25 (iii) -2/-3 and 2/3      (iv) -3/5 and -12/20

(v) 8/-5 and -24/15     (vi) 1/3 and -1/9          (vii) -5/-9 and 5/-9

(i) -7/21 and 3/9

-7/21 = -1/3                               [7 and 21 are divided by 7]

3/ 9 = 1/3                                   [3 and 9 are divided by 3]

Since -1/3 ≠ 1/3

So, -7/21 ≠ 3/9

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

-16/20 = -4/5                            [16 and 20 are divided by 4]

20/-25 = -20/25 = -4/5            [20 and 25 are divided by 5]

Since, -4/5 = -4/5

So, -16/20 = 20/-25

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

-2/-3 = 2/3

2/3 = 2/3

Since, 2/3 = 2/3

So, -2/-3 = 2/3

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

-3/5 = -3/5

-12/20 = -3/5                              [12 and 20 are divided by 4]

Since, -3/5 = -3/5

Hence, -3/5 = -12/20

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

8/-5 = -8/5

-24/15 = -8/5                             [24 and 15 are divided by 3]

Since, -8/5 = -8/5

Hence, 8/-5 = -24/15

(vi) 1/3 and -1/9

1/3 = 1/3

-1/9 = -1/9

Hence, 1/3 ≠ -1/9

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

-5/-9 = 5/9

5/-9 = -5/9

Since, 5/9 ≠ -5/9

Hence, -5/-9 ≠ 5/-9

Question 7:

Rewrite the following rational numbers in the simplest form:

(i) -8/6                   (ii) 25/45                   (iii) -44/72              (iv) -8/10

(i) -8/6 = -4/3                      [8 and 6 are divided by 2]

(ii) 25/45 = 5/9                   [25 and 45 are divided by 5]

(iii) -44/72 = -11/18          [44 and 72 are divided by 4]

(iv) -8/10 = -4/5                [8 and 10 are divided by 2]

Question 8:

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

(i)  -5/7      2/3     (ii)-4/5      -5/7     (iii) -7/8      14/-16         (iv) -8/5     -7/4

(v) 1/-3    -1/4     vi) 5/-11    -5/11  vii) 0      -7/6

(i)  -5/7           2/3 since the positive number is greater than the negative number.

(ii) (-4 * 7)/(5 * 7) = -28/35

(-5 * 5)/(7 * 5) = -25/35

So, -4/5          -5/7

(iii) (-7 * 2)/(8 * 2) = -14/16

So, -7/8         14/-16

(iv)( -8 * 4)/(5 * 4) = -32/20

(-7 * 5)/(4 * 5) = -35/20

So, -8/5          -7/4

(v) (1 * 4)/(-3 * 4) = 4/-12 = -4/12

( -1 * 3)/(4 * 3) = -3/13

So, 1/-3         -1/4

1. vi) 5/-11 = -5/11

Hence, 5/-11         -5/11

vii) 0          -7/6 since 0 is greater than negative number.

Question 9:

Which is greater in each of the following:

(i) 2/3, 5/2    (ii) -5/6, -4/3     (iii)-3/4, 2/-3     (iv)-1/4, 1/4      (v) -3  , -3

(i) 2/3 = (2 * 2)/(3 * 2) = 4/6

5/2 = (5 * 3)/(2 * 3) = 15/6

Since, 4/6 < 15/6

Hence, 2/3 < 5/2

(ii) -5/6 = (-5 * 1)/(6 * 1) = -5/6

(-4 * 2)/(3 * 2) = -8/6

Since, -5/6 > -8/6

Hence, -5/6 > -4/3

(iii)-3/4 = (-3 * 3)/(4 * 3) = -9/12

2/-3 = -2/3 = (-2 * 4)/(3 * 4) = -8/12

Since -9/12 < -8/12

Hence, -3/4 < 2/-3

(iv)-1/4 < 1/4  since a positive number is greater than a negative number.

(v) -3  =  -(3 + 2/7) = -{(3 * 7 + 2)/7} = -(21 + 2)/7 = -23/7

Now, -23/7 = (-23 * 5)/(7 * 5) = -115/35

-3  = -(3 + 4/5) = -{(3 * 5 + 4)/5} = -(15 + 4)/5 = -19/5

Now, -19/5 = (-19 * 7)/(5 * 7) = -133/15

Since, -115/35 > -133/35

Hence, -3  > -3

Question 10:

Write the following rational numbers in ascending order:

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

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

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

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

=> -3/5 < -2/5 < -1/5

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

=> 3/9, -2/9, -12/9                            [Converting into the same denominator]

=> -12/9, -2/9, 3/9

=> -4/3, -2/9, 1/3

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

=> -3/2 < -3/4 < -3/7