Chapter 8: Rational Numbers

Solution:-

We have:

= (5/4) – (11/4)

= [(5 – 11)/4] … [∵ denominator is same in both the rational numbers]

= (-6/4)

= -3/2 … [∵ Divide both numerator and denominator by 3]

(ii) (5/3) + (3/5)

Solution:-

Take the LCM of the denominators of the given rational numbers.

The LCM of 3 and 5 is 15

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(5/3)= [(5×5)/ (3×5)] = (25/15)

(3/5)= [(3×3)/ (5×3)] = (9/15)

Then,

= (25/15) + (9/15) … [∵ denominator is same in both the rational numbers]

= (25 + 9)/15

= 34/15

(iii) (-9/10) + (22/15)

Solution:-

Take the LCM of the denominators of the given rational numbers.

The LCM of 10 and 15 is 30

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-9/10)= [(-9×3)/ (10×3)] = (-27/30)

(22/15)= [(22×2)/ (15×2)] = (44/30)

Then,

= (-27/30) + (44/30) … [∵ denominator is same in both the rational numbers]

= (-27 + 44)/30

= (17/30)

(iv) (-3/-11) + (5/9)

Solution:-

We have,

= 3/11 + 5/9

Take the LCM of the denominators of the given rational numbers.

The LCM of 11 and 9 is 99

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(3/11)= [(3×9)/ (11×9)] = (27/99)

(5/9)= [(5×11)/ (9×11)] = (55/99)

Then,

= (27/99) + (55/99) … [∵ denominator is same in both the rational numbers]

= (27 + 55)/99

= (82/99)

(v) (-8/19) + (-2/57)

Solution:-

We have

= -8/19 – 2/57

Take the LCM of the denominators of the given rational numbers.

The LCM of 19 and 57 is 57

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-8/19)= [(-8×3)/ (19×3)] = (-24/57)

(-2/57)= [(-2×1)/ (57×1)] = (-2/57)

Then,

= (-24/57) – (2/57) … [∵ denominator is same in both the rational numbers]

= (-24 – 2)/57

= (-26/57)

(vi) -2/3 + 0

Solution:-

We know that if any number or fraction is added to zero, the answer will be the same number or fraction.

Hence,

= -2/3 + 0

= -2/3

(vii) +

Solution:-

First, we have to convert mixed fractions into improper fractions.

=
= -7/3

=
= 23/5

We have, -7/3 + 23/5

Take the LCM of the denominators of the given rational numbers.

The LCM of 3 and 5 is 15

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-7/3)= [(-7×5)/ (3×5)] = (-35/15)

(23/5)= [(23×3)/ (15×3)] = (69/15)

Then,

= (-35/15) + (69/15) … [∵ denominator is same in both the rational numbers]

= (-35 + 69)/15

= (34/15)

Solution:-

Take the LCM of the denominators of the given rational numbers.

The LCM of 24 and 36 is 72

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(7/24)= [(7×3)/ (24×3)] = (21/72)

(17/36)= [(17×2)/ (36×2)] = (34/72)

Then,

= (21/72) – (34/72) … [∵ denominator is same in both the rational numbers]

= (21 – 34)/72

= (-13/72)

(ii) 5/63 – (-6/21)

Solution:-

We can also write -6/21 = -2/7

= 5/63 – (-2/7)

We have,

= 5/63 + 2/7

Take the LCM of the denominators of the given rational numbers.

The LCM of 63 and 7 is 63

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(5/63)= [(5×1)/ (63×1)] = (5/63)

(2/7)= [(2×9)/ (7×9)] = (18/63)

Then,

= (5/63) + (18/63) … [∵ denominator is same in both the rational numbers]

= (5 + 18)/63

= 23/63

(iii) -6/13 – (-7/15)

Solution:-

We have,

= -6/13 + 7/15

The LCM of 13 and 15 is 195

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-6/13)= [(-6×15)/ (13×15)] = (-90/195)

(7/15)= [(7×13)/ (15×13)] = (91/195)

Then,

= (-90/195) + (91/195) … [∵ denominator is same in both the rational numbers]

= (-90 + 91)/195

= (1/195)

(iv) -3/8 – 7/11

Solution:-

Take the LCM of the denominators of the given rational numbers.

The LCM of 8 and 11 is 88

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-3/8)= [(-3×11)/ (8×11)] = (-33/88)

(7/11)= [(7×8)/ (11×8)] = (56/88)

Then,

= (-33/88) – (56/88) … [∵ denominator is same in both the rational numbers]

= (-33 – 56)/88

= (-89/88)

(v) – – 6

Solution:-

First, we have to convert the mixed fraction into an improper fraction,

–
= -19/9

We have, -19/9 – 6

Take the LCM of the denominators of the given rational numbers.

The LCM of 9 and 1 is 9

Express each of the given rational numbers with the above LCM as the common denominator.

Now,

(-19/9)= [(-19×1)/ (9×1)] = (-19/9)

(6/1)= [(6×9)/ (1×9)] = (54/9)

Then,

= (-19/9) – (54/9) … [∵ denominator is same in both the rational numbers]

= (-19 – 54)/9

= (-73/9)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

The above question can be written as (9/2) × (-7/4)

We have,

= (9×-7)/ (2×4)

= -63/8

(ii) (3/10) × (-9)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

The above question can be written as (3/10) × (-9/1)

We have,

= (3×-9)/ (10×1)

= -27/10

(iii) (-6/5) × (9/11)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

We have,

= (-6×9)/ (5×11)

= -54/55

(iv) (3/7) × (-2/5)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

We have,

= (3×-2)/ (7×5)

= -6/35

(v) (3/11) × (2/5)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

We have,

= (3×2)/ (11×5)

= 6/55

(vi) (3/-5) × (-5/3)

Solution:-

The product of two rational numbers = (product of their numerator)/(product of their denominator)

We have,

= (3×-5)/ (-5×3)

On simplifying,

= (1×-1)/ (-1×1)

= -1/-1

= 1

Solution:-

We have,

= (-4/1) × (3/2) … [∵ reciprocal of (2/3) is (3/2)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-4×3) / (1×2)

= (-2×3) / (1×1)

= -6

(ii) (-3/5) ÷ 2

Solution:-

We have,

= (-3/5) × (1/2) … [∵ reciprocal of (2/1) is (1/2)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-3×1) / (5×2)

= -3/10

(iii) (-4/5) ÷ (-3)

Solution:-

We have,

= (-4/5) × (1/-3) … [∵ reciprocal of (-3) is (1/-3)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-4× (1)) / (5× (-3))

= -4/-15

= 4/15

(iv) (-1/8) ÷ 3/4

Solution:-

We have,

= (-1/8) × (4/3) … [∵ reciprocal of (3/4) is (4/3)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-1×4) / (8×3)

= (-1×1) / (2×3)

= -1/6

(v) (-2/13) ÷ 1/7

Solution:-

We have,

= (-2/13) × (7/1) … [∵ reciprocal of (1/7) is (7/1)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-2×7) / (13×1)

= -14/13

(vi) (-7/12) ÷ (-2/13)

Solution:-

We have,

= (-7/12) × (13/-2) … [∵ reciprocal of (-2/13) is (13/-2)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (-7× 13) / (12× (-2))

= -91/-24

= 91/24

(vii) (3/13) ÷ (-4/65)

Solution:-

We have,

= (3/13) × (65/-4) … [∵ reciprocal of (-4/65) is (65/-4)]

The product of two rational numbers = (product of their numerator)/(product of their denominator)

= (3×65) / (13× (-4))

= 195/-52

= -15/4