Chapter 1: Rational Numbers

Solution:-

From the question it is given that,

Total monthly allowance received by Huma, Hubna and Seem = ₹ 2,016

from their mother

Seema gets allowance = ½ of Huma’s share

Hubna gets allowance = of Seema’s share

= 5/3 of Seema’s share

= 5/3 of ½ of Huma’s share … [∵ given]

= 5/3 × ½ of Huma’s share

= 5/6 of Huma’s share

So,

Huma’s share + Hubna’s share + Seema’s share = ₹ 2,016

Let Huma’s share be 1,

1 + (5/6) Huma’s share + ½ Huma’s share = ₹ 2,016

(1 + (5/6) + ½) = ₹ 2,016

The LCM of the denominators 1, 6 and 2 is 6

(1/1) = [(1×6)/ (1×6)] = (6/6)

(5/6) = [(5×1)/ (6×1)] = (5/6)

(1/2) = [(1×3)/ (2×3)] = (3/6)

Then,

(6/6) + (5/6) + (3/6) = ₹ 2,016

(6 + 5 + 3)/ 6 = ₹ 2,016

(14/6) = ₹ 2,016

So, Huma’s share = ₹ 2,016 ÷ (14/6)

= 2,016 × (6/14)

= 144 × 6

∴Huma’s Share is = ₹ 864

Seema’s share = ½ Huma’s share

= ½ × 864

= ₹ 432

Hubna’s share = 5/6 of Huma’s share

= 5/6 × 864

= 5 × 144

= ₹ 720

Solution:-

From the question it is given that,

A mother and her two daughters got a room constructed for = ₹ 62,000

Let us assume mother’s share be x,

Then,

The elder daughter’s contribute = 3/8 of her mother’s share

= 3/8 x

The younger daughter’s contribute = ½ of her mother’s share

= ½ x

So, mother’s share + elder daughter’s share + younger daughter’s share = ₹ 62,000

x + (3/8) x + ½ x = ₹ 62,000

The LCM of the denominators 1, 8 and 2 is 8

(1/1) = [(1×8)/ (1×8)] = (8/8)

(3/8) = [(3×1)/ (8×1)] = (3/8)

(1/2) = [(1×4)/ (2×4)] = (4/8)

Then,

(8/8) x + (3/8) x + (4/8) x = 62,000

(8x + 3x + 4x)/8 = 62,000

(15x/8) = 62,000

15x = 62,000 × 8

X = 496000/15

X = ₹ 33,066.6

∴Mother’s share = ₹ 33,066.6

Elder daughter’s share = 3/8 of her mother’s share

= 3/8 x

= 3/8 × 33066.6

= ₹ 12,400

Younger daughter’s share = ½ of her mother’s share

= ½ x

= ½ × 33066.6

= ₹ 16,533.3

Solution:-

(2/3) × [¾ × (5/7)] and [(2/3) × (5/7)] × ¾ this can be compared with associative property and commutative property.

Solution:-

The above rational number is in the form of Commutative property over multiplication.

(ii) (-2/3) × [(3/4) + (-½)] = [(-2/3) × (3/4)] + [(-2/3) × (½)]

Solution:-

The above rational number is in the form of Distributive property over addition.

(iii) (1/3) + [(4/9) + (-4/3)] = [(1/3) + (4/9)] + [-4/3]

Solution:-

The above rational number is in the form of Associative property over addition.

(iv) (-2/7) + 0 = 0 + (-2/7) = (-2/7)

Solution:-

The above rational number is in the form of Additive identity of rational number.

(v) (3/8) × 1 = 1 × (3/8) = (3/8)

Solution:-

The above rational number is in the form of Multiplicative identity of rational number.

Solution:-

The given number can be written as = -9/8

The multiplicative inverse = -8/9

(ii)

Solution:-

The given number
can be written as = 10/3

The multiplicative inverse = 3/10

Solution:-

The LCM of the denominators 4, 16 and 8 is 16

∴ ¼ = [(1×4)/ (4×4)] = (4/16)

(13/16) = [(13×1)/ (16×1)] = (13/16)

(5/8) = [(5×2)/ (8×2)] = (10/16)

Now, 13 < 10 < 4

⇒ (13/16) > (10/16) > (4/16)

Hence, (13/16) > (5/8) > (¼)

Descending order 13/16, 5/8, 1/4

Solution:-

Let us assume the other number be y.

Given, product of two rational number = -14/27

One number = 7/9

Then,

= y × (7/9) = -14/27

= y = (-14/27)/ (7/9)

= y = (-14/27) × (9/7)

= y = (-2/3) × (1/1)

= y = -2/3

So, the other number is -2/3

Solution:-

Let us assume the other number be y.

Given, product of two rational number = -5/7

One number = -15/20

Then,

= y × (-15/20) = -5/7

= y = (-5/7)/ (-15/20)

= y = (-5/7) × (-20/15)

= y = (-1/7) × (-20/3)

= y = -20/21

So, the other number is -20/21

Solution:-

Let us assume the other number be y.

Given, product of two rational number = 24

One number = -8/13

Then,

= y × (-8/13) = 24

= y = 24/ (-8/13)

= y = (24/1) × (-13/8)

= y = (3/1) × (-13/1)

= y = -39

So, the other number is -39

Solution:-

Let us assume the other number be y.

Given, product of two rational number = -7

One number = -5

Then,

= y × (-5) = -7

= y = -7/ (-5)

= y = 7/5

So, the other number is 7/5