Class 8 - Maths - Rational Numbers

**Exercise 1.1**

**Question 1:**

Using appropriate properties, find:

(i) -2/3 * 3/5 + 5/2 – 3/5 * 1/6 (ii) 2/5 * (3/-7) – 1/6 * 3/2 + 1/14 * 2/5

Answer:

(i) -2/3 * 3/5 + 5/2 – 3/5 * 1/6

= -2/3 * 3/5 – 3/5 * 1/6 + 5/2 [Using associative property]

= 3/5 * (-2/3 – 1/6) + 5/2 [Using distributive property]

= 3/5 * {(-4 - 1)/6} + 5/2 [LCM (3, 2) = 6]

= 3/5 * (-5/6) + 5/2

= -3/6 + 5/2

= -1/2 + 5/2

= (-1 + 5)/2

= 4/2

= 2

(ii) 2/5 * (3/-7) – 1/6 * 3/2 + 1/14 * 2/5

= 2/5 * (-3/7) + 1/14 * 2/5 – 1/6 * 3/2 [Using associative property]

= 2/5 * (-3/7 + 1/14) – 1/2 * 1/2 [Using distributive property]

= 2/5 * {(-6 + 1)/14} – 1/4 [LCM (7, 14) = 14]

= 2/5 * (-5/14) – 1/4

= -1/7 – 1/4

= (-4 -7)/28 [LCM (7, 4) = 28]

= -11/28

**Question 2:**

Write the additive inverse of each of the following:

(i) 2/8 (ii) -5/9 (iii) -6/-5 (iv) 2/-9 (v) 19/-6

Answer:

We know that additive inverse of a rational number a/b is (-a/b) such that a/b + (-a/b) = 0

(i) Additive inverse of 2/8 = -2/8

(ii) Additive inverse of -5/9 = 5/9

(iii) -6/-5 = 6/5

Additive inverse of 6/5 = -6/5

(iv) 2/-9 = -2/9

Additive inverse of -2/9 = 2/9

(v) 19/-6 = -19/6

Additive inverse of -19/6 = 19/6

**Question 3:**

Verify that -(-x) = x for:

(i) x = 11/15 (ii) x = -13/17

Answer:

(i) Putting x = 11/15 in -(-x) = x, we get

=> -(-11/15) = 11/15

=> 11/15 = 11/15

=> LHS = RHS

Hence, verified.

(i) Putting x = -13/17 in -(-x) = x, we get

=> -{-(-13/17)} = -13/17

=> -(13/17) = -13/17

=> -13/17 = -13/17

=> LHS = RHS

Hence, verified.

**Question 4:**

Find the multiplicative inverse of the following:

(i) -13 (ii) -13/19 (iii) 1/5 (iv) (-5/8)*(-3/7) (v) -1 * (-2/5) (vi) -1

Answer:

We know that multiplicative inverse of a rational number a is 1/a such that a * 1/a = 1

(i) Multiplicative inverse of -13 = -1/13

(ii) Multiplicative inverse of -13/19 = -19/13

(iii) Multiplicative inverse of 1/5 = 5

(iv) (-5/8)*(-3/7) = (5 * 3)/(8 * 7) = 15/56

Multiplicative inverse of 15/56 = 56/15

(v) -1 * (-2/5) = 2/5

Multiplicative inverse of 2/5 = 5/2

(vi) Multiplicative inverse of -1 = 1/-1 = -1

**Question 5:**

Name the property under multiplication used in each of the following:

(i) -4/5 * 1 = 1 * -4/5

(ii) -13/17 * -2/7 = -2/7 * -13/17

(iii) -19/29 * 29/-19 = 1

Answer:

(i) 1 is the multiplicative identity.

(ii) Commutative property.

(iii) Multiplicative Inverse property.

**Question 6:**

Multiply 6/13 by the reciprocal of -7/16

Answer:

The reciprocal of -7/16 = -16/7

Now, 6/13 * (-16/7) = -(6 * 16)/(13 * 7) = -96/91

**Question 7:**

Tell what property allows you to compute 1/3 * (6 * 4/3) as (1/3 * 6) * 4/3

Answer:

By using associative property of multiplication, a * (b * c) = (a * b) * c

**Question 8:**

Is 8/9 the multiplicative inverse of -1? Why or why not?

Answer:

Since multiplicative inverse of a rational number a is 1/a if a * 1/a = 1

Therefore, 8/9 * (-1) = 8/9 * -9/8 = -1

But its product must be positive.

So, 8/9 is not multiplicative inverse of (-1)

**Question 9:**

Is 0.3 the multiplicative inverse of 3? Why or why not?

Answer:

Since multiplicative inverse of a rational number a is 1/a if a * 1/a = 1

Therefore, 0.3 * 3 = 3/10 * 10/3 = (3 * 10)/(10 * 3) = 30/30 = 1

So, 0.3 is the multiplicative inverse of 3

**Question 10:**

Write:

(i) The rational number that does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

Answer:

(i) 0 (ii) 1 and -1 (iii) 0

**Question 11:**

Fill in the blanks:

(i) Zero has _______________ reciprocal.

(ii) The numbers _______________ and _______________ are their own reciprocals.

(iii) The reciprocal of -5 is _______________.

(iv) Reciprocal of 1/x where x ≠ 0 is _______________.

(v) The product of two rational numbers is always a _______________.

(vi) The reciprocal of a positive rational number is _______________.

Answer:

(i) No (ii) 1, -1 (iii) -1/5 (iv) x

(v) Rational Number (vi) Positive rational number

**Exercise 1.2**

**Question 1:**

Represent these numbers on the number line:

(i) 7/4 (ii) -5/6

Answer:

(i) 7/4 = 1

Here, P is 1

(ii)- 5/6

Here, M is -5/6

**Question 2:**

Represent -2/11, -5/11, -9/11 on the number line.

Answer:

Here, B = -2/11, C = -5/11, D = -9/11

**Question 3:**

Write five rational numbers which are smaller than 2.

Answer:

2 can be represented as 14/7

Hence, five rational numbers smaller than 2 are:

13/7, 12/7, 11/7, 10/7 and 9/7

**Question 4:**

Find ten rational numbers between -2/5 and 1/2

Answer:

Given rational numbers are -2/5 and 1/2

Here, LCM of 5 and 2 is 10

So, -2/5 = (-2 * 2)/(5 * 2) = -4/10

and 1/2 = (1 * 5)/(2 * 5) = 5/10

Again, -4/10 = (-4 * 2)/(10 * 2) = -8/20

and 5/10 = (5 * 2)/(10 * 2) = 10/20

Hence, ten rational numbers between -2/5 and 1/2 are:

-7/20, -6/20, -5/20, -7/20, -4/20, -3/20, -2/20, -1/20, 0, 1/20, 2/20

**Question 5:**

Find five rational numbers between:

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

Answer:

(i) Given rational numbers are 2/3 and 4/5

Here, LCM of 3 and 5 is 15

So, 2/3 = (2 * 5)/(3 * 5) = 10/15

and 4/5 = (4 * 3)/(5 * 3) = 12/15

Again, 10/15 = (10 * 4)/(15 * 4) = 40/60

and 4/5 = (12 * 4)/(15 * 4) = 48/60

Hence, ten rational numbers between 2/3 and 4/5 are:

14/60, 42/60, 43/60, 44/60, 45/60

(ii) Given rational numbers are -3/2 and 5/3

Here, LCM of 2 and 3 is 6

So, -3/2 = (-3 * 3)/(2 * 3) = -9/6

and 5/3 = (5 * 2)/(3 * 2) = 10/6

Hence, ten rational numbers between -3/2 and 5/3 are:

-8/7, -7/6, 0, 1/6, 2/6

(iii) Given rational numbers are 1/4 and 1/2

Here, LCM of 4 and 2 is 4

So, 1/4 = (1 * 1)/(4 * 1) = 1/4

and 1/2 = (1 * 2)/(2 * 2) = 2/4

Again, 1/4 = (1 * 8)/(4 * 8) = 8/32

and 2/4 = (2 * 8)/(4 * 8) = 16/32

Hence, ten rational numbers between 1/4 and 1/2 are:

9/32, 10/32, 11/32, 12/32, 1/32

**Question 6:**

Write 5 rational numbers greater than -2.

Answer:

-2 can be represented as -14/2

Therefore, five rational numbers greater than -2 are:

-13/7, -12/7, -11/7, -10/7, -9/7

**Question 7:**

Find ten rational numbers between 3/5 and 3/4.

Answer:

(iii) Given rational numbers are 3/5 and 3/4

Here, LCM of 5 and 4 is 20

So, 3/5 = (3 * 4)/(5 * 4) = 12/20

and 3/4 = (3 * 5)/(4 * 5) = 15/20

Again, 12/20 = (12 * 8)/(20 * 8) = 96/160

and 15/20 = (15 * 8)/(20 * 8) = 120/160

Hence, ten rational numbers between 3/5 and 3/4 are:

97/160, 98/160, 99/160, 100/160, 101/160, 102/160, 103/160, 104/160, 105/160, 106/160,

.