GSEB Solutions Class 8 Maths Chapter 1 Rational Numbers Ex 1.1

Gujarat Board GSEB Textbook Solutions Class 8 Maths Chapter 1 Rational Numbers Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 8 Maths Chapter 1 Rational Numbers Ex 1.1

Question 1.
Using appropriate properties find:
(i) $-\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6}$
(ii) $\frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5}$
Solution:
(i) $-\frac{2}{3} \times \frac{3}{5}+\frac{5}{2}-\frac{3}{5} \times \frac{1}{6}$
= $\left[-\frac{2}{3} \times \frac{3}{5}+\left(\frac{-3}{5}\right) \times \frac{1}{6}\right]+\frac{5}{2}$
(Using commutative)
= $\left[-\frac{2}{3} \times \frac{3}{5}+\left(\frac{-3}{5}\right) \times \frac{1}{6}\right]+\frac{5}{2}$ = $\frac { 5 }{ 2 }$
= $\left(\frac{-3}{5}\right)\left[\frac{2}{3}+\frac{1}{6}\right]+\frac{5}{2}$ (Using distributivity)
= $\left(\frac{-3}{5}\right)\left[\frac{4+1}{6}\right]+\frac{5}{2}=\left(\frac{-3}{5}\right)\left[\frac{5}{6}\right]+\frac{5}{2}$
= $\frac{-3}{5} \times \frac{5}{6}+\frac{5}{2}=\frac{-1}{2}+\frac{5}{2}=\frac{-1+5}{2}=\frac{4}{2}$ = 2

(ii) $\frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5}$
= $\frac{2}{5} \times\left(\frac{-3}{7}\right)-\frac{1}{4}+\frac{1}{14} \times \frac{2}{5}$
= $\frac{2}{5} \times\left(\frac{-3}{7}\right)+\frac{1}{14} \times \frac{2}{5}-\frac{1}{4}$
(Using commutative)
= $\frac{2}{5}\left[\frac{-3}{7}+\frac{1}{14}\right]-\frac{1}{4}$ (Using distributivity

Question 2.
Write the additive inverse of each of the following:

1. $\frac { 2 }{ 8 }$
2. $\frac { -5 }{ 9 }$
3. $\frac { -6}{ -5 }$
4. $\frac { 2 }{ -9 }$
5. $\frac { 19 }{ -6 }$

Solution:

Question 3.
Verify that – (-x) = x for:

1. x = $\frac { 11 }{ 15 }$
2. x = – $\frac { 13 }{ 17 }$

Solution:

Question 4.
Find the multiplicative inverse of the following:

1. – 13
2. $\frac { -13 }{ 19 }$
3. $\frac { 1 }{ 5 }$
4. $\frac { -5 }{ 8 }$ × $\frac { -3 }{ 7 }$
5. -1 × $\frac { -2 }{ 5 }$
6. -1

Solution:

Question 5.
Name the property under multiplication used in each of the following:

1. $\frac { 1 }{ 2 }$ × 1 = 1 × $\frac { -4 }{ 5 }$ = – $\frac { 4 }{ 5 }$
2. – $\frac { 13 }{ 17 }$ × $\frac { 13 }{ 17 }$
3. $\frac{-19}{29} \times \frac{29}{-19}$ = 1

Solution:

Question 6.
Multiply $\frac { 6 }{ 13 }$ by the reciprocal of $\frac { -7 }{ 16 }$
Solution:

Question 7.
Tell what property allows you to compute
$\frac { 1 }{ 3 }$ ×$\left(6 \times \frac{4}{3}\right)$ as $\left(\frac{1}{3} \times 6\right) \times \frac{4}{3}$
Solution:
In computing $\frac { 1 }{ 3 }$ ×$\left(6 \times \frac{4}{3}\right)$ as $\left(\frac{1}{3} \times 6\right) \times \frac{4}{3}$ , we use the associativity.

Question 8.
Is $\frac { 8 }{ 9 }$ the multiplicative invers of -1$\frac { 1 }{ 8 }$ ?
Why or Why not?
Solution:
Since, -1$\frac { 1 }{ 8 }$ = $\frac { -9 }{ 8 }$ and $\frac { 8 }{ 9 }$ × $\frac { 8 }{ 9 }$ × $\frac { -9 }{ 8 }$ = -1
[Which is not equal to 1]
∴ $\frac { 8 }{ 9 }$ is not the multiplicative invers of $\frac { -9 }{ 8 }$
[∴ The product of $\frac { -9 }{ 8 }$ and its multiplication invers must be equal to 1]

Question 9.
Is 0.3 the multiplicative inverse of 3 $\frac { 1 }{ 3 }$ ? Why or why not?
Solution:
0.3 = $\frac { 3 }{ 10 }$ and 3$\frac { 1 }{ 3 }$ = $\frac { 10 }{ 3 }$
and, multiplicative invers of 3 $\frac { 1 }{ 3 }$ or $\frac { 10 }{ 3 }$ = $\frac { 3 }{ 10 }$ = 0.3
∴the multiplicative inverse of 3$\frac { 1 }{ 3 }$ is 0.3

Question 10.
Write:

1. The rational number that does not have a reciprocal.
2. The rational numbers that are equal to their reciprocals.
3. The rational number that is equal to its negative.

Solution:

1. The rational number zero (0) does not have a reciprocal.
2. The rational numbers 1 and (-1) are equal to their reciprocals respectively.
3. ∵ [A rational number] + [Negative of the rational number] = 0 [ ∵ [0] = [0] = 0 ]
   So, Negative of 0 is 0. Hence, 0 is equal to its negative.

Question 11.
Fill in the blanks:

1. Zero has _______ reciprocal
2. The numbers _______ and _____ are their own reciprocals.
3. The reciprocal of -5 is ______
4. Reciprocal of $\frac { 1 }{ x }$ , where x ≠ 0 is ______.
5. The reciprocal of a positive rational number is always a _______
6. The reciprocal of a positive rational number is ______

Solution:

1. Zero has no reciprocal.
2. The numbers 1 and -1 are their own reciprocals.
3. The reciprocal of -5 is $\frac { -1 }{ 5 }$
4. The reciprocal of $\frac { 1 }{ x }$ where x ≠ 0 is x.
5. The product of two rational numbers is always a rational number.
6. The reciprocal of a positive rational number is positive.