Maths M.C.Q. [1 Marks Each]

$ -(-x) = x$
Negative of negative rational number is equal to positive rational number.

 We know that, the product of two rational numbers is $1,$ taken they are multiplication inverse of each other, e.g.
Suppose, $p $ is negative rational number, i.e.
$\frac{1}{\text{p}}$ is the multiplicative inverse of $-p,$
Then, $-\text{p}\times\frac{1}{-\text{p}}= 1$
Hence, multiplicative inverse of a negative rational number is a negative rational number.

 Let $x$ and $y$ be two numbers.
Case-I If $x < y$
Then, $\frac{\text{x}+\text{y}}{2}$ lies in between $x$ and such that

Case-II If $x < y$
Then, $\frac{\text{x}+\text{y}}{2}$ lies in between $x $ and $y$ such that

Sum of two numbers $=\frac{-4}{3}$
One number $= -5$
Second number $=\frac{-4}{3}-(-5)$
$=\frac{-4}{3}+\frac{5}{1}$
$=\frac{-4+15}{3}$
$=\frac{11}{3}$

 Given, $-\frac{3}{8}+\frac{1}{7}=+\Big(\frac{-3}{8}\Big)$
Let two rational number, $\text{a}=\frac{-3}{8},\ \text{b}=\frac{1}{7}$
$\therefore\text{a}+\text{b}=\frac{-3}{8}+\frac{1}{7}$
$=\frac{-21+8}{56}$
$=\frac{-13}{56}$
and
$\text{b}+\text{a}=\frac{1}{7}+\frac{-3}{8}$
$=\frac{8-21}{56}$
$=\frac{-13}{56}$
Clearly, $a + b = b + a$
So, addition is communucation for rational numbers.

Reciprocal of $'0' \frac{1}{0}$ which does not exist.
Therefore, Zero has no reciprocal.

Sum $= -3$
One number $=\frac{-10}{3}$
$\therefore$ Second number $=-3-\Big(\frac{-10}{3}\Big)$
$=-3+\frac{10}{3}$
$=\frac{-9+10}{3}$
$=\frac{1}{3}$

The $1$ and $-1$ are the two numbers which having reciprocal of its own. Except $1$ and $-1$ no other numbers are not having its own reciprocal.

$ 7$ and $8$ can be written as $\frac{7}{1}$ and $\frac{8}{1}$
As we need to find five rational numbers between two consecutive integers, multiply the numerator and denominator of both the fractions by $6,$
$\frac{7\times6}{1\times6}=\frac{42}{6}\text{and }\frac{8\times6}{1\times6}=\frac{48}{6}$
So, the $5$ rational numbers between $7$ and $8$ are $\frac{43}{6},\frac{22}{6},\frac{15}{6},\frac{23}{6},\frac{47}{6}$
After simplifying we get, $\frac{43}{6},\frac{22}{6},\frac{15}{6},\frac{23}{6},\frac{47}{6}$

The reciprocal of $-1$ is the number itself.

 Rational numbers are not not closed under division.
As, $1$ and $0$ are the rational number but $\frac{1}{0}$ is not defined.

  Associative Property of Addition $- $ The sum of three or more numbers remains the same irrespective of the way numbers are grouped.
$(A + B) + C = A + (B + C)$

The additive identity is $0$
The multiplicative identity is $1$
So in respectively $0$ and $1.$

The reciprocal of $0$ is not defined.

 It is known that $1$ and $0$ are rational numbers but $\frac{1}{0}$ is not defined.
Therefore, rational numbers are not closed under division.

Zero $(0)$ is the identity for addition of rational numbers.
That means,
If $a$ is a rational number.
Then, $a + 0 = 0 + a = a$
Note: Zero $(0)$ is also the additive identity for integers and whole number as well.

As per associative property:
$A + (B + C) = (A + B) + C$
$A × (B × C) = (A × B) × C$

Let $x$ is required rational
$\therefore\frac{4}{9}\div\text{x}=\frac{-8}{15}$
$\Rightarrow\frac{4}{9}\times\frac{1}{\text{x}}=\frac{-8}{15}$
$\Rightarrow\frac{1}{\text{x}}=\frac{-8}{15}\times\frac{9}{4}$
$=\frac{-72}{60}$
$=\frac{-72\div12}{60\div12}$
$=\frac{-6}{5}$
$\therefore\text{x}=\frac{-5}{6}$

The two given integers are $-7$ and $-10.$
The given integers can be written as $\frac{-7}{1}$ and $\frac{-10}{1}$
Multiplying numerator and denominator of both the fractions by $4$.
$\frac{-7\times4}{1\times4}=\frac{-28}{4}\text{ and }\frac{-10\times4}{1\times4}=\frac{-40}{4}$
The six. rational numbers in between $-7$ and $-10$ are $\frac{-29}{4},\frac{-30}{4},\frac{-31}{4},\frac{-32}{4},\frac{-33}{4},\frac{-34}{4}$