Question 11 Mark
Name the property used in each of the following. $\frac{3}{8}\times1=1\times\frac{3}{8}=\frac{3}{8}$
Answer
View full question & answer→Existence of multiplicative identity.
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Question 21 Mark
Find the multiplicative inverse of: $-1\frac{1}{8}.$
Answer
View full question & answer→Given number is $-1\frac{1}{8},$ i.e. $\frac{-9}{8}.$The multiplicative inverse of $\frac{-9}{8}$ is $\frac{-8}{9}.$
Question 31 Mark
Name the property used in each of the following. $-\frac{2}{3}\times\Big[\frac{3}{4}+\frac{-1}{2}\Big]=\Big[\frac{-2}{3}\times\frac{3}{4}\Big]+\Big[\frac{-2}{3}\times\frac{-1}{2}\Big]$
Answer
View full question & answer→Distributive property over addition.
Question 41 Mark
The multiplicative inverse of $\frac{-3}{5}$ is $\frac{5}{3}.$
Answer
View full question & answer→False. Solution: The multiplicative inverse of $\frac{-3}{5}$ is $\frac{1}{\Big(\frac{-3}{5}\Big)},$ i.e. $\frac{-5}{3}.$
Question 51 Mark
The negative of a negative rational number is always a _____ rational number.
Answer
View full question & answer→The negative of a negative rational number is always a Positive rational number.
Solution:
Let $x$ be a positive rational number.
Then, $-x$ be a negative rational number.
Now, negative of a negative rational number $= -(-x) = x =$ positive rational number.
Solution:
Let $x$ be a positive rational number.
Then, $-x$ be a negative rational number.
Now, negative of a negative rational number $= -(-x) = x =$ positive rational number.
Question 61 Mark
Simplify: $\frac{7}{8}+\frac{1}{16}-\frac{1}{12}$
Answer
View full question & answer→Given, $\frac{7}{8}+\frac{1}{16}-\frac{1}{12}$$=\frac{14+1}{16}-\frac{1}{12}$
$=\frac{15}{16}-\frac{1}{12}$
$=\frac{45-4}{48}$
$=\frac{41}{48}$
$=\frac{15}{16}-\frac{1}{12}$
$=\frac{45-4}{48}$
$=\frac{41}{48}$
Question 71 Mark
Every fraction is a rational number.
Answer
View full question & answer→A fraction is a part or portion of the whole which can be expressed in the form of $\frac{\text{p}}{\text{q}}.$ (Positive rational number) where $p,$ $\text{q}\in\text{I}^{+}.$
Hence, every fraction is a rational number but vich-versa is true.
Hence, every fraction is a rational number but vich-versa is true.
Question 81 Mark
Find: $0\div\frac{2}{3}$
Answer
View full question & answer→$0\div\frac{2}{3}$ $=\frac{0}{2}\times3$ $=0$
Question 91 Mark
Rational numbers are closed under addition and multiplication but not under subtraction.
Answer
View full question & answer→ False.
Solution:
Rational numbers are closed under addition, subtraction and multiplication.
Solution:
Rational numbers are closed under addition, subtraction and multiplication.
Question 101 Mark
Tell which property allows you to compute. $\frac{1}{5}\times\Big[\frac{5}{6}\times\frac{7}{9}\Big]$ as $\Big[\frac{1}{5}\times\frac{5}{6}\Big]\times\frac{7}{9}$
Answer
View full question & answer→$\frac{1}{5}\times\Big[\frac{5}{6}\times\frac{7}{9}\Big]$ Can be written as $\Big[\frac{1}{5}\times\frac{5}{6}\Big]\times\frac{7}{9}$ by the help of associative property for multiplication.
Question 111 Mark
Every whole number is an integer.
Answer
View full question & answer→$W($whole numbers$) = \{0,1, 2, 3\}$
$Z($integers$) = \{… -3, -2, -1, 0, 1, 2, 3, …\}$
Every whole number is an integer, but every integer is not a whole number.
$Z($integers$) = \{… -3, -2, -1, 0, 1, 2, 3, …\}$
Every whole number is an integer, but every integer is not a whole number.
Question 121 Mark
The rational numbers can be represented on the number line.
Answer
View full question & answer→ True.
Solution:
The rational numbers can be represented on the number line.
Solution:
The rational numbers can be represented on the number line.
Question 131 Mark
The negative of $0$ does not exist.
Answer
View full question & answer→True.
Solution:
Since, zero is neither a positive integer nor a negative integer.
Solution:
Since, zero is neither a positive integer nor a negative integer.
Question 141 Mark
Subtraction of rational number is commutative.
Answer
View full question & answer→Subtraction of rational numbers is not commutative, i.e. $\text{a}-\text{b}\neq\text{b}-\text{a}$
Where, $a$ and $b$ are rational numbers.
Where, $a$ and $b$ are rational numbers.
Question 151 Mark
For every rational number $x, x + 1 = x.$
Answer
View full question & answer→For every rational number, $x + 0 = x$
Question 161 Mark
The reciprocal of a positive rational number is _____.
Answer
View full question & answer→The reciprocal of a positive rational number is $\frac{\text{p}}{\text{q}}.$
Solution:
The Positive rational number is of the form $\frac{\text{p}}{\text{q}},$ where $p$ and $q$ both belongs to $I^+$ (positive integers or $I^-$ (negative integers).
Hence, the reciprocal is of the form $\frac{\text{q}}{\text{p}},$ where $p$ and $q$ both belong to $I^+$ or $I^-$.
Solution:
The Positive rational number is of the form $\frac{\text{p}}{\text{q}},$ where $p$ and $q$ both belongs to $I^+$ (positive integers or $I^-$ (negative integers).
Hence, the reciprocal is of the form $\frac{\text{q}}{\text{p}},$ where $p$ and $q$ both belong to $I^+$ or $I^-$.
Question 171 Mark
The rational number $\frac{57}{23}$ lies to the left of zero on the number line.
Answer
View full question & answer→False. Solution: Since, $-\frac{57}{23}$ is a positive rational number. So, it lies on the right of zero on the number line.
Question 181 Mark
Between any two rational numbers there are exactly ten rational numbers.
Answer
View full question & answer→False. Solution: There are infinite rational numbers between any two rational numbers.
Question 191 Mark
The reciprocal of $x^{-1}$ is $\frac{1}{\text{x}}.$
Answer
View full question & answer→$\text{x}^{-1}=\frac{1}{\text{x}}$
$\therefore\ $Reciprocal of $\frac{1}{\text{x}}$ is $x.$
$\therefore\ $Reciprocal of $\frac{1}{\text{x}}$ is $x.$
Question 201 Mark
The reciprocal of $\frac{-5}{7}$ is _____.
Answer
View full question & answer→The reciprocal of $\frac{-5}{7}$ is $\frac{-7}{5}.$ Solution: The reciprocal of $\frac{-5}{7}$ is $\frac{1}{\Big(\frac{-5}{7}\Big)},$ i.e. $\frac{-7}{5}.$
Question 211 Mark
If $\frac{\text{x}}{\text{y}}$ is a rational number, then $y$ is always a whole number.
Answer
View full question & answer→If $\frac{\text{x}}{\text{y}}$ is a rational number. Then; $x$ and $y$ are integers, where $\text{y}\neq0$ Hence, $y$ is always a non-zero integer.
Question 221 Mark
For every rational number $x, x \times 0 = x.$
Answer
View full question & answer→For every rational number $x, x \times 0 = 0.$
Question 231 Mark
The two rational numbers lying between $-2$ and $-5$ with denominator as $1$ are _____ and _____.
Answer
View full question & answer→The two rational numbers lying between $-2$ and $-5$ with denominator as $1$ are $-3$ and $-4.$
Solution:
$-3$ and $-4$ are the two rational numbers lie between $-2$ and $-5$ with denominator $1.$
Solution:
$-3$ and $-4$ are the two rational numbers lie between $-2$ and $-5$ with denominator $1.$
Question 241 Mark
If $\frac{\text{x}}{\text{y}}$ is the additive inverse of $\frac{\text{c}}{\text{d}},$ then $\frac{\text{x}}{\text{y}}-\frac{\text{c}}{\text{d}}=0.$
Answer
View full question & answer→ False.
Solution:
If $\frac{\text{x}}{\text{y}}$ is the additive inverse of $\frac{\text{c}}{\text{d}},$
i.e. $\frac{\text{x}}{\text{y}}-\frac{\text{c}}{\text{d}}=0.$
Solution:
If $\frac{\text{x}}{\text{y}}$ is the additive inverse of $\frac{\text{c}}{\text{d}},$
i.e. $\frac{\text{x}}{\text{y}}-\frac{\text{c}}{\text{d}}=0.$
Question 251 Mark
The reciprocal of $\frac{2}{5}\times\Big(\frac{-4}{9}\Big)$ is _____.
Answer
View full question & answer→The reciprocal of $\frac{2}{5}\times\Big(\frac{-4}{9}\Big)$ is $\frac{-45}{8}$ Solution: Here, $\frac{2}{5}\times\Big(\frac{-4}{9}\Big)$ $=\frac{-8}{45}$ Hencem, the reciprocal of $-\frac{8}{45}$ is $\frac{-45}{8}.$
Question 261 Mark
Every integer is a rational number.
Answer
View full question & answer→True. Solution: Every integer is a rational number whose denominator remain $1.$
Question 271 Mark
$\frac{5}{10}$ lies between $\frac{1}{2}$ and $1.$
Answer
View full question & answer→First, we convert the given rational numbers with denominator as $10,$
we get $\frac{1}{2}=\frac{1}{2}\times\frac{5}{5}=\frac{5}{10}$
$1=1\times\frac{10}{10}=\frac{10}{10}$
$\frac{1}{2}$ is equal to $\frac{5}{10}.$
Therefore, $\frac{5}{10}$ does not lie between $\frac{1}{2}$ and $1.$
we get $\frac{1}{2}=\frac{1}{2}\times\frac{5}{5}=\frac{5}{10}$
$1=1\times\frac{10}{10}=\frac{10}{10}$
$\frac{1}{2}$ is equal to $\frac{5}{10}.$
Therefore, $\frac{5}{10}$ does not lie between $\frac{1}{2}$ and $1.$
Question 281 Mark
$\frac{1}{5}\times\Big[\frac{2}{7}+\frac{3}{8}\Big]=\Big[\frac{1}{5}\times\frac{2}{7}\Big]+$ _____.
Answer
View full question & answer→$\frac{1}{5}\times\Big[\frac{2}{7}+\frac{3}{8}\Big]=\Big[\frac{1}{5}\times\frac{2}{7}\Big]+\frac{1}{5}\times\frac{3}{8}.$ Solution: $\frac{1}{5}\times\Big[\frac{2}{7}+\frac{3}{8}\Big]=\frac{1}{5}\times\frac{2}{7}+\frac{1}{5}\times\frac{3}{8}.$
Question 291 Mark
$\frac{5}{6}$ lies between $\frac{2}{3}$ and $1.$
Answer
View full question & answer→ First, we convert the given rational numbers with denominator as $6,$
we get $\frac{2}{3}=\frac{2}{3}\times\frac{2}{2}=\frac{4}{6}$
$1=1\times\frac{6}{6}=\frac{6}{6}$
$\because\frac{4}{6}<\frac{5}{6}<\frac{6}{6}$
$\therefore\frac{2}{3}<\frac{5}{6}<1$
Therefore, $\frac{5}{6}$ lies between $\frac{2}{3}$ and $1.$
Note: we kno9w that, if $a$ and $b$ are two rational number,
then $\frac{\text{a}+\text{b}}{2}$ is rational number between $a$ and $b$
such that $\text{a}<\frac{\text{a}+\text{b}}{2}<\text{b}.$
we get $\frac{2}{3}=\frac{2}{3}\times\frac{2}{2}=\frac{4}{6}$
$1=1\times\frac{6}{6}=\frac{6}{6}$
$\because\frac{4}{6}<\frac{5}{6}<\frac{6}{6}$
$\therefore\frac{2}{3}<\frac{5}{6}<1$
Therefore, $\frac{5}{6}$ lies between $\frac{2}{3}$ and $1.$
Note: we kno9w that, if $a$ and $b$ are two rational number,
then $\frac{\text{a}+\text{b}}{2}$ is rational number between $a$ and $b$
such that $\text{a}<\frac{\text{a}+\text{b}}{2}<\text{b}.$
Question 301 Mark
The negative of the negative of any rational number is the number itself.
Answer
View full question & answer→Let $x$ be a positive rational number. Then, $-x$ be a negative rational number.
Now, negative of negative rational number $= -(-x) = x =$ Positive rational number.
Now, negative of negative rational number $= -(-x) = x =$ Positive rational number.
Question 311 Mark
Find: $\frac{1}{3}\times\frac{-5}{7}\times\frac{-21}{10}$
Answer
View full question & answer→$\frac{1}{3}\times\frac{-5}{7}\times\frac{-21}{10}$ $=\frac{1}{3}\times\frac{3}{2}$ $=\frac{1}{2}$
Question 321 Mark
Zero has _____ reciprocal.
Answer
View full question & answer→Zero has no reciprocal. Solution: The reciprocal of $0$ is $\frac{1}{0}$ and $\frac{1}{0}$ is not defined.
Question 331 Mark
Select those which can be written as a rational number with denominator $4$ in their lowest form: $\frac{7}{8},\frac{64}{16},\frac{36}{-12},\frac{-16}{17},\frac{5}{-4},\frac{140}{28}$
Answer
View full question & answer→From the given rational numbers, the number with denominator $4$ in their lowest form is $-\frac{5}{-4}$
Question 341 Mark
$0$ is a rational number.
Answer
View full question & answer→True.
Solution:
$0=\frac{0}{1}$ is rational number.
Solution:
$0=\frac{0}{1}$ is rational number.
Question 351 Mark
Simplify each of the following by using suitable property. Also name the property. $\frac{-3}{5}\times\bigg\{\frac{3}{7}+\Big(\frac{-5}{6}\Big)\bigg\}$
Answer
View full question & answer→Given, $\frac{-3}{5}\times\bigg\{\frac{3}{7}+\Big(\frac{-5}{6}\Big)\bigg\}$ $=\frac{-3}{5}\times\frac{3}{7}+\Big(\frac{-3}{2}\Big)\times\Big(\frac{-5}{6}\Big)$ $=\frac{-9}{35}+\frac{15}{30}$ [Using distributive property of multiplication over addition] $=\frac{-54+105}{210}$ $=\frac{51}{210}$ $=\frac{17}{70}$
Question 361 Mark
For rational numbers $x$ and $y,$ if $x < y$ then $x - y$ is a positive rational number.
Answer
View full question & answer→For rational number $x$ and $y,$ If $x < y,$ then $x - y$ is a negative rational number. e.g.
Let $\text{x}=\frac{1}{2},\text{y}=\frac{1}{3}$ are two rational numbers.
Then, according to equation, $\text{x}-\text{y}=\frac{1}{2}-\frac{1}{3}$
$=\frac{3-2}{6}=\frac{1}{6}.$
Let $\text{x}=\frac{1}{2},\text{y}=\frac{1}{3}$ are two rational numbers.
Then, according to equation, $\text{x}-\text{y}=\frac{1}{2}-\frac{1}{3}$
$=\frac{3-2}{6}=\frac{1}{6}.$
Question 371 Mark
For all rational numbers $x$ and $y, x - y = y - x.$
Answer
View full question & answer→For all rational numbers $x$ and $y, x - y = -(y - x).$
Question 381 Mark
$1$ is the only number which is its own reciprocal.
Answer
View full question & answer→Reciprocal of $1$ is $1$ and reciprocal of $-1$ is $-1.$
Question 391 Mark
The reciprocal of a negative rational number is _____.
Answer
View full question & answer→The reciprocal of a negative rational number is $\frac{\text{q}}{\text{p}}.$ Solution: The negative rational number is of the form $\frac{\text{p}}{\text{q}},$ where $\text{p}\in\text{I}^{+},\text{q}\in\text{I}^{-}$ or $\text{q}\in\text{I}^{+}$ Hence, the reciprocal is of the form $\frac{\text{q}}{\text{p}},$ where $\text{p}\in\text{I}^{+},\text{q}\in\text{I}^{-1}$ or $\text{p}\in\text{I}^{-},\text{q}\in\text{I}^{+}$
Question 401 Mark
For all rational numbers $a, b$ and $c, a(b + c) = ab + bc.$
Answer
View full question & answer→False.
Solution:
As, addition is not distributive over multiplication.
Solution:
As, addition is not distributive over multiplication.
Question 411 Mark
$\frac{-7}{2}$ lies between $-3$ and $-4.$
Answer
View full question & answer→First, we convert the given rational numbers with denominator as $2,$ we get
$-3=-3\times\frac{2}{2}=\frac{-6}{2}$
$-4=-4\times\frac{2}{2}=\frac{-8}{2}$
$\because\frac{-8}{2}<\frac{-7}{2}<\frac{-6}{2}$
$\therefore-4<\frac{-7}{2}<-3$
Therefore, $\frac{-7}{2}$lies between $-3$ and $-4.$
$-3=-3\times\frac{2}{2}=\frac{-6}{2}$
$-4=-4\times\frac{2}{2}=\frac{-8}{2}$
$\because\frac{-8}{2}<\frac{-7}{2}<\frac{-6}{2}$
$\therefore-4<\frac{-7}{2}<-3$
Therefore, $\frac{-7}{2}$lies between $-3$ and $-4.$
Question 421 Mark
Using suitable rearrangement and find the sum: $\frac{4}{7}+\Big(\frac{-4}{9}\Big)+\frac{3}{7}+\Big(\frac{-13}{9}\Big)$
Answer
View full question & answer→Here, $\frac{4}{7}+\Big(\frac{-4}{9}\Big)+\frac{3}{7}+\Big(\frac{-13}{9}\Big)$ $=\frac{4}{7}+\frac{3}{7}+\Big(\frac{-4}{9}\Big)+\Big(\frac{-13}{9}\Big)$ $=\frac{7}{7}-\frac{17}{9}$ $=1-\frac{17}{9}$ $=\frac{9-17}{9}$ $=\frac{-8}{9}$
Question 431 Mark
All positive rational numbers lie between $0$ and $1000.$
Answer
View full question & answer→Infinite positive rational numbers lie on the right side of $0$ on the number line.
Question 441 Mark
For any rational number $x, x + (-1) = -x.$
Answer
View full question & answer→For every rational number $x,$
$x × (-1) = -x.$
$x × (-1) = -x.$
Question 451 Mark
If $x + y = 0,$ then $-y$ is known as the negative of $x,$ where $x$ and $y$ are rational numbers.
Answer
View full question & answer→If $x$ and $y$ are rational numbers and $x + y = 0.$ Then, y is known as the negative of $x.$
Question 461 Mark
Verify $-(-x) = x$ for:
$\text{x}=\frac{13}{-15}$
$\text{x}=\frac{13}{-15}$
Answer
View full question & answer→Given, $\text{x}=\frac{13}{-15}$
$\Rightarrow-\text{x}=-\Big(\frac{13}{-15}\Big)$
$\Rightarrow-\text{x}=\frac{13}{15}$
$\Rightarrow-(-\text{x})=\frac{-13}{15}$
$=\text{x}$
$\Rightarrow-\text{x}=-\Big(\frac{13}{-15}\Big)$
$\Rightarrow-\text{x}=\frac{13}{15}$
$\Rightarrow-(-\text{x})=\frac{-13}{15}$
$=\text{x}$
Question 471 Mark
If $x$ and $y$ are negative rational numbers, then so is $x + y.$
Answer
View full question & answer→e.g. $\Big(-\frac{1}{2}\Big)+\Big(-\frac{1}{2}\Big)=-1,$
which is again a negative rational number.
Note: Sum of two negative rational number is equal to a negative rational number.
which is again a negative rational number.
Note: Sum of two negative rational number is equal to a negative rational number.
Question 481 Mark
Every whole number is a rational number.
Answer
View full question & answer→Every whole number can be written in the form of $-\frac{\text{p}}{\text{q}},$ where $p, q$ are integers and $\text{a}\neq0.$ Hence, every whole number is a rational number.
Question 491 Mark
Simplify: $\frac{3}{7}+\frac{28}{15}\div\frac{14}{5}$
Answer
View full question & answer→Given, $\frac{3}{7}+\frac{28}{15}\div\frac{14}{5}$$=\frac{4}{5}+\frac{14}{5}$
$=\frac{4}{5}\times\frac{5}{14}$
$=\frac{2}{7}$
$=\frac{4}{5}\times\frac{5}{14}$
$=\frac{2}{7}$
Question 501 Mark
Verify $-(-x) = x$ for:
$\text{x}=\frac{3}{5}$
$\text{x}=\frac{3}{5}$
Answer
View full question & answer→Given, $\text{x}=\frac{3}{5}$
$\Rightarrow-\text{x}=\frac{-3}{5}$
$\Rightarrow-(-\text{x})=-\Big(\frac{-3}{5}\Big)$
$\Rightarrow-(-\text{x})=\frac{-7}{9}$
$=\text{x}$
$\Rightarrow-\text{x}=\frac{-3}{5}$
$\Rightarrow-(-\text{x})=-\Big(\frac{-3}{5}\Big)$
$\Rightarrow-(-\text{x})=\frac{-7}{9}$
$=\text{x}$