3 Marks Question

Question 13 Marks

If a property holds for rational number, will it also hold for integers? For whole numbers? Which will? Which will not?

Answer

All properties of operations on rational numbers also hold in case of integers except the following property: $a+b$ is a rational number, if $b \neq 0$ but $a+b$ is not necessarily an integer.
All properties of operations on rational numbers also hold in case of whole numbers except the following properties.
(i) If $a$ and $b$ are rational numbers, then $(a-b)$ may or may not be a whole number.
(ii) If $a$ and $b$ are rational numbers, then $a+b$ (where, $b \neq 0$ ) is not necessarily a whole number.

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Question 23 Marks

Name the property under multiplication used in each of the following
(i) $\frac{-4}{5} \times 1=1 \times\left(\frac{-4}{5}\right)=-\frac{4}{5}$
(ii) $\frac{-13}{17} \times\left(\frac{-2}{7}\right)=\frac{-2}{7} \times\left(\frac{-13}{17}\right)$
(iii) $\frac{-19}{29} \times\left(\frac{29}{-19}\right)=1$

Answer

(i) We have, $\frac{-4}{5} \times 1=1 \times\left(\frac{-4}{5}\right)=-\frac{4}{5}$
This is of the form of $a \times 1=1 \times a=a$, where $a=-\frac{4}{5}$
So, 1 is multiplicative identity.
(ii) We have, $\frac{-13}{17} \times\left(\frac{-2}{7}\right)=\frac{-2}{7} \times\left(\frac{-13}{17}\right)$
This is of the form of $a \times b=b \times a$, where $a=\frac{-13}{17}$ and $b=\frac{-2}{7}$, i.e. commutative property of multiplication.
So, property used is commutativity.
(iii) We have, $\frac{-19}{29} \times \frac{29}{-19}=1$
This is of the form of $\frac{a}{b} \times \frac{b}{a}=1$, where $\frac{a}{b}=\frac{-19}{29}$ and $\frac{b}{a}=\frac{29}{-19}$
So, property used is multiplicative inverse.

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Question 33 Marks

Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.

Answer

We know that rational numbers are closed under addition, subtraction and multiplication. We can understand this from the following examples:
Rational numbers are closed under
Addition
e.g. $\frac{4}{7}+\frac{1}{2}=\frac{8+7}{14}=\frac{15}{14}$ which is a rational number.
Subtraction
e.g. $\frac{4}{7}-\frac{1}{2}=\frac{8-7}{14}=\frac{1}{14}$ which is a rational number.
Multiplication
e.g. $\frac{4}{7} \times \frac{1}{2}=\frac{4}{14}=\frac{2}{7}$ which is a rational number.
But rational numbers are not closed under division. If zero is excluded from the collection of rational numbers, then we can say that rational numbers are closed under division.
Now, we see the examples given below
$\frac{4}{7}+\frac{1}{2}=\frac{4}{7} \times 2=\frac{8}{7}$ which is a rational number.
But $\frac{4}{7}\div0=\frac{4}{7} \times \frac{1}{0}$
which is not defined and so, it is not a rational number.
Also, $\frac{1}{2}\div0=\frac{1}{2} \times \frac{1}{0}$,
which is not defined and also not a rational number.

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Question 43 Marks

Simplify each of the following by using suitable property. Also, name the properties.
(i) $\left[\frac{1}{2} \times \frac{1}{4}\right]+\left[\frac{1}{2} \times 6\right]$
(ii) $\left[\frac{1}{5} \times \frac{2}{15}\right]-\left[\frac{1}{5} \times \frac{2}{5}\right]$
(iii) $\frac{-3}{5} \times\left[\frac{3}{7}+\left(\frac{-5}{6}\right)\right]$

Answer

(i) $\left[\frac{1}{2} \times \frac{1}{4}\right]+\left[\frac{1}{2} \times 6\right]=\frac{1}{2} \times\left[\frac{1}{4}+6\right]\quad$ [by distributive property over addition]
$=\frac{1}{2} \times\left[\frac{1+24}{4}\right]=\frac{25}{8}$
(ii) $\left[\frac{1}{5} \times \frac{2}{15}\right]-\left[\frac{1}{5} \times \frac{2}{5}\right]=\left[\frac{1}{5} \times \frac{2}{15}\right]+\left[\frac{1}{5} \times\left(\frac{-2}{5}\right)\right]$
$=\frac{1}{5} \times\left[\frac{2}{15}-\frac{2}{5}\right]\quad$ [by distributive property over addition]
$=\frac{1}{5} \times\left[\frac{2-6}{15}\right]=\frac{-4}{75}$
(iii) $\frac{-3}{5} \times\left\{\frac{3}{7}+\left(\frac{-5}{6}\right)\right\}=\frac{-3}{5} \times \frac{3}{7}+\left(\frac{-3}{5}\right) \times\left(\frac{-5}{6}\right)\quad$ [by distributive property]
$=\frac{-9}{35}+\frac{15}{30}=\frac{-54+105}{210}=\frac{51}{210}=\frac{17}{70}$

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Question 53 Marks

Robin writes the properties of rational numbers as
Property 1 For any two rational number$x$ and $y$,
$ x+y=y+x $
Property 2 For any two rational number $x$ and $y,$
$x-y=y-x$
Property 3 For any two rational number $x$ and $y,$
$x / y=y / x$
Property 4 For any two rational number $x$ and $y,$
$x \times y=y \times x$
Which properties written by Robin are incorrect? Why are they incorrect?

Answer

Property 2 and Property 3 written by Robin are incorrect
i.e. for any two rational numbers $x$ and $y$,
$x-y \neq y-x$
and $ \frac{x}{y} \neq \frac{y}{x} $
e.g. Let $\frac{3}{4}$ and $\frac{6}{5}$ be any two rational numbers.
Then, $\frac{3}{4}-\frac{6}{5}=\frac{3 \times 5-6 \times 4}{20}=\frac{15-24}{20}=\frac{-9}{20}$
and $\frac{6}{5}-\frac{3}{4}=\frac{6 \times 4-3 \times 5}{20}=\frac{24-15}{20}=\frac{9}{20}$
$\therefore \quad \frac{3}{4}-\frac{6}{5} \neq \frac{6}{5}-\frac{3}{4}$
$\therefore$ Property 2 does not hold true.
Also, $\frac{3}{4} \div \frac{6}{5}=\frac{3}{4} \times \frac{5}{6}=\frac{5}{8}$ and $\frac{6}{5} \div \frac{3}{4}=\frac{6}{5} \times \frac{4}{3}=\frac{8}{5}$
$\therefore \quad \frac{3}{4} \div \frac{6}{5} \neq \frac{6}{5} \div \frac{3}{4}$
Hence, Property 3 does not hold true.

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Question 63 Marks

State which property is in the following and verify it.
(i) $\frac{2}{5}+\left\{\left(-\frac{6}{7}\right)+\frac{7}{6}\right\} \neq\left\{\frac{2}{5}+\left(\frac{-6}{7}\right)\right\} + \frac{7}{6}$
(ii) $\frac{3}{4}-\left(\frac{9}{16}-\frac{19}{8}\right) \neq\left(\frac{3}{4}-\frac{9}{16}\right)-\frac{9}{8}$
(iii) $4 \div(8\div16) \neq(4 \div 8) \div 16$

Answer

(i) $\frac{2}{5} \div\left[\left(\frac{-6}{7}\right) \div \frac{7}{6}\right] \neq\left[\frac{2}{5} \div\left(\frac{-6}{7}\right)\right] \div \frac{7}{6}$
Rational numbers are not associative under division.
LHS $=\frac{2}{5} \div\left(\frac{-6}{7} \times \frac{6}{7}\right)=\frac{2}{5} \times\left(\frac{-49}{36}\right)=\frac{-98}{180}$
RHS $=\left[\frac{2}{5}+\left(\frac{-6}{7}\right)\right]+\frac{7}{6}$
$=\left(\frac{2}{5} \times \frac{-7}{6}\right) \div \frac{7}{6}=\frac{2 \times(-7)}{5 \times 6} \times \frac{6}{7}=\frac{-2}{5}$
$\Rightarrow LHS\neq RHS$
(ii) $\frac{3}{4}-\left(\frac{9}{16}-\frac{19}{8}\right) \neq\left(\frac{3}{4}-\frac{9}{16}\right)-\frac{9}{8}$
Rational numbers are not associative under subtraction
LHS $=\frac{3}{4}-\left(\frac{9-19 \times 2}{16}\right)=\frac{3}{4}-\left(\frac{-29}{16}\right)$
$=\frac{3}{4}+\frac{29}{16}=\frac{12+29}{16}=\frac{41}{16}$
RHS $=\left(\frac{3}{4}-\frac{9}{16}\right)-\frac{9}{8}$
$=\left(\frac{12-9}{16}\right)-\frac{9}{8}=\frac{3}{16}-\frac{9}{8}=\frac{3-18}{16}=\frac{-15}{16}$
$\Rightarrow$ $LHS\neq RHS$
(iii) $4 \div(8 \div 16) \neq(4 \div 8) \div 16$
Rational numbers are not associative under division.
LHS $=4 \div(8 \div 16)$
$=4 \div\left(8 \times \frac{1}{16}\right)=4 \div\left(\frac{1}{2}\right)=4 \times 2=8$
RHS $=(4 \div 8) \div 16$
$=\left(4 \times \frac{1}{8}\right) \div 16=\frac{1}{2} \div 16=\frac{1}{2} \times \frac{1}{16}=\frac{1}{32}$
$\Rightarrow \quad$ LHS $\neq$ RHS

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Question 73 Marks

Is the statement 'All integers are rational numbers but all rational numbers are not integers' correct? Give examples to support your answer.

Answer

We know that integers are the collection of whole numbers and negative numbers i.e. . ...., $-3,-2,-1,0,1,2,3, \ldots$
Also, any number of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ is rational number.
Thus, all integers are rational number because any integer $x$ can be written as $\frac{x}{1}$, which is rational number.
But not all rational numbers are integers.
e.g. $\frac{3}{4}$ is a rational number, but not an integer.
Also, $\frac{-5}{4}$ is a rational number, but not an integer.

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Question 83 Marks

Name the property under multiplication used in each of the following
(i) $\frac{-8}{13} \times \frac{13}{-8}=1$
(ii) $\frac{5}{29} \times \frac{-19}{5}=\frac{-19}{5} \times \frac{5}{29}$
(iii) $\frac{-37}{50} \times 1=1 \times \frac{-37}{50}=\frac{-37}{50}$

Answer

(i) Multiplicative inverse
(ii) Commutativity under multiplication
(iii) Multiplicative identity

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Question 93 Marks

In each of the following write the name of property which allows you to compute the expression.
(i) $4 \times 2+4 \times \frac{-5}{6}$ as $4 \times\left[2+\left(\frac{-5}{6}\right)\right]$
(ii) $\frac{9}{10} \times\left(\frac{10}{9} \times 30\right)$ as $\left(\frac{9}{10} \times \frac{10}{9}\right) \times 30$
(iii) $9 \times \frac{13}{16}-9 \times \frac{5}{16}$ as $9 \times\left(\frac{13}{16}-\frac{5}{16}\right)$

Answer

(i) Distributive property over addition, $\frac{14}{3}$
(ii) Associative property over multiplication, 30
(iii) Distributive property over subtraction, $\frac{9}{2}$

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Question 103 Marks

Find the following expression using the appropriate property.
(i) $0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)$
(ii) $\frac{16}{9} \times\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right) \times \frac{51}{49}$
(iii) $\frac{4}{9} \times \frac{19}{20}-\frac{4}{9} \times \frac{1}{20}$
(iv) $\frac{5}{2} \times \frac{1}{10}+\frac{2}{7}-\frac{9}{4} \times \frac{1}{10}$

Answer

(i) $0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)$
Since, $\frac{2}{3}$ and $\frac{9}{16}$ are rational numbers. So, $\frac{2}{3} \times \frac{9}{16}$ will be a rational number.
$[\because$ product of 2 rational numbers is also a rational number]
$\because 0 \div\left(\frac{p}{q}\right)=0, \quad$ [where $\frac{p}{q}$ is a rational number]
and when we divide 0 by any non-zero rational number, we get 0
$\Rightarrow 0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)=0$
(ii) $\frac{16}{9} \times\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right) \times \frac{51}{49}$
$=\frac{16}{9} \times\left[\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right)\right] \times \frac{51}{49}$
$=\frac{16}{9} \times\left[\left(\frac{-27}{4}\right) \times\left(\frac{-14}{17}\right)\right] \times \frac{51}{49}\quad$ [by commutative property]
$=\frac{16}{9} \times\left(\frac{-27}{4}\right) \times\left[\left(\frac{-14}{17}\right) \times \frac{51}{49}\right]\quad$ [by associative property]
$=\frac{16 \times(-27)}{9 \times 4} \times \frac{(-14) \times 51}{17 \times 49}=\frac{4 \times(-3)}{1} \times \frac{(-2) \times 3}{7}=\frac{72}{7}$
(iii) $\frac{4}{9} \times \frac{19}{20}-\frac{4}{9} \times \frac{1}{20}=\frac{4}{9} \times\left(\frac{19}{20}-\frac{1}{20}\right)\quad$ [by distributive property over subtraction of rational numbers]
$=\frac{4}{9} \times\left(\frac{19-1}{20}\right)$
$=\frac{4}{9} \times \frac{18}{20}=\frac{4 \times 18}{9 \times 20}=\frac{2}{5}$
(iv) $\frac{5}{2} \times \frac{1}{10}+\frac{2}{7}-\frac{9}{4} \times \frac{1}{10}$
$=\frac{5}{2} \times \frac{1}{10}+\left[\frac{2}{7}+\left(\frac{-9}{4}\right) \times \frac{1}{10}\right]$
$=\frac{5}{2} \times \frac{1}{10}+\left[\left(\frac{-9}{4}\right) \times \frac{1}{10}+\frac{2}{7}\right]\quad$ [by commutative property for addition of rational numbers]
$=\left[\frac{5}{2} \times \frac{1}{10}+\left(\frac{-9}{4}\right) \times \frac{1}{10}\right]+\frac{2}{7}\quad$ [by associative property]
$=\left(\frac{10-9}{4 \times 10}\right)+\frac{2}{7}$
$=\frac{1}{40}+\frac{2}{7}=\frac{7+40 \times 2}{40 \times 7}=\frac{87}{280}$

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Question 113 Marks

State which property is in the following and verify it.
(i) $\frac{19}{50} \times\left(\frac{50}{38} \times \frac{5}{19}\right)=\left(\frac{19}{50} \times \frac{50}{38}\right) \times \frac{5}{19}$
(ii) $\frac{-6}{5}+\left\{\frac{2}{15}+\left(\frac{-9}{25}\right)\right\}=\left\{\left(\frac{-6}{5}\right)+\frac{2}{15}\right\}+\left(\frac{-9}{25}\right)$
(iii) $\frac{6}{19} \times\left\{\frac{4}{15}-\frac{2}{5}\right\}=\frac{6}{19} \times \frac{4}{15}-\frac{6}{19} \times \frac{2}{5}$

Answer

(i) $\frac{19}{50} \times\left(\frac{50}{38} \times \frac{5}{19}\right)=\left(\frac{19}{50} \times \frac{50}{38}\right) \times \frac{5}{19}$
This statement follow associative property undet multiplication
LHS $=\frac{19}{50} \times\left(\frac{50}{38} \times \frac{5}{19}\right)=\frac{19}{50} \times \frac{50 \times 5}{38 \times 19}=\frac{5}{38}$
RHS $=\left(\frac{19}{50} \times \frac{50}{38}\right) \times \frac{5}{19}=\frac{1}{2} \times \frac{5}{19}=\frac{5}{38}$
(ii) $\frac{-6}{5}+\left[\frac{2}{5}+\left(\frac{-9}{25}\right)\right]=\left[\left(\frac{-6}{5}\right)+\frac{2}{5}\right]+\left(\frac{-9}{25}\right)$
This statement follow associative property under addition.
LHS $=\frac{-6}{5}+\left[\frac{2}{5}+\left(\frac{-9}{25}\right)\right]$
$=\frac{-6}{5}+\left(\frac{2}{5}-\frac{9}{25}\right)=\frac{-6}{5}+\left(\frac{10-9}{25}\right)=\frac{-6}{5}+\frac{1}{25}$
$=\frac{-6}{5}+\frac{1}{25}=\frac{-30+1}{25}=\frac{-29}{25}$
RHS $=\left[\left(\frac{-6}{5}\right)+\frac{2}{5}\right]+\left(\frac{-9}{25}\right)$
$=\left(\frac{-6}{5}+\frac{2}{5}\right)-\frac{9}{25}=\frac{-4}{5}-\frac{9}{25}=\frac{-20-9}{25}=\frac{-29}{25}$
(iii) $\frac{6}{19} \times\left(\frac{4}{15}-\frac{2}{5}\right)=\frac{6}{19} \times \frac{4}{15}-\frac{6}{19} \times \frac{2}{5}$
This statement follow distributive property over subtraction.
LHS $=\frac{6}{19} \times\left(\frac{4}{15}-\frac{2}{5}\right)$
$=\frac{6}{19} \times\left(\frac{4-6}{15}\right)=\frac{6}{19} \times \frac{(-2)}{15}=\frac{2 \times(-2)}{19 \times 5}=\frac{-4}{95}$
RHS $=\frac{6}{19} \times \frac{4}{15}-\frac{6}{19} \times \frac{2}{5}$
$=\frac{8}{95}-\frac{12}{95}=\frac{8-12}{95}=\frac{-4}{95}$

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Question 123 Marks

(i) If $x=6, y=\frac{1}{9}, z=0\quad$ (ii) If $x=\frac{4}{5}, y=\frac{-9}{10}, z=\frac{43}{15}$
Then, verify the following properties and name them
(a) $x \times(y+z)=x \times y+x \times z$
(b) $x \times(y \times z)=(x \times y) \times z$
(c) $x \times y=y \times x$
(d) $x \times(y-z)=x \times y-x \times z$

Answer

(i) For $x=6, y=\frac{1}{9}$ and $z=0$
(a) $x \times(y+z)=x \times y+x \times z$
This statement follow distributive property over addition.
LHS $=x \times(y+z)=6 \times\left(\frac{1}{9}+0\right)=6 \times \frac{1}{9}=\frac{2}{3}$
RHS $=x \times y+x \times z=6 \times \frac{1}{9}+6 \times 0=\frac{2}{3}+0=\frac{2}{3}$
(b) $x \times(y \times z)=(x \times y) \times z$
This statement follow associative property under multiplication
LHS $=x \times(y \times z)=6 \times\left(\frac{1}{9} \times 0\right)=6 \times 0=0$
$RHS =(x \times y) \times z=\left(6 \times \frac{1}{9}\right) \times 0=\frac{2}{3} \times 0=0$
(c) $x \times y=y \times x$
This statement follow commutative property under multiplication.
LHS $=x \times y=6 \times \frac{1}{9}=\frac{2}{3}$
$RHS =y \times x=\frac{1}{9} \times 6=\frac{2}{3}$
(d) $x \times(y-z)=x \times y-x \times z$
This statement follow distributive property over subtraction.
$LHS =x \times(y-z)=6 \times\left(\frac{1}{9}-0\right)=6 \times \frac{1}{9}=\frac{2}{3}$
$RHS =x \times y-x \times z=6 \times \frac{1}{9}-6 \times 0$
$=\frac{2}{3}-0=\frac{2}{3}$
(ii) $x=\frac{4}{5}, y=\frac{-9}{10}, z=\frac{43}{15}$
(a) $x \times(y+z)=x \times y+x \times z$
This statement follow distributive property over addition.
$LHS =x \times(y+z)=\frac{4}{5} \times\left(\frac{-9}{10}+\frac{43}{15}\right)$
$=\frac{4}{5} \times \frac{(-27+86)}{30}=\frac{4}{5} \times \frac{59}{30}=\frac{118}{75}$
$RHS =x \times y+x \times z=\frac{4}{5} \times\left(\frac{-9}{10}\right)+\frac{4}{5} \times \frac{43}{15}$
$=-\frac{18}{25}+\frac{172}{75}=\frac{-54+172}{75}=\frac{118}{75}$
(b) $x \times(y \times z)=(x \times y) \times z$
This statement follow associative property under multiplication.
LHS $=x \times(y \times z)$
$=\frac{4}{5} \times\left\{\left(\frac{-9}{10}\right) \times \frac{43}{15}\right\}=\frac{4}{5} \times\left(\frac{-129}{50}\right)=-\frac{516}{250}$
RHS $=(x \times y) \times z$
$=\left\{\frac{4}{5} \times\left(\frac{-9}{10}\right)\right\} \times \frac{43}{15}=\frac{-18}{25} \times \frac{43}{15}$
$=\frac{-258}{125}=\frac{-516}{250}$
(c) $x \times y=y \times x$
This statement follow commutative property under multiplication.
LHS $=x \times y=\frac{4}{5} \times\left(\frac{-9}{10}\right)=\frac{-18}{25}$
RHS $=y \times x=\frac{-9}{10} \times \frac{4}{5}=\frac{-18}{25}$
(d) $x \times(y-z)=x \times y-x \times z$
This statement follow distributive property under subtraction.
LHS $x \times(y-z)=\frac{4}{5} \times\left(\frac{-9}{10}-\frac{43}{15}\right)=\frac{4}{5} \times\left(\frac{-27-86}{30}\right)$
$=\frac{4}{5} \times\left(-\frac{113}{30}\right)=\frac{-226}{75}$
$RHS =x \times y-x \times z$
$=\frac{4}{5} \times \frac{-9}{10}-\frac{4}{5} \times \frac{43}{15}=\frac{-18}{25}-\frac{172}{75}$
$=\frac{-54-172}{75}=\frac{-226}{75}$

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Question 133 Marks

Rearrange suitably and find the sum in each of the following
(i) $\frac{2}{3}+\frac{9}{2}+\frac{7}{4}+\left(\frac{-6}{3}\right)+\frac{-3}{2}$
(ii) $\frac{-5}{7}+\frac{5}{6}+\frac{1}{7}+3+\left(\frac{-13}{6}\right)$

Answer

(i) $\frac{41}{12}\quad$ (ii) $\frac{23}{21}$

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MATHS 3 Marks Question

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