[5 marks sum] - Page 2 - MATHEMATICS STD 9 Questions - Vidyadip

Questions · Page 2 of 2

[5 marks sum]

Question 515 Marks

Express the following decimal as a rational number.$0.017$

Answer

Let $x=0.0 \overline{17}$
$=0.01717 . .$
Here, only numbers $17$ is being repeated,
so first we need to remove $0$ which proceeds $17.$
We multiply by $10$ so that the recurring digits remain after decimal.
$\therefore 10 x=0.1717 \ldots (1)$
The number of digits recurring equation $(1)$ is $2$,
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 1000 x=100 \times 0.1717 \ldots$
$=17.1717 \ldots \ldots(2)$
On subtracting $(1)$ from $(2)$, we get
$990 x=17$
$\therefore x=\frac{17}{990}$
$\therefore 0.017=(17) /(990)$

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Question 525 Marks

Express the following decimal as a rational number.$4.6724$

Answer

Let $x=4 . \overline{6724}$
$=4.6724724 \ldots$
Here, only numbers $724$ is being repeated,
so first we need to remove $6$ which proceeds $724 .$
We multiply by $10$ so that only the recurring digits remain after decimal.
$\therefore 10 x=46.724724 \ldots (1)$
The number of digits recurring in equation $(1)$ is $3$ ,
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 10000 x =1000 \times 46.724724 \ldots$
$=46724.724 \ldots \ldots(2)$
On subtracting $(1)$ from $(2)$, we get
$9990 x=4678$
$\therefore x=\frac{46678}{9990}$
$=\frac{23339}{4995}$
$\therefore 4 . \overline{6724}=\frac{763}{999}$
$=\frac{23339}{4995}$

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Question 535 Marks

Express the following decimal as a rational number.$2.67$

Answer

Let $x=2 . \overline{67}$
Then, $x =2.676767 \ldots (1)$
Here, the number of digits recurring is $2$ ,
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x =100 \times 2.676767 \ldots $
$=267.8989 \ldots \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$99 x=265 $
$\therefore x=\frac{265}{99} $
$\therefore 2 . \overline{67}=\frac{265}{99}$

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Question 545 Marks

Express the following decimal as a rational number.$0.763$

Answer

Let $x=0 . \overline{763}$
Then, $x=0.763763 \ldots (1)$
Here, the number of digits recurring is $3,$
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 1000 x =1000 \times 0.763763 \ldots $
$=763.763 \ldots . . \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$999 x=763 $
$\therefore x=\frac{763}{999} $
$\therefore 0 . \overline{763}=\frac{763}{999}$

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Question 555 Marks

Express the following decimal as a rational number.$0.057$

Answer

Let $x=0 . \overline{057}$
Then, $x=0.057057 \ldots (1)$
Here, the number of digits recurring is $3,$
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 1000 x =1000 \times 0.057057 \ldots . $
$=57.057 \ldots \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$999 x=57 $
$\therefore x=\frac{57}{999} $
$=\frac{19}{333} $
$\therefore 0 . \overline{057}=\frac{19}{333}$

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Question 565 Marks

Express the following decimal as a rational number.$0.35$

Answer

Let $x=0 . \overline{35}$
Then, $x=0.353535 \ldots$
Here, the number of digits recurring is $2,$
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x =100 \times 0.353535 \ldots$
$=35.3535 \ldots$
On subtracting $(1)$ from $(2),$ we get
$95 x=35 $
$\therefore x=\frac{35}{99} $
$\therefore 0.35=\frac{35}{99}$

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Question 575 Marks

Find the value of:$1.35 + 1.5$

Answer

Let $x=1.3 \overline{5} $
$\Rightarrow 10 x=13 . \overline{5} .....(i)$
$\Rightarrow 100 x=135 . \overline{5} .....(ii)$
Subtracting $(i) a=$ from $(ii),$
$90 x =122 $
$\Rightarrow x =\frac{122}{90} $
$=\frac{61}{45}$
Let $y=1 . \overline{5} .....(iii)$
$\Rightarrow 10 y =15 . \overline{5} .....(iv)$
$\Rightarrow 9 y =14 $
$\Rightarrow y =\frac{14}{9} $
$\therefore 1.3 \overline{5}+1 . \overline{5}= x + y $
$=\frac{61}{45}+\frac{14}{9} $
$=\frac{61 \times 1+14 \times 5}{45} $
$=\frac{61+70}{45} $
$=\frac{131}{45} $
$=2.91$

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Question 585 Marks

Find the value of:$2.12 - 0.45$

Answer

Let $x=2 . \overline{12} \ldots(i) $
$\Rightarrow 100 x=212.12 ........(ii)$
Subtracting $(i)$ from $(ii),$
$99 x =210 $
$\Rightarrow x =\frac{210}{99} $
$=\frac{70}{33} $
Let $y =0.4 \overline{5} $
$\Rightarrow 10 y =4 . \overline{5} ......(iii) $
$\Rightarrow 100 y =45 . \overline{5} .......(iv)$
Subtracting $(iii)$ from $(iv),$
$90 y =41 $
$\Rightarrow y =\frac{41}{90} $
$\therefore 2 . \overline{12}-0.4 \overline{5}= x - y $
$=\frac{70}{33}-\frac{41}{90} $
$=\frac{70 \times 30-41 \times 11}{990} $
$=\frac{2100-451}{990} $
$=\frac{1649}{9 \underline{90}} $
$=1.665$

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Question 595 Marks

Find the value of:$1. 32 - 0.91$

Answer

Let $x =1 . \overline{32} \ldots( i )$
$\Rightarrow 100 x =132 . \overline{32} ...... (ii)$
Subtracting $(i)$ from $(ii),$
$99 x =131$
$\Rightarrow x =\frac{131}{99}$
Let $y =0.9 \overline{1}$
$ \Rightarrow 10 y =9 . \overline{1} \ldots . . . \text { (iii) }$
$\Rightarrow 100 y =9 . \overline{1} \quad \ldots \text { (iv) }$
Subtracting $(iii)$ from $(iv),$
$ 90 y=82$
$\Rightarrow y=\frac{82}{90}$
$=\frac{41}{45}$
$\therefore 1.32-0.91=x-y$
$=\frac{131}{99}-\frac{41}{45}$
$=\frac{131 \times 5-41 \times 11}{495}$
$=\frac{655-451}{495}$
$=\frac{204}{495}$
$=0.412$

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Question 605 Marks

Find the value of:$2.65 + 1.25$

Answer

Let $x=2.6 \overline{5}=2.6555 \ldots $
$\Rightarrow 10 x=26 . \overline{5} \ldots . .( i ) $
$\Rightarrow 100 x=265 . \overline{5} \ldots \text { (ii) }$
Subtracting $(i)$ and $(ii),$
$90 x =239 $
$\Rightarrow x =\frac{239}{90}$
Let $y =1 . \overline{25} ........ (iii)$
$\Rightarrow 100 y =125 . \overline{25} ........ (iv)$
Subtracting $(iii)$ from $(iv),$
$99 y =124 $
$\Rightarrow y =\frac{214}{99} $
$\therefore 2.65+1.25= x + y $
$=\frac{239}{90}+\frac{124}{99} $
$=\frac{239 \times 11+124 \times 10}{990} $
$=\frac{2629+1240}{990} $
$=\frac{3869}{990} $
$=3.908$

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Question 615 Marks

Arrange the following rational numbers in descending order.$\frac{7}{13}, \frac{8}{15}$, and $\frac{3}{5}$

Answer

Given numbers : $\frac{7}{13}, \frac{8}{15}$, and $\frac{3}{5}$
The $\text{L.C.M.}$ of $13,15$ and $5$ is $195.$
Thus, numbers are :
$\frac{7}{13} $
$=\frac{7 \times 15}{13 \times 15} $
$=\frac{105}{195} $
$\frac{8}{15} $
$=\frac{8 \times 13}{15 \times 13} $
$=\frac{104}{195} $
$\frac{3}{5} $
$=\frac{3 \times 39}{5 \times 39} $
$=\frac{117}{195}$
Since $117>105>104$,
we have $\frac{3}{5}>\frac{7}{13}>\frac{8}{15}$.

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Question 625 Marks

Arrange the following rational numbers in ascending order.$\frac{7}{4}, \frac{-6}{5}$ and $\frac{-5}{2}$

Answer

Given number : $\frac{7}{4}, \frac{-6}{5}$ and $\frac{-5}{2}$
The $\text{L.C.M.}$ of $4,5$ and $2$ is $20.$
Thus, numbers are:
$\frac{7}{4} $
$=\frac{7 \times 5}{4 \times 5} $
$=\frac{35}{20} $
$\frac{-6}{5} $
$=\frac{-6 \times 4}{5 \times 4} $
$=\frac{-36}{20} $
$\text { and } \frac{-5}{2} $
$=\frac{-5 \times 10}{2 \times 10} $
$=\frac{-50}{20}$
Since $-50<-36<35$,
we have $\frac{-5}{2}<\frac{-6}{5}<\frac{7}{4}$.

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Question 635 Marks

Arrange the following rational numbers in ascending order.$\frac{10}{9}, \frac{13}{12}$ and $\frac{19}{18}$

Answer

Given number : $\frac{10}{9}, \frac{13}{12}$ and $\frac{19}{18}$
The $\text{L.C.M}.$ of $9,12$ and $18$ is $36.$
Thus, number are:
$\frac{10}{9} $
$=\frac{10 \times 4}{9 \times 4} $
$=\frac{40}{36} $
$\frac{13}{12} $
$=\frac{13 \times 3}{\frac{12}{39} \times 3} $
$=\frac{3^{36}}{19} $
$\text { and } \frac{19}{18} $
$=\frac{19 \times 2}{18 \times 2} $
$=\frac{38}{36}.$
Since $38<39<40$,
we have $\frac{19}{18}<\frac{13}{12}<\frac{10}{9}$.

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Question 645 Marks

Arrange the following rational numbers in ascending order.$\frac{-7}{12}, \frac{-3}{10}$ and $\frac{-2}{5}$

Answer

Given number: $\frac{-7}{12}, \frac{-3}{10}$ and $\frac{-2}{5}$
The $\text{L.C.M.}$ of $12,10$ and $5$ is $60.$
Thus, numbers are:
$\frac{-7}{12} $
$=\frac{-7 \times 5}{12 \times 5} $
$=\frac{-35}{60} ; $
$\frac{-3}{10} $
$=\frac{-3 \times 6}{10 \times 6} $
$=\frac{-18}{60} ; $
$\frac{-2}{5} $
$=\frac{-2 \times 10}{5 \times 10} $
$=\frac{-20}{60}$
Since $-35<-20<-18$,
we have $\frac{-7}{12}<\frac{-2}{5}<\frac{-3}{10}$.

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Question 655 Marks

Arrange the following rational numbers in ascending order.$\frac{4}{5}, \frac{6}{7}$ and $\frac{7}{10}$

Answer

Given number : $\frac{4}{5}, \frac{6}{7}$ and $\frac{7}{10}$
The $\text{L.C.M.}$ of $5, 7$ and $10$ is $70.$
Thus, numbers are :
$\frac{4}{5} $
$=\frac{4 \times 14}{5 \times 14} $
$=\frac{56}{70} ;$
$\frac{6}{7} $
$=\frac{6 \times 10}{7 \times 10} $
$=\frac{60}{70} $
and $\frac{7}{10}$
$=\frac{7 \times 7}{10 \times 7} $
$=\frac{49}{70} .$
Since $49<56<60$,
we have $\frac{7}{10}<\frac{4}{5}<\frac{6}{7}$.

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[5 marks sum] - Page 2 - MATHEMATICS STD 9 Questions - Vidyadip