Question

Ques 4 : Express in the form of $\displaystyle\frac{p}{q}$.
(i) 0.90
(ii) 1.0001
(iii) $0.\overline{621}$
(iv) $0.4\overline{7}$
(v) $0.4 \overline{7}$​

Answer 1

Step-by-step explanation:

$(1) \: \: \displaystyle{0.9} \\ \\ \bold{no. \: of \: numbers \: after \: decimal \: point = 1} \\ \\ multiply \: and \: divide \: with \: 10 \\ \\ \bold{0.9 = 0.9 \times \frac{10}{10} } \\ \\ \bold{0.9 = \frac{9}{10} } \\ \\ \color{blue}\large{ \bold{1 \: .\: .\: \frac{9}{10} }}$

$(2) \: \: \displaystyle{1.0001} \\ \\ \bold{no. \: of \: numbers \: after \: decimal \: point = 4} \\ \\ multiply \: and \: divide \: with \: 10000 \\ \\ \bold{1.0001 = 1.0001 \times \frac{10000}{10000} } \\ \\ \bold{1.0001 = \frac{10001}{10000} } \\ \\ \color{blue}\large{ \bold{2 \: .\: .\: \frac{10001}{10000} }}$

$(3) \: \: \displaystyle{0. \overline{621}} \\ \\ \bold{let \: \: x = 0. \overline{621}..... < 1 > } \\ \\ \bold{1000x = 621. \overline{621}..... < 2 > } \\ \\ < 2 > - < 1 > \\ \\ \bold{1000x - x = 621. \overline{621} - 0. \overline{621}} \\ \\ \bold{999x = 621} \\ \\ \bold{x = \frac{621}{999} } \\ \\ \bold{x = \frac{3 \times 207}{3 \times 333} } \\ \\ \bold{x = \frac{207}{333} } \\ \\ \large \color{blue} \bold{0. \overline{621} = \frac{207}{333} }$

$(4) \: \: \displaystyle{0.4 \overline{7}} \\ \\ \bold{let \: x =0.4 \overline{7}} \\ \\ \bold{100x = 47. \overline{7}..... < 3 > } \\ \\ \bold{10x = 4 .\overline{7}..... < 4 > } \\ \\ < 3 > - < 4 > \\ \\ \bold{100x - 10x = 4 7.\overline{7} - 4 .\overline{7}} \\ \\ \bold{90x = 43} \\ \\ \bold{x = \frac{43}{90}} \\ \\ \large \color{blue} \bold{0 .4\overline{7} = \frac{43}{90} }$

$(5) \: \: 0 .\overline{47} \\ \\ \bold{let \: x = 0.\overline{47}..... < 5 > } \\ \\ \bold{100x = 47 .\overline{47}..... < 6 > } \\ \\ < 6 > - < 5 > \\ \\ \bold{100x - x = 47 .\overline{47} - 0 .\overline{47}} \\ \\ \bold{99x = 47} \\ \\ \bold{x = \frac{47}{99}} \\ \\ \large \color{blue} \bold{0 .\overline{47} = \frac{47}{99}}$

$<marquee> HOPE IT HELPS$

Answer 2

⠀ ⠀ ⠀ ⠀ ⠀⠀${\huge {\underline{\underline {\frak {\red {Answer}}}}}}$

Here in the given question, we have to express the given values in the form of $\dfrac{p}{q}$

⠀ ⠀ ⠀ ❇0.90

The values are after two places of decimal.

So in fraction,

= ${\pink{\boxed{\bf{\orange{\underline{ \dfrac{90}{100}}}}}}}$

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⠀ ⠀ ⠀ ❇1.0001

The question the values are after four places of decimal.

So in fraction,

= ${\pink{\boxed{\bf{\orange{\underline{ \dfrac{10001}{10000}}}}}}}$

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⠀ ⠀ ⠀ ❇$0.\overline{621}$

Let x = $0.\overline{621}$ ⠀ ⠀- (1)

On multiplying equation (1) by 100

=${\bf { 1000x =621.621....}}$ ⠀ ⠀- (2)

Now, equation(2) - equation(1)

=${\bf { 1000x =621.621....}}$ ⠀ ⠀

=${\bf { (-)x =(-)0.621....}}$ ⠀

=${\bf { 999x =621.00....}}$ ⠀

=${\pink{\boxed{\bf{\orange{\underline { x =\dfrac{621}{999}}}}}}}$

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⠀ ⠀ ⠀ ❇ $0.4\overline{7}$

Let x = $04.\overline{7}$ ⠀ ⠀- (1)

On multiplying equation (1) by 10

=${\bf { 10x =4.7....}}$ ⠀ ⠀

=${\bf { 10x =4.7777...}}$ ⠀ ⠀- (2)

On multiplying equation (2) by 10

=${\bf { 100x =47.7777...}}$ ⠀ ⠀- (3)

Now, equation(3) - equation(2)

=${\bf { 100x =47.7777....}}$ ⠀ ⠀

=${\bf { (-)10x =(-)4.7777....}}$ ⠀

=${\bf { 90x =43.0000....}}$ ⠀

=${\pink{\boxed{\bf{\orange{\underline { x =\dfrac{43}{90}}}}}}}$

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⠀ ⠀ ⠀ ❇ $0.4\overline{7}$

Let x = $04.\overline{7}$ ⠀ ⠀- (1)

On multiplying equation (1) by 10

=${\bf { 10x =4.7....}}$ ⠀ ⠀

=${\bf { 10x =4.7777...}}$ ⠀ ⠀- (2)

On multiplying equation (2) by 10

=${\bf { 100x =47.7777...}}$ ⠀ ⠀- (3)

Now, equation(3) - equation(2)

=${\bf { 100x =47.7777....}}$ ⠀ ⠀

=${\bf { (-)10x =(-)4.7777....}}$ ⠀

=${\bf { 90x =43.0000....}}$ ⠀

=${\pink{\boxed{\bf{\orange{\underline { x =\dfrac{43}{90}}}}}}}$

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