Question

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Answer 1

QUESTION :

Express in the form p/q form.

SOLUTION :

(i) $\bf 0.\bar{9}$

Let x = 0.9999.....   (I)

multiplying 10 to both the sides in equation (I), we get,

10x = 9.9999......    (II)

Subtracting equation (I) from (II), we get,

10x = 9.999.....

-   x = 0.999.....

______________

 9x = 9.000.....

______________

 

⇒ 9x = 9

⇒ x = 9/9

⇒ x = 1

_________________________

(ii) $\bf 0.\overline{001}$

Let x = 0.001001001...  (I)

multiplying 1000 to both the sides in equation (I), we get,

1000x = 1.001001001... (II)

Subtracting equation (I) from (II) we get,

1000x = 1.001001001.....

-       x = 0.001001001.....

______________

 999x = 1

______________

⇒ x = 1/999

________________________

(iii) $\bf 0.3\overline{23}$

Let x = 0.323232323.....   (I)

multiplying 10 to both the sides in equation (I), we get

10x = 3.23232323.....   (II)

multiplying 100 to both the sides in equation (II),we get

1000x = 323.232323....  (III)

Subtracting equation (II) from (III) we get

1000x = 323.232323.....

-   10x =      3.232323.....

______________

990x = 320

______________

 

⇒ x = 320/990

⇒ x = 32/99

Answer 2

To express in $\frac{p}{q}$ form.

(i) $0\overline{.9}$

Let $0\overline{.9}$ be equal to z.

                                              z = 0.999... → (1)

(1)*10                                   10z = 9.999... → (2)

(2) - (1)                                  9z = 9

                                              z = 9/9

∴ $0\overline{.9}$ = 9/9

(ii) $0\overline{.001}$

Let $0\overline{.001}$ be equal to p.

                                             p = 0.001001... → (1)

(1)*1000                         1000p = 1.001001... → (2)

(2) - (1)                             999p = 1

                                             p = 1/999

∴ $0\overline{.001}$ = 1/999

(iii) $0.3\overline{23}$

Let $0.3\overline{23}$ be equal to y.

                                            y = 0.32323... → (1)

(1)*10                                 10y = 3.2323... → (2)

(2)*100                         1000y = 323.2323... → (3)

(3) - (2)                           990y = 320

                                            y = 320/990 = 32/99

∴ $0.3\overline{23}$ = 32/99