Q. 3

Express the following decimal expansions into rational numbers.i. ii. iii. iv. v. vi.

i. let

100x = 18.181818 … = 18 + 0.181818…

100x = 18 + x

99x = 18

ii. let

1000x = 427.427427 … = 18 + 0.427427427…

1000x = 427 + x

999x = 427

iii. let

10000x = 1.00010001 … = 1 + 0.00010001…

10000x = 1 + x

9999x = 1

iv. As,

Let x = 0.454545…

1 + x = 1 + 0.454545..

1+ x = 1.45454545

100 + 100x = 145.454545… = 145 + 0.454545…

100 + 100x = 145 + x

99x = 45

And

v. As,

Let x = 0.3333…

7 + x = 7 + 0.3333…

7 + x = 7.3333…

70 + 10x = 73.3333… = 73 + 0.3333…

70 + 10x = 73 + x

9x = 3

And

vi. let

1000x = 416.416416 … = 416 + 0.416416…

1000x = 416 + x

999x = 416

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