# Without actually

(i). We have Therefore, it clearly is non-terminating decimal. (since, it is recurring or repeating).

Justification:

First, converting it in the form p/q, where p and q are integers and q ≠ 0.

So, let .

That is, x = 0.55555… …(i)

Multiply 10 by equation (i),

10 × x = 10 × 0.55555…

10x = 5.55555… …(ii)

Subtracting equations (i) from (ii), we get

10x – x = 5.5555… - 0.5555…

9x = 5

x = 5/9

Now, by factorizing the denominator, we get

9 = 3 × 3 = 32

Denominator is not in the form of 2n × 5n. is non-terminating.

(ii). We have 11/18.

Now, without actual division, we got to factorize the denominator of the rational number.

By factorizing the denominator, we get

18 = 2 × 3 × 3

18 = 2 × 32

Denominator is not in the form of 2n × 5n.

11/18 is non-terminating.

(iii). We have 13/20.

Now, without actual division, we got to factorize the denominator of the rational number.

By factorizing the denominator, we get

20 = 2 × 2 × 5

20 = 22 × 51

Denominator is in the form of 2n × 5n (i.e., 22 × 51)

13/20 is terminating decimal.

(iv). We have 41/42.

Now, without actual division, we got to factorize the denominator of the rational number.

By factorizing the denominator, we get

42 = 2 × 3 × 7

Denominator is not in the form of 2n × 5n.

41/42 is non-terminating.

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