Discussion

Explanation:

Let x =  $0.21\overline{7}$

Multiplying both sides with 10² (there are 2 non repeating decimal digits i.e., 2 and 1)

⇒ 100x = $21.\overline{7}$   ...(1)

Now, again multiply with 10¹ (one repeating digit i.e., 7)

⇒ 1000x = $217.\overline{7}$   ...(1)

(2) - (1)

⇒ 1000x - 100x = $217.\overline{7}$ - $21.\overline{7}$ = 217 - 21 = 196

∴ x = $\frac{196}{900}$

Alternately,
The shortcut way of writing a recurring decimal in fractional form is

Numerator = (Write the whole number without decimal with recurring set of digits appearing once) - (Write the whole number without decimal and recurring set of digits)
Numerator = 217 - 21 = 196

Denominator = 9..(Number of 9's is same as the number of recurring digits) 0...(Number of 0's is same as the number of non recurring digits after decimal)
Denominator = 900 (There is one recurring digits i.e., 7 and two non recurring digits i.e., 2 and 1)

∴ x = $\frac{196}{900}$

Hence, option (b).

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