Recurring Decimals | Algebra - Number Theory

Answer: Option D

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Explanation :

Let x =  $0.777\overline{7}$   ...(1)

Multiplying both sides with 10

⇒ 10x = $7.777\overline{7}$   ...(2)

(2) - (1)

⇒ 10x - x = $7.777\overline{7}$ - $0.777\overline{7}$ = 7

∴ x = $\frac{7}{9}$

Alternately,
A number of the form $a.\overline{b}$ can be written as $\frac{ab-a}{9}$

Here, a = 0 and b = 7, hence the fractional form of the given number is $\frac{7-0}{9}$ = $\frac{7}{9}$

Hence, option (d).

Answer: Option B

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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).

Answer: Option A

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Explanation :

Let x = $21.3\overline{76}$

Muliplyting both sides with 10¹ (There is one non recurring digit after decimal)

⇒ 10x = $213.\overline{76}$   ...(1)

Muliplyting both sides with 10² (There are two recurring digits after decimal)

⇒ 1000x = $21376.\overline{76}$   ...(2)

(2) - (1)

⇒ 1000x -10x = $21376.\overline{76}$ - $213.\overline{76}$ = 21376 - 213 = 21163

∴ x = $\frac{21163}{990}$

Alternately,
Numerator of the fraction = 21376 - 213 = 21163

Denominator = 990

∴ x = $\frac{21163}{990}$

Hence, option (a).

Answer: Option C

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Explanation :

Let x = $0.222\overline{35}$

Numerator of the fraction = 22235 - 222 = 22013

Denominator = 99000

∴ x = $\frac{22013}{99000}$

Hence, option (c).