(i) Let x =\(0.\overline4\)

Now, x = \(0.\overline4\) = 0.444… (1)

Multiplying both sides of equation (1) by 10, we get,

10x = 4.444… (2)

Subtracting equation (1) by (2)

10x – x = 4.444… - 0.444…

\(9\times = 4\)

x = \(\frac{4}{9}\)

Hence, \(0.\overline4 = \frac{4}{9}\)

(ii) Let x = \(0.\overline{37}\)

Now, x = 0.3737…. (1) 

Multiplying equation (1) by 10 

10x = 3.737…. (2) 

Multiplying equation (2) by 10 

100x = 37.3737…. (3) 

100x – x = 37 

99x = 37

\(x = \frac{37}{99}\)

Hence, \(0.\overline{37}\) \(= \frac{37}{99}\)

(iii) Now x \(= 0.\overline{54}\)

= 0.5454… (i) 

Multiplying both sides of equation (i) by 100, we get 

100x = 54.5454…. (ii) 

Subtracting (i) by (ii), we get 

100x – x = 54.5454…. – 0.5454…. 

99x = 54

\(x = \frac{54}{99}\)

(iv) Now x \(= 0\overline{621}\)

= 0.621621…. (i) 

Multiplying both sides by 1000, we get 

1000x = 621.621621…. (ii) 

Subtracting (i) by (ii), we get 

1000x – x = 621.621621…. – 0.621621…. 

999x = 621

\(x = \frac{621}{999}\) \(= \frac{23}{37}\)

(v) Now x = \(125.\overline3\)

 = 125.3333…. (i)

Multiplying both sides of equation (i) by 10, we get 

10x = 1253.3333…. (ii) 

Subtracting (i) by (ii), we get 

10x – x = 1253.3333…. – 125.3333…. 

9x = 1128 

x = 1128 / 9 = 376/3 

(vi) Now x \(= 4.\overline7\)

= 4.7777…. (i) 

Multiplying both sides of equation (i) by 10, we get 

10x = 47.7777…. (ii) 

10x – x = 47.7777…. – 4.7777….

\(9\times = 43\)

\(x = \frac{43}{9}\)

(vii) Now, x = \(0.4\overline7\)

 = 0.47777…. 

Multiplying both sides by 10, we get 

10x = 4.7777…. (i) 

Multiplying both sides of equation (i) by 10, we get 

100x = 47.7777…. (ii) 

Subtracting (i) from (ii), we get 

100x – 10x = 47.7777…. – 4.7777….

\(90\times = 43\)

x \(= \frac{43}{90}\)