(i) Let x =

Now, x = 0.4̅ = 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…

9x = 4

x =

Hence, =

(ii) Let x =

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 =

Hence, =

(iii) Now x =

= 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 =

(iv) Now x =

= 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 = =

(v) Now x =

= 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 =

= 4.7777…. (i)

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

10x = 47.7777…. (ii)

10x – x = 47.7777…. – 4.7777….

9x = 43

x =

(vii) Now, x =

= 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….

90x = 43

x =