Let x = 0.00010001 ............. \(\rightarrow\) (1)
10000x = 1.00010001 .......... \(\rightarrow\) (2)
(2) - (1) \(\Rightarrow\) 10000x - x = 1.00010001 ......... (-)
                                   0.00010001 .........
                                 ____________
                     9999x = 1
      \(x=\frac{1}{9999}\)      

Let x = 17.2151515 ......  (1)
Here period of decimal is 3, multiply equation (1) by 1000
1000 x = 17215.151515
(2) - (1)
999x = 17197.936
\( x=\frac{17197.936}{999} \\ x=\frac{17197936}{999000} \\ x=\frac{5681}{330} \)

Let x = \(3.1\overline { 7 } \) = 3.1777 .......  (1)
Here period of decimal is 2, multiply equation (1) by 100
10x = 31.777 .........  (2)
(2)-(1)

\( =\frac{143}{45} \)

Let \(2.\overline { 327 } \) = 2.327327327 ..... \(\rightarrow\) (1)
Here period of decimal is 3, multiply equation (1) by 1000
1000 x = 2327.327327 ..... \(\rightarrow\) (2)
(2) - (1) \(\Rightarrow\) 1000x - x = 999x = 2325
 \(x=\frac{2325}{999}\) (or) \(775\over 333\)

Let x = 0.242424..... \(\rightarrow\) (1)
100 x = 24.2424 ...... \(\rightarrow\) (2)
(2) - (1) \(\Rightarrow\) 100x - x = 24.2424 
99x = 24
x = \(24\over 99\) (or) \(8\over 33\)
x = \(8\over 33\)

Relational between \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 5 } \) = \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
= \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
Rational numbers between\(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \)
=\(\frac { 1 }{ 2 } \left( \frac { 10+7 }{ 20 } \right) \)
=\(\frac { 17 }{ 40 } \)
Rational number between \(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \) =\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } \times \frac { 17 }{ 40 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 20+17 }{ 40 } \right) \)
=\(\frac { 37 }{ 80 } \)
Thus the rational numbers are \(\frac { 7 }{ 20 } ,\frac { 17 }{ 40 } and\frac { 37 }{ 80 } \)

Let us convert the given rational numbers having the same denominators.
L.C.M of 8 and 6 is 24.
\(\frac { 5 }{ 8 } \times \frac { 5\times 3 }{ 8\times 3 } =\frac { 15 }{ 24 } \)
\(\frac { 5 }{ 8 } \times \frac { 5\times 3 }{ 8\times 3 } =\frac { 15 }{ 24 } \)
Now the rational numbers between \(-\frac { 202 }{ 24 } and\frac { 15 }{ 24 } are-\frac { 19 }{ 24 } ,-\frac { 18 }{ 24 } ,-\frac { 17 }{ 24 } ,....,\frac { 0 }{ 24 } ,\frac { 1 }{ 24 } ,\frac { 22 }{ 24 } ,...,\frac { 14 }{ 24 } \)
we can take any seven of them \(\frac { 1 }{ 24 } ,\frac { 2 }{ 24 } ,\frac { 3 }{ 24 } ,\frac { 4 }{ 24 } ,\frac { 5 }{ 24 } ,\frac { 6 }{ 24 } ,\frac { 7 }{ 24 } \)

Let x = 0.474747 ....... \(\rightarrow\) (1)
100x = 47.4747 ......... \(\rightarrow\) (2)

 \(x={47\over 99}\)
\(\therefore 0.\overline {47}={47\over 99}\)