A rational number between \(\frac { 1 }{ 2 } \) and \(\frac { 2 }{ 3 } \) = \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 2 }{ 3 } \right) =\frac { 1 }{ 2 } \left( \frac { 3+4 }{ 6 } \right) =\frac { 1 }{ 2 } \left( \frac { 7 }{ 6 } \right) =\frac { 7 }{ 12 } \)
A rational number between \(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 12 } \) = \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 7 }{ 12 } \right) =\frac { 1 }{ 2 } \left( \frac { 6+7 }{ 12 } \right) =\frac { 1 }{ 2 } \left( \frac { 13 }{ 12 } \right) =\frac { 13 }{ 24 } \)
Hence two rational numbers between \(\frac { 1 }{ 2 } \) and \(\frac { 2 }{ 3 } \) are \(\frac { 7 }{ 12 } \) and \(\frac { 13 }{ 24 } \) (of course, there are many more!)
There is an interesting result that could help you to write instantly rational numbers between any two given rational numbers.

Two rational numbers between the given two irrational numbers are 0.5152 and 0.5352

Yes, it is possible to write.

(i) \(3\over 4\) = 0.75

(ii) \(5\over 8\) = 0.625

(iii) \(9\over 25\) = 0.36

(i) Given that a = 2  + \(\sqrt { 3 } \) , b = 2 - \(\sqrt { 2 } \) 
x = a + b  = ( 2 + \(\sqrt { 3 } \)) + (2 - \(\sqrt { 3 } \)) = 4 a rational number
y = a - b = (2 +  \(\sqrt { 3 } \) ) - (2 - \(\sqrt { 3 } \)) = 2\(\sqrt { 3 } \)  an irrational number
(ii) Given that a  = \(\sqrt { 2 } \) + 7 , b = \(\sqrt { 2 } \) - 7
x = a + b = (\(\sqrt { 2 } \) + 7) + (\(\sqrt { 2 } \) - 7) = 2\(\sqrt { 2 } \) an irrational number.
y = a -b  = (\(\sqrt { 2 } \) + 7 ) - (\(\sqrt { 2 } \) - 7) = 14 a rational number
(iii) Given that a = \(\sqrt { 75 } \) , b = \(\sqrt { 3 } \)
x = ab = \(\sqrt { 75 } \times \sqrt { 3 } =\sqrt { 75\times 3 } =\sqrt { 5\times 5\times 3\times 3 } =5\times 3=15\) a rational number
\(y=\frac { a }{ b } =\frac { \sqrt { 75 } }{ \sqrt { 3 } } =\sqrt { \frac { 75 }{ 3 } } =\sqrt { 25 } =5\), rational number
(iv) Given that a = \(\sqrt { 18 } \), b = \(\sqrt { 3 } \)
\(x=ab=\sqrt { 18 } \times \sqrt { 3 } =\sqrt { 18\times 3 } =\sqrt { 6\times 3\times 3 } =3\sqrt { 6 } ,\) an irrational number,
\(y=\frac { a }{ b } =\frac { \sqrt { 18 } }{ \sqrt { 3 } } =\sqrt { \frac { 18 }{ 3 } } =\sqrt { 6 } \) an irrational number

(i) Let \(x=0.\overline { 3 } \)= 0.3333   (1)
(Here period of decimal is 1, multiply equation (1) by 10)
10x = 3.3333....    (2)
(2) - (1) : 9x = 3  or \(x=\frac { 1 }{ 3 } \)
(ii) Let \(x=2.\overline { 124 } \) = 2.124124124…            (1)
(Here period of decimal is 3, multiply equation (1) by 1000)
1000 x = 2124.124124124… (2)
(2)–(1): 999 x = 2122  \(x=\frac { 2122 }{ 999 } \)
(iii) Let x = \(0.4\overline { 5 } \) = 0.45555…              (1)
(Here the repeating decimal digit is 5, which is the second digit after the decimal point, multiply equation (1) by 10)
10 x = 4.5555…  (2)
(Now period of decimal is 1, multiply equation (2) by 10)
100 x = 45.5555…              (3)
(3) – (2): 90 x = 41 or \(x=\frac { 41 }{ 90 } \)
(iv) Let x = \(0.5\overline { 68 } \)=  0.5686868…  (1)
(Here the repeating decimal digit is 68, which is the second digit after the decimal point, so multiply equation (1) by 10)
10 x = 5.686868…                (2)
(Now period of decimal is 2, multiply equation (2) by 100)
1000 x = 568.686868… (3)
3) – (2): 990 x = 563 or \(x=\frac { 563 }{ 990 } \)

(i) 0.35 = \(\frac { 35 }{ 100 } =\frac { 7 }{ 20 } \)
(ii) 2.176 = \(\frac { 2176 }{ 1000 } =\frac { 272 }{ 125 } \)
(iii) -0.0028 =\(\frac { -28 }{ 10000 } =\frac { -7 }{ 2500 } \)

We know that \(\frac { 1 }{ 3 } =0.\overline { 3 } \)
Therefore \(\frac { 1 }{ 27 } =\frac { 1 }{ 9 } \times \frac { 1 }{ 3 } \times \frac { 1 }{ 9 } \times 0 333... = 0.037037..\ =\ 0.\overline { 037 } \)
Also, \(\frac { 59 }{ 27 } =2\frac { 5 }{ 27 } =2+\frac { 5 }{ 27 } =2+\left( 5\times \frac { 1 }{ 27 } \right) \)
\(=2+(5\times 0.\overline { 037 } )=2+(5\times 0.037037037..)\)
\(=2+0.185185...=2.185185..\quad =2.\overline { 185 } \)

(i)  \(\frac { 1 }{ 4 } \) and \(1\over 5\)
Take LCM = 200
\( \frac{1}{4} \times \frac{50}{50}=\frac{50}{200} \)
\( \frac{1}{5} \times \frac{40}{40}=\frac{40}{200}\)
The numbers are \(\frac{41}{200}, \frac{42}{200} \ldots \ldots \ldots \frac{49}{200}\)
(ii) 0.1 and 0.11
Change decimal into fraction \(0.1=\frac{1}{10} \text { and } 0.11=\frac{11}{100}\)
Take LCM = 100
\(\frac{1}{10} \times \frac{10}{10}=\frac{10}{100}, \frac{11}{100} \times \frac{1}{1}=\frac{11}{100}\)
Take LCM = 600
\(\frac{10}{100} \times \frac{6}{6}=\frac{60}{600} \text { and } \frac{11}{100} \times \frac{6}{6}=\frac{66}{600}\)
A rationalnumberbetween and \(\frac{60}{600} \text { and } \frac{66}{600}\)
are \(\frac{61}{600}, \frac{62}{600}, \frac{63}{600}, \frac{64}{600}, \frac{65}{600}\)
(iii) -1 and -2
Take LCM = 10 
\(\frac{-1}{1} \text { and } \frac{-2}{1}\)
\(\frac{-1}{1} \times \frac{10}{10}=\frac{-10}{10} \text { and } \frac{-2}{1} \times \frac{10}{10}=\frac{-20}{10}\)
The Numbers are \(\frac{-11}{10}, \frac{-12}{10} \ldots \ldots \ldots \ldots . \frac{-19}{10}\)

6.34 \(\times\) 10⁴