Find five rational numbers between.
(i) $ \frac{2}{3} $ and $ \frac{4}{5} $
(ii) $ \frac{-3}{2} $ and $ \frac{5}{3} $
(iii) $ \frac{1}{4} $ and $ \frac{1}{2} $.

To find: 

We need to find five rational numbers between.

(i) \( \frac{2}{3} \) and \( \frac{4}{5} \)
(ii) \( \frac{-3}{2} \) and \( \frac{5}{3} \)
(iii) \( \frac{1}{4} \) and \( \frac{1}{2} \).

Solution: 

To solve this question, first, we need to convert them into like fractions.

(i) LCM of denominators (3 and 5) is 15. Now we have to change the fractions in such a way that denominators become 15.

To convert into like fractions we will multiply the numerator and denominator of $\frac{2}{3}$ with 5.

$\frac{2}{3} \ =\ \frac{2}{3}\ \times\ \frac{5}{5}\ =\ \frac{10}{15}$

We will multiply the numerator and denominator of $\frac{4}{5}$ with 3.

$\frac{4}{5}\ =\ \frac{4}{5}\ \times\ \frac{3}{3}\ =\ \frac{12}{15}$

Now, our numbers are $\frac{10}{15}$ and $\frac{12}{15}$.

We can find 5 rational numbers between $\frac{10}{15}$ and $\frac{12}{15}$ by multiplying them with ($5+1=6$).

$\frac{10}{15}\ \times\ \frac{6}{6}\ =\ \frac{60}{90}$

And,

$\frac{12}{15}\ \times\ \frac{6}{6}\ =\ \frac{72}{90}$

Therefore,

Five rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$ are: 

$\frac{61}{90},\ \frac{62}{90},\ \frac{63}{90},\ \frac{64}{90}\ and\ \frac{65}{90}$. 

 (ii) LCM of denominators (2 and 3) is 6. Now we have to change the fractions in such a way that denominators become 6.

To convert into like fractions we will multiply the numerator and denominator of $\frac{-3}{2}$ with 3.

$\frac{-3}{2} \ =\ \frac{-3}{2}\ \times\ \frac{3}{3}\ =\ \frac{-9}{6}$

We will multiply the numerator and denominator of $\frac{5}{3}$ with 2.

$\frac{5}{3}\ =\ \frac{5}{3}\ \times\ \frac{2}{2}\ =\ \frac{10}{6}$

Now, our numbers are $\frac{-9}{6}$ and $\frac{10}{6}$.

We can find 5 rational numbers between $\frac{-9}{6}$ and $\frac{10}{6}$.

Therefore,

Five rational numbers between $\frac{-3}{2}$ and $\frac{5}{3}$ are: 

$\frac{-8}{6},\ \frac{-7}{6},\ \frac{-6}{6}=-1,\ \frac{-5}{6}\ and\ \frac{-4}{6}$. 

(iii) LCM of denominators (4 and 2) is 4. Now we have to change the fractions in such a way that denominators become 4.

To convert into like fractions we will multiply the numerator and denominator of $\frac{1}{2}$ with 2.

$\frac{1}{2} \ =\ \frac{1}{2}\ \times\ \frac{2}{2}\ =\ \frac{2}{4}$

Now, our numbers are $\frac{1}{4}$ and $\frac{2}{4}$.

We can find 5 rational numbers between $\frac{1}{4}$ and $\frac{2}{4}$ by multiplying them with ($5+1=6$).

$\frac{1}{4}\ \times\ \frac{6}{6}\ =\ \frac{6}{24}$

And,

$\frac{2}{4}\ \times\ \frac{6}{6}\ =\ \frac{12}{24}$

Therefore,

Five rational numbers between $\frac{1}{4}$ and $\frac{1}{2}$ are: 

$\frac{7}{24},\ \frac{8}{24},\ \frac{9}{24},\ \frac{10}{24}\ and\ \frac{11}{24}$.