Question

How to find two rational numbers between two rational numbers?

Answer 1

Rational nos. between two rational no.s
1st method - Take average Eg. - $\frac{1}{2}\&1$
$Averageof\frac{1}{2}\&1=\frac{\frac{1}{2}+1}{2}=\frac{3}{4}$
$\therefore \frac{3}{4}willliebetween\frac{1}{2}\&1$
$\frac{1}{2},\frac{3}{4},1$
Again take the average of these two $\frac{\frac{1}{2}+\frac{3}{4}}{2}=\frac{5}{8}$
$\therefore \frac{1}{2},\frac{5}{8},\frac{3}{4},1$
Again take angle here & you can continue.
2. Given two fractions $\frac{a}{b}\&\frac{c}{d}$
One rational no. lying between them is $\frac{a+c}{b+d}$
Eg - $\frac{2}{3},\frac{4}{9}\to \frac{2+4}{3+9}=\frac{6}{12}$ will lie between them.
$\frac{2}{3},\frac{6}{12},\frac{4}{9}$
Apply the same method for these too. You will get one more number.
3. By taking LCM of denominator
Eg - $\frac{1}{12}\&\frac{4}{12}$
$\frac{1\times 100}{12\times 100}\frac{4\times 100}{12\times 100}$
$\frac{100}{1200}\frac{400}{1200}$
$So,\frac{101}{1200},\frac{102}{1200},\frac{103}{1200}\cdots \frac{398}{1200},\frac{399}{1200}$ all lie b/w them