Question

Insert six rational numbers between $\frac{5}{6}$ and $\frac{8}{9}$.

Answer 1

LCM of denominators 6 and 9 is 18 Make, denominator of each rational number equal to 18 i.e, the LCM
$\frac{5}{6}=\frac{5\times 3}{6\times 3}=\frac{15}{18}and\frac{8}{9}=\frac{8\times 2}{9\times 2}=\frac{16}{18}\text{Since, six rational numbers are required, multiply the numerator and denominator of each rational numberby 6+1=7}\therefore \frac{15}{18}=\frac{15\times 7}{18\times 7}=\frac{105}{126}and\frac{16}{18}=\frac{16\times 7}{18\times 7}=\frac{112}{126}\therefore \text{Required rational numbers between}\frac{5}{6}and\frac{8}{9}are=\frac{106}{126},\frac{107}{126},\frac{108}{126},\frac{109}{126},\frac{110}{126},\frac{111}{126}=\frac{53}{63},\frac{107}{126},\frac{6}{7},\frac{109}{126},\frac{55}{63},\frac{55}{63},\frac{37}{42}$