Question

Find a rational number exactly halfway between:

${\displaystyle \frac{5}{-13}}$ and ${\displaystyle \frac{-7}{9}}$

Answer 1

We know that, a rational number, which is halfway between two rational number i.e.a and b = ${\displaystyle \frac{a+b}{2}}$

Given rational numbers are ${\displaystyle \frac{5}{-13}}$ and ${\displaystyle \frac{-7}{9}}$

Here, a = ${\displaystyle -\frac{5}{13}}$ and b = ${\displaystyle -\frac{7}{9}}$

∴ ${\displaystyle \frac{a+b}{2}=\frac{\frac{-5}{13}+(-\frac{7}{9})}{2}}$

= ${\displaystyle \frac{\frac{-5}{13}-\frac{7}{9}}{2}}$

= ${\displaystyle \frac{\frac{-5\times 9}{13\times 9}-\frac{7\times 13}{9\times 13}}{2}}$

= ${\displaystyle \frac{\frac{-45}{177}-\frac{91}{177}}{2}}$

= ${\displaystyle \frac{\frac{-45-91}{177}}{2}}$

= ${\displaystyle \frac{-136}{117\times 2}}$

= ${\displaystyle \frac{-136}{234}}$

Hence, the exactly of halfway between ${\displaystyle \frac{5}{-13}}$ and ${\displaystyle \frac{-7}{9}}$ is ${\displaystyle -\frac{136}{234}}$.