Question

What is a rational number halfway between 6/7 and 5/6

Answer 1

Well hmmm let's see hmm
our denominators are 7 and 6

so.. their LCD or a GCF for that matter..  is just 7 * 6 or 42

so let's make both fractons the same denominator,
that is, the LCD of 42

for the 7, we need to multiply by 6
we do that to the denominator, we have to
also do it for its numerator

for the 6, we multiply by 7
we do that for the denominator, we need to
also do it for its numerator

so, we end up with 
$\bf \begin{cases} \cfrac{6}{7}\cdot \cfrac{6}{6}\implies \cfrac{6\cdot 6}{7\cdot 6}\implies \cfrac{36}{42} \\ \quad \\ \cfrac{5}{6}\cdot \cfrac{7}{7}\implies \cfrac{5\cdot 7}{6\cdot 7}\implies \cfrac{35}{42} \end{cases} \\ \quad \\ \textit{so.. hmmm they're next to one another, kinda}\\\\ \textit{let us use }\frac{35}{42}+0.5\textit{ or half a bit more}\\\\ \textit{pass 35 and below 36} \\ \quad \\ \cfrac{35}{42}+0.5\implies \cfrac{35}{42}+\cfrac{1}{2}$

so.. that rational is, passed 35/42 and  before 36/42
so. is between, half-way really, since we used 1/2

now, we could have use some other fraction of 1,
say 1/25 or 1/7 or 3/23  and those would have also worked,
because they're passed 35/42 and before 36/42