What common fraction is exactly half-way between $\frac{2}{3}$ and $\frac{4}{5}$?
One way to approach this problem is to find a common denominator for the two fractions and then average their numerators. The least common multiple of 3 and 5 is 15, so I can rewrite the fractions as $\frac{10}{15}$ and $\frac{12}{15}$. To find the average of their numerators, I can add them and divide by 2: $\frac{10 + 12}{2} = 11$. So the fraction that is exactly half-way between $\frac{2}{3}$ and $\frac{4}{5}$ is $\frac{11}{15}$.

