between a sixth and a twelfth

Answer: $\frac19$

Finding the distance 

The distance between $\frac16$ and $\frac1{12}$ is $\frac16-\frac1{12}=\frac{2}{12}-\frac1{12}=\frac1{12}$

Length of each of the 3 sections is equal to $\frac13$ of $\frac1{12}=\frac1{36}$

Image

So the number indicated is $\frac1{12}+\frac1{36}=\frac3{36}+\frac1{36}=\frac4{36}=\frac19$

Using a weighted average

To find the point half way between $\frac16$ and $\frac1{12}$, we would add $\frac16$ and $\frac1{12}$ and divide by $2.$

We want the point that is twice as close to $\frac1{12}$ as it is to $\frac16$ - so give twice as much importance to $\frac1{12}$ as to $\frac16.$ This is called a weighted average.

$$\begin{split}\left(\tfrac16+2\times\tfrac1{12}\right)\div3&=\left(\tfrac16+\tfrac16\right)\div3\\

&=\tfrac26\div3\\&=\tfrac13\div3\\&=\tfrac19\end{split}$$