# between a sixth and a twelfth

The space on a number line between a sixth and a twelfth is split into 3 equal parts. Find the number indicated.

On this number line, the space between $\frac16$ and $\frac1{12}$ is split into 3 equal parts.

What number is indicated by the arrow?

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*This problem is taken from the World Mathematics Championships*

**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 $\frac1{12}=\frac1{36}$

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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}$$