3050.12 – Bugs on a Branch

There are two bugs. Let’s call them Peter and Quincy. They are sitting on a tree branch (with endpoints A and B), and both of them are on the same half of the branch. Peter notices that, from where he sits, he divides the entire branch in the ratio of 2:32:3, while Quincy notices that, from where he sits, he divides the entire branch in the ratio 3:43:4. The distance between Peter and Quincy is 2 inches.

How long is the tree branch?

Solution

Call the points where Peter and Quincy are sitting P and Q. The figure below assumes P is to the left of Q but it might be the reverse. Let M be the midpoint.

Then, AP=25ABAP = \dfrac{2}{5} AB and AQ=37AB.AQ = \dfrac{3}{7} AB. 25\dfrac{2}{5} is less than 37\dfrac{3}{7}, which explains the relative positions of the points in the diagram (which is not drawn to scale).

Continuing:

37AB−25AB=235⋅(37AB−25AB)=35⋅2=7015⋅AB−14⋅AB=70AB=70\begin{aligned}\dfrac{3}{7} AB - \dfrac{2}{5} AB &= 2 \\[0.5em]35 \cdot (\dfrac{3}{7} AB - \dfrac{2}{5} AB) &= 35 \cdot 2 = 70 \\[0.5em]15 \cdot AB - 14 \cdot AB &= 70 \\AB &= 70\end{aligned}