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 2:3, while Quincy notices that, from where he sits, he divides the entire branch in the ratio 3:4. 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, $AP = \frac{2}{5} AB$ and $AQ = \frac{3}{7} AB.$ $\frac{2}{5}$ is less than $\frac{3}{7}$, which explains the relative positions of the points in the diagram (which is not drawn to scale).

Continuing:

$\frac{3}{7} AB - \frac{2}{5} AB = 2,$

$35 \cdot (\frac{3}{7} AB - \frac{2}{5} AB) = 35 \cdot 2 = 70,$

$15 \cdot AB - 14 \cdot AB = 70,$

$AB = 70.$