Draw diagrams of both options. Which distances are the same and which are different?
The second option, leave your lunch at home and eat at D.
Practice makes perfect
We have been asked to determine which distance is shorter, going back home to get your lunch or leaving the lunch at home and having lunch at D. We will determine each distance separately and then compare them.
Going Back Home
If you decide to go back home to get your lunch, you travel from A to M and then back again.
Then you have to travel to work, going the full length of AB, and then home again, also the length of AB.
The total distance you have to travel that day would then be the sum of these four lengths.
AM+AM+AB+AB=2AM+2AB
Having Lunch at D
If you instead choose to go to work and eat at D, you first travel from A to B.
You then go from B to D and back to B again after your lunch break is over.
Last, you travel home, from B to A.
Your total distance traveled will be the sum of these four lengths.
AB+BD+BD+AB=2AB+2BD
Comparing the distances
Which distance is the shortest? In either alternative you travel the distance AB twice. This means that everything comes down to if AM is shorter than BD or vice versa.
Since M is the midpoint, AM and MB have the same length. However, D is between M and B, so the distance between B and D is shorter than the distance between A and M.
AM > DB
Therefore, you should leave your lunch at home and eat at D.
