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Five friends live on the same street. Their houses are at points A, B, C, D, and E, with the distances shown.

 

 

 

The five friends decide to meet at point P so that the total walking distance for all five friends is minimized. What is AP?

 Sep 26, 2021
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If everybody meets at 

A, the total distance walked by A is 0, by B is 5, by C is 9, by D is 11, by E is 14, total 39  

B, the total distance walked by A is 5, by B is 0, by C is 4, by D is 6, by E is 9, total 24  

C, the total distance walked by A is 9, by B is 4, by C is 0, by D is 2, by E is 5, total 20  

D, the total distance walked by A is 11, by B is 6, by C is 2, by D is 0, by E is 3, total 22  

E, the total distance walked by A is 14, by B is 9, by C is 5, by D is 3, by E is 0, total 31 

 

Let's try to improve on the 20 total  

 

If everybody meets halfway between  

B and C, the total distance walked by A is 7, by B is 2, by C is 2, by D is 4, by E is 7, total 22  

C and D, the total distance walked by A is 10, by B is 5, by C is 1, by D is 1, by E is 4, total 21  

 

It appears that if you move left of C, the total distance increases, 

and likewise, if you move right of C, the total distance increases.  

 

I don't know if this is the right way to find the answer, but I'd put my money on P is at the home of C  

Therefore A to P is 9      I'm not going to highlight this in red because I'm not relatively confident.  

.

 Sep 26, 2021

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