BdMO National 2013: Primary 7

BdMO
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BdMO National 2013: Primary 7

Arefin, Farhad, Himu, Mahdi, Rachi, Sadia and Tusher are seven friends who live in Gulshan by the side of same linear road. Distances of others home from Tusher home are given below. Tusher’s home is at the starting point of that road. They want to meet at the same place on that road every evening for gossiping. Find a place on that road so that the sum of distances of that place from everyone’s home is minimum. Write down your logic and distance of that place from Tusher’s home.
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TANISHA
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Re: BdMO National 2013: Primary 7

ans of this

*Mahi*
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Re: BdMO National 2013: Primary 7

The answer should be $5$, i.e at Mahdi's home.
But I am surprised they asked for the logic behind this in a primary category question. This basically is the question $\text{why median minimizes the absolute deviation}$, and a rigorous proof (which I saw) of the general form of this used induction.
I think this can be showed like this- first find the distance from everyone's house at $5$, then show what happens when you increase or decrease that value.
Use $L^AT_EX$, It makes our work a lot easier!