Assume that we have a set of real positive numbers:
a1,a2,a3,...,a9,...,a17.
Of course, a9 is the middle number.
The average of all 17 is
10.9=17∑i=117ai↔∑i=117ai=17⋅10.9=185.3.The average of the first nine numbers is
10.5=9∑i=19ai=9∑i=18ai+a9.The average of the last nine numbers is 1/9th of the sum of all 17 minus the sum of the first 8:
11.4=9(∑i=117ai−∑i=18ai).
Look at the last two equations: we have all we need. The sum of all 17 is 185.3, now put
∑i=18ai=x,a9=y.The two equations become a system with two undefined variables:
11.4=9(185.3−x), 10.5=9x+y. The solution for this simple system is
x=82.7,y=11.8. Thus, a9=y=11.8.
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