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Arithmetic Mean is calculated by adding all the values of the ratings together and dividing it by the number of values in the group. Therefore, if we represent each rating by #x_a, x_b, x_c, .......#, according to the problem statement:

Arithmetic Mean of the first #10# ratings #=(x_1+x_2+x_3+........+x_10)/10=75#, #color(red)(Equation - 1)#

Similarly:

Arithmetic Mean of the first #20# ratings would be #(x_1+x_2+x_3+......+x_20)/20=85#, #color(red)(Equation - 2)#

We have to consider the best case scenario of the ratings number #12# through #20# to be the highest, i.e. #100#.

If that occurs what would rating number #11# have to be to result in Arithmetic Mean of #85# for the first #20# ratings?

We can rewrite #color(red)(Equation - 2)# as:

#((x_1+x_2+x_3+......+x_10)/20)+((x_11+x_12+.......+x_20)/20)=85#

We need to calculate the first piece from #color(red)(Equation - 1)#:

#((x_1+x_2+x_3+......+x_10)/20)=((x_1+x_2+x_3+........+x_10)/10)(1/2)=75(1/2)=75/2#

We now substitute this into #color(red)(Equation - 2)#, and plug in #100# for #(x_12+.....+x_20)# to account for the best case scenario:

#75/2+((x_11+9(100))/20)=85#

#75/2+(x_11+900)/20=85#

#(750+x_11+900)/20=85#

#750+x_11+900=1700#

#x_11=50#