.
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
