Suppose there are n groups containing \({a_1},{a_2},{a_3}, \ldots .{a_n}\) items and averages of the  groups are \({x_1},{x_2},{x_3}, \ldots .{x_n}\), then the  average of all the items in all the group put together is known as weighted average. If the weighted average is \(\bar x\), then \[\color{blue}{\bar x =\frac{{{a_1}{x_1} + {a_2}{x_2} + {a_3}{x_3} + ....{a_n}{x_n}}}{{{a_1} + {a_2} + {a_3} + ....{a_n}}}}\]

Example 01: The average weights of the students in the classes \(A, B\) and \(C\) are 45, 55 and 60kg. If the numbers of students in the three classes are 50, 60 and 75, find the average weight of the students in all the three classes put together.

Solution: Applying the formula for weighted average  =\(\frac{{45 \times 50 + 55 \times 60 + 60 \times 75}}{{50 + 60 + 75}}\)

=\(\frac{{2250 + 3300 + 4500}}{{185}}\)=\(\frac{{10050}}{{185}}\) =\(54\frac{{12}}{{37}}\)kg

Example 02: Three math classes: \(X, Y\) and \(Z\) take an algebra test. The average score in class \(X\) is 83. The average score in class \(Y\) is 76. The average score in class \(Z\) is 85. The average score of all students in classes \(X\) and \(Y\) together is 79. The average score of all students in classes \(Y\) and \(Z\) together is  81. What is the average for all the three classes ?

Solution: Suppose in classes \(X, Y\) and \(Z\) number of students are  \(x\,,\,y\,\,and\,\,z\) , then we have

\[\frac{{83x\, + \,76y}}{{x\, + \,y}} = 79\] \[\frac{{76y\, + \,85z}}{{y\, + \,z}} = 81\]

Solving these equations we get,

 \(83x + 76y = 79 (x + y)\) or \(4x = 3y\) or 

\(x : y = 3 : 4\)

Similarly \(4z = 5y\) or \(y : z = 4 : 5\)

Hence \(x : y : z = 3 : 4 : 5\)

So the average for all the three classes =\(\cfrac{{83x\, + \,76y\, + 85z}}{{x\, + \,y\, + z}}\)=\(\cfrac{{83 \times 3\, + 76 \times 4\, + 85 \times 5}}{{3 + 4 + 5}} = 81.5\)