# Difficult Averages Solved QuestionAptitude Discussion

 Q. 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?
 ✖ A. 81 ✔ B. 81.5 ✖ C. 82 ✖ D. 84.5

Solution:
Option(B) is correct

Average score of class $X$ is 83 and that of class $Y$ is 76 and the combined average of $X$ and $Y$ is 79.

By rule of alligation ratio of students in $X : Y$ is given by

$X$        :          $Y$

79
/       \

83            76

3        :        4

Similarly, average score of class $Y$ is 76 and that of class $Z$ is 85 and the combined average is 81.

By rule of alligation ratio of students in $Y : Z$ is

$Y$           :          $Z$
81
/           \
76                 85
4          :        5

$X:Y:Z=3:4:5$

Total average for $X, Y$ and $Z$

$=\dfrac{3\times 83+4\times 76+5\times 85}{3+4+5}$

$=81.5$