Problem:
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84, and the afternoon class mean score is 70. The ratio of the number of students in the morning class to the number of students in the afternoon class is 43. What is the mean of the scores of all the students?
Answer Choices:
A. 74
B. 75
C. 76
D. 77
E. 78
Solution:
Let there be 3x students in the morning class and 4x students in the afternoon class. The total number of students is 3x+4x=7x. The average is 7x3x⋅84+4x⋅70=76. Therefore, the answer is (C) 76 .
OR
Suppose the morning class has m students and the afternoon class has a students. We have the following table:
MorningAfternoon# of StudentsmaMean8470Total84m70a
We are also given that am=43, which rearranges as m=43a.
The mean of the scores of all the students is
m+a84m+70a=43a+a84(43a)+70a=47a133a=133⋅74=(C) 76 .
OR
Of the average, 3+43=73 of the scores came from the morning class and 74 came from the afternoon class. The average is 73⋅84+74⋅70=(C) 76 .
The problems on this page are the property of the MAA's American Mathematics Competitions