SOLUTION: A total of 50 juniors and seniors were given a mathematics test. The 35 juniors attained an average score of 80 while the 15 seniors attained an average of 70. What was the averag

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Question 484087: A total of 50 juniors and seniors were given a mathematics test. The 35 juniors
attained an average score of 80 while the 15 seniors attained an average of 70. What was the average score for all 50 students who took the test?
A. 73
B. 75
C. 76
D. 77
E. 78
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
First, you need to be careful that you don't average the two averages. By this I mean that you don't add the average score of the juniors (80) with the average score of the seniors (70) to get:
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%2880+%2B+70%29%2F2+=+150%2F2
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This would give you an "overall average" of 75 and it would be wrong.
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Next, you can get a sense of the answer by recognizing that more students scored near 80 points than the number of students that scored near 70 points. Therefore, the "overall average" will be closer to 80 than it is to 70. So you can suspect that the answer of 73 (answer A) is also wrong.
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So how do you solve this problem?
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To have the 35 juniors get an average score of 80, the total of all the scores by juniors divided by the number of juniors (35) must be 80. So you can ask yourself what would be the total of the junior's scores (call it TJ) that when divided by 35 gives an answer of 80? In equation form this would be:
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%28TJ%29%2F35+=+80
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This can be solved by multiplying both sides of this equation by 35 as follows:
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%28TJ%29%2Across%2835%29%2Fcross%2835%29+=+80%2A35
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Multiply out the right side and you have:
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TJ+=+2800
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So the juniors scored a total of 2800 points.
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How many total points did the seniors score? (Call this total TS.) Using the same process for the 15 seniors as we just did for the juniors we would get:
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%28TS%29%2F15+=+70
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Multiply both sides of this equation by 15 as follows:
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%28TS%29%2Across%2815%29%2Fcross%2815%29+=+70%2A15
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Multiply out the right side and you have:
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TS+=+1050
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So the seniors scored a total of 1050 points.
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Now when you combine the points scored by the juniors and seniors, the total number of points scored by the two classes was:
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2800%2B1050+=+3850
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So the 50 students scored a total of 3850 points. This means that the average score for the 50 students is the total points scored by all 50 students divided by the total number of students who took the test. In equation form this can be written as:
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3850%2F50+=+77
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The correct answer to this problem is answer D.
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I hope that this helps you to understand the problem and also helps you get a sense of how to work with averages. Good luck!
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