There are at least three times as many boys as girls in a class.

Question

There are at least three times as many boys as girls in a class. On a test, the boys’ average score is 78, and the girls’ average score is 94. Which of the following could be the average test score for the entire class?

I. 84
II. 82
III. 80

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

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g1006 · Answer

Let the average score of boys, B=78 and let the average score of girls, G=94

It is given that there are least three times as many boys as girls in a class. Since the average score of boys is lower than that of girls, the maximum class average will occur when the number of boys is the least possible.

If we assume there is only one girl in the class, then the least number of boys in the class will be 3.
Class average = (No. of boys x Average score of boys + No. of girls x Average score of girls)/(No. of boys + No. of girls)
Therefore, class average = (3 x 78 + 1 x 94)/(3 + 1) = (234 + 94)/4 = 328/4 =  82

84 cannot be an answer, so choices A and E are wrong.

Now, we need to check whether 80 can be the average score of the class for positive integer values of the number of boys and girls.

Let m be the number of boys and n be the number of girls.
Assuming the average is 80. Therefore, 80 = (m x 78 + n x 94)/(m + n)
Multiplying both sides by (m + n) we get, 80(m + n) = 78m + 94n, or 80m + 80n = 78m + 94n, or 2m=14n, or 1m=7n

For n=1,2,3,etc we get m=7,14,21,etc. As we have positive integer values for both m and n, the class average can take 80 as a value.

Both 82 and 80 are possible values.  Therefore, choice D is correct

$!vakumar.m · Answer

Which of the following could be the average test score for the entire class?
I. 84 II. 82 III. 80

Let no of girls be x then minimum no of boys is 3x and maximum no of boys is not given.
girls average score = 94 and boys average score is 78
Note that its a weighted average question 
 class average (max) = \frac{(94x + 3x*78)}{(x+3x)}  =  \frac{(94x + 234x)}{4x}  =  82  (since we are taking min no of boys so the class average is max)
 So its clear that if we increase the number of boys the class average will drop  
therefore any value below 82 could be the average test score for the entire class as upper limit on number of boys is not given.
Both 82 and 80 could be the class average 
Answer D

ChakAvi · Answer

Boys:girls = 3:1 or higher. So, overall average will be closer to the boys' average. 
When 3:1 ratio, average= (0.25*(94-78)) +78 = 82.
If the ratio increases, the average may move below 82, which can be 80.

So, 80 & 82 ..option D

mayankkamal999 · Answer

I solved via weighted averages. I hope the approach is right.

We know that number of boys >=3 times number of girls. So we should get b/g >=3
1. For average as 84: b/g= /  = 10/6 =1.67 
2. For average as 82: b/g= /  = 12/4 = 3 
3. For average as 80: b/g= /  = 14/2 = 7

So 82 and 80 should be possible.

There are at least three times as many boys as girls in a class.

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There are at least three times as many boys as girls in a class.

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