At a national robotics competition, the average score of participants

Question

At a national robotics competition, the average score of participants who received a grant was 20 points higher than the average score of those who did not. What was the average score of all participants?

(1) The average score of those who received a grant was 192.

(2) 20% of participants received a grant.

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Bunuel · Accepted Answer

Official Solution:

At a national robotics competition, the average score of participants who received a grant was 20 points higher than the average score of those who did not. What was the average score of all participants? 
 
Let the fraction of participants who received a grant be  p , and the average score of those participants be  x . Then the overall average is:  Overall\ average = p * x + (1 - p) * (x - 20) = x - 20 + 20p 
 
(1) The average score of those who received a grant was 192.

This gives the value of  x , but the value of  p  is unknown. Not sufficient.
 
(2) 20% of participants received a grant.

This gives the value of  p , but the value of  x  is unknown. Not sufficient.
 
(1)+(2) We now know both  x = 192  and  p = 0.2 , so we can compute the average:  Overall\ average = 192 - 20 + 20 * 0.2 = 172 + 4 = 176 . Sufficient.
 
Alternatively, since 80% of participants did not receive a grant and 20% did, the ratio of group sizes is 4:1. So the distances from the individual group averages to the overall average must be in the inverse ratio, 1:4. That means the 20-point range between the two group averages must be divided in a 4:16 split. Thus, the overall average is 16 points below the average of those who received a grant:  Overall\ average = 192 - 16 = 176 . Sufficient.

Answer: E

ManifestDreamMBA · Answer

Ag = Ang+20
To calculate avg, I need the weights of grant vs non grant students and actual avg of either of them

Statement1
Ag = 192
Ang = 172
We don't know the ratio of these students, so cannot calculate overall average
Insufficient

Statement2
Ag/Ang = 20/80 = 1/4
We don't know Ag or Ang, so cannot calculate overall average
Insufficient

Combined, we have Ag/Ang = 1/4 and Ag = 192, Ang = 172. Sufficient

gmatophobia · Answer

The number of students who received a grant = x
The number of students who did not receive a grant = y

Statement 1
  
(1) The average score of those who received a grant was 192.

Average of the group =  \frac{192 x + (192 + 20) y }{ x + y}  =  \frac{192 (x + 20y)}{x+y}

The statement alone is not sufficient to find the average of the group. Eliminate A and D.

Statement 2
  
(2) 20% of participants received a grant.

x = 0.2(x + y)

0.8x = 0.2y

y = 4x

While we have a relationship between x and y, we don't know the average score of each target group. Hence, we cannot find the average score of the whole group.

The statement alone is not sufficient to answer the question asked.

Eliminate B.

Combined

From (1) →  \frac{192 (x + 20y)}{x+y}

From (2) →  y = 4x

Combined

\frac{192 (x + 20*4x)}{x+4x}

The x from numerator and denominator can be eliminated and we will have a definite answer.

Option C

At a national robotics competition, the average score of participants

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