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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.
(1) The average score of those who received a grant was 192.
(2) 20% of participants received a grant.
30 Apr 2025, 13:23
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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: C
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: C
10 Sep 2025, 17:47
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Hey Bunuel
I have one doubt regarding the formula used for calculating overall average.
It looks like the calculation of Total Scores and not the average score.
Eg Average = Total Score/No. of students -> x = Total Score/p -> Total Score = px ?
Pls help.
I have one doubt regarding the formula used for calculating overall average.
It looks like the calculation of Total Scores and not the average score.
Eg Average = Total Score/No. of students -> x = Total Score/p -> Total Score = px ?
Pls help.
Bunuel
