Question: The arithmetic mean of the scores of a group of students in a test was 52. The top 20% of the students obtained a mean score of 80, while the bottom 25% obtained a mean score of 31. What was the mean score of the remaining 55% of the students?
(A) 45%
(B) 50%
(C) 51.4%
(D) 54.6%
(E) 58.2%
Correct Answer: C
Solution and Explanation
Approach Solution 1:
The question tries to throw you off by mentioning "top" and "bottom" scorers. Distribution does not matter in the mean, it is simply an average. The information that the question provides is that the OVERALL (100%) mean of 52 is made up of 3 smaller components. 25% of the 100% has a mean of 31 and 20% of the 100% has a mean of 80. The remaining 55% has a mean of "x", which is what we are trying to find.
Let us consider the remaining mean score as “x”. Considering this, the simple equation stands as:
Overall mean = 20%*(Mean of 80) + 25% (mean of 31) + 55% (mean of x)
52 = 20/100 (80) +25/100 (31) + 55/100 (x)
52 = 0.2(80) + 0.25(31) + 0.55(x)
Multiply using mental math:
52 = 16 + 7.75 + 0.55x
52 = 23.75 + 0.55x
28.25 = 0.55x
Multiply the entire equation by 100 (shift decimals to the right by two places) and…
28.25*100 = 0.55x
2825 = 55x
Therefore x = 2825/55 = 565/11
If we divide this, the result comes to 51.36. However, the nearest option is 51.4 and hence, C is the correct answer.
Approach Solution 2:
The following information is given in the problem statement:
- The arithmetic mean of the scores of a group of students in a test was 52.
- The top 20% of the students obtained a mean score of 80, while the bottom 25% obtained a mean score of 31.
Candidates need to find out:
What was the mean score of the remaining 55% of the students?
Solution:
Let us consider the mean score of the remaining 55% of students as X
We can derive the equation from the above information:
Total mean score = (20% of the students* Mean score of 80) + (25% of the students * Mean score of 31) + (55% of the students* Mean score X)
If we put the values the equation becomes:
52 = 20%*80+25%*31+55%*x
52 = 20/100*80 + 25/100*31 + 55/100*X
52 = 0.2*80 + 0.25*31 + 0.55*X
52 = 16+7.75+.55X
X=51.36%
Considering the above number, the correct answer should be 51.4 as it is the nearest one.
Approach Solution 3:
Mean score of remaining 55%
=[100×52−(20×80+25×31)]/55
=5200−(1600+775)/55
=5200−2375/55= 2825/55= 51.36%
=51.4%(approx)
“The arithmetic mean of the scores of a group of students in a test was 52.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Reviews". To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.
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