After a 16-member team at Company A saw a certain software demonstration, each of the team members gave the software...
GMAT Two Part Analysis : (TPA) Questions
After a 16-member team at Company A saw a certain software demonstration, each of the team members gave the software exactly one of the ratings 1, 2, 3, 4, or 5, where greater numbers indicate higher ratings. The team's manager intended to record the 16 ratings but accidentally recorded one team member's rating twice. The ratings she recorded show two ratings of 1, two ratings of 2, two ratings of 3, two ratings of 4, and nine ratings of 5, for a total of 17 ratings.
The manager determined that the extra rating was \(\mathrm{P}\), and after this extra rating was removed, the average (arithmetic mean) of the remaining 16 ratings was exactly \(\mathrm{Q}\). Select for P and for Q the options that create a statement that is consistent with the information provided.
Phase 1: Owning the Dataset
Visual Representation
Let's create a table showing the recorded ratings:
| Rating | Count Recorded |
| 1 | 2 |
| 2 | 2 |
| 3 | 2 |
| 4 | 2 |
| 5 | 9 |
| Total | 17 |
Key insight: One rating was recorded twice, so we have 17 entries instead of 16.
Phase 2: Understanding the Question
We need to find:
- P: The rating value that was recorded twice (the extra rating)
- Q: The average of the 16 ratings after removing the extra
Let's calculate the sum of all 17 recorded ratings:
- Sum = \(2(1) + 2(2) + 2(3) + 2(4) + 9(5)\)
- Sum = \(2 + 4 + 6 + 8 + 45 = 65\)
Phase 3: Finding the Answer
Systematic Approach
After removing rating P, we'll have 16 ratings with sum = \(65 - \mathrm{P}\)
The average Q = \(\frac{65 - \mathrm{P}}{16}\)
Since Q must be one of our answer choices (1, 2, 3, 4, or 5), let's check each possibility:
If P = 1:
- Q = \(\frac{65 - 1}{16} = \frac{64}{16} = 4\) ✓
- 4 is in our answer choices!
If P = 2:
- Q = \(\frac{65 - 2}{16} = \frac{63}{16} = 3.9375\)
- Not a whole number, not in choices
If P = 3:
- Q = \(\frac{65 - 3}{16} = \frac{62}{16} = 3.875\)
- Not a whole number, not in choices
If P = 4:
- Q = \(\frac{65 - 4}{16} = \frac{61}{16} = 3.8125\)
- Not a whole number, not in choices
If P = 5:
- Q = \(\frac{65 - 5}{16} = \frac{60}{16} = 3.75\)
- Not a whole number, not in choices
Stop here - we found our answer.
Verification
If P = 1 (the extra rating), after removal we have:
- 1 rating of 1 (not 2)
- 2 ratings of 2
- 2 ratings of 3
- 2 ratings of 4
- 9 ratings of 5
- Total: 16 ratings ✓
Sum = \(1 + 2(2) + 2(3) + 2(4) + 9(5) = 1 + 4 + 6 + 8 + 45 = 64\)
Average = \(\frac{64}{16} = 4\) ✓
Phase 4: Solution
Final Answer:
- P = 1 (the extra rating that was removed)
- Q = 4 (the resulting average)
The only combination that produces a valid average from our answer choices is when the extra rating of 1 is removed, resulting in an average of exactly 4.