Question:A data set consists of 15 distinct positive integers. The arithmetic mean of the 15 integers is 40. When one...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A data set consists of \(\mathrm{15}\) distinct positive integers. The arithmetic mean of the \(\mathrm{15}\) integers is \(\mathrm{40}\). When one integer, \(\mathrm{x}\), is removed from the set, the arithmetic mean of the remaining integers is \(\mathrm{38.5}\). When \(\mathrm{x}\) is returned to the set and a different integer, \(\mathrm{y}\), is removed, the arithmetic mean of the remaining integers is \(\mathrm{40.5}\). What is the value of \(\mathrm{y}\)?
- 28
- 33
- 35
- 42
- 61
1. TRANSLATE the problem information
- Given information:
- 15 distinct positive integers
- Mean of all 15 integers = 40
- When x removed: mean of remaining 14 = 38.5
- When y removed instead: mean of remaining 14 = 40.5
- What this tells us: We need to use the relationship between mean, sum, and count
2. INFER the solution strategy
- Key insight: If we know the mean and count, we can find the sum
- Strategy: Find the original sum, then find what's left when y is removed
- The difference will give us y directly
3. SIMPLIFY to find the original sum
- Using \(\mathrm{Sum} = \mathrm{Mean} \times \mathrm{Count}\):
- Original sum = \(15 \times 40 = 600\)
4. SIMPLIFY to find sum without y
- Sum when y is removed = \(14 \times 40.5 = 567\) (use calculator)
5. INFER the final calculation
- Since y was removed from the original sum:
- \(\mathrm{y} = \mathrm{Original\ sum} - \mathrm{Sum\ without\ y} = 600 - 567 = 33\)
Answer: 33 (Choice B)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students often get confused by the multiple scenarios (removing x vs removing y) and try to work with x first, even though x isn't needed to find y.
They might calculate \(\mathrm{x} = 600 - 539 = 61\) and then get stuck trying to figure out how x relates to y. This leads to confusion and guessing rather than recognizing they can work directly with the y scenario.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors with the decimal multiplication \(14 \times 40.5\), getting 566 instead of 567, leading them to calculate \(\mathrm{y} = 600 - 566 = 34\), which isn't among the answer choices.
This causes them to second-guess their approach and potentially select Choice (C) (35) as the closest value.
The Bottom Line:
This problem tests whether students can navigate multiple scenarios without getting sidetracked, and whether they recognize that the most direct path to y doesn't require finding x first.