Data set A consists of the heights of 75 buildings and has a mean of 32 meters. Data set B...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
Data set A consists of the heights of \(\mathrm{75}\) buildings and has a mean of \(\mathrm{32}\) meters. Data set B consists of the heights of \(\mathrm{50}\) buildings and has a mean of \(\mathrm{62}\) meters. Data set C consists of the heights of the \(\mathrm{125}\) buildings from data sets A and B. What is the mean, in meters, of data set C?
1. TRANSLATE the problem information
- Given information:
- Data set A: 75 buildings with mean height 32 meters
- Data set B: 50 buildings with mean height 62 meters
- Data set C: all 125 buildings combined \(75 + 50 = 125\)
- What we need: mean height of data set C
2. INFER the approach
- To find the mean of the combined data set, we need the total sum of all heights divided by the total count
- We can't just average the two means \((32 + 62)/2 = 47\) because the data sets have different sizes
- Strategy: Use the mean formula backwards to find individual sums, then combine them
3. SIMPLIFY to find the sum of data set A
- Using sum = mean × count:
Sum of A = \(32 \times 75 = 2,400\) meters
4. SIMPLIFY to find the sum of data set B
- Using sum = mean × count:
Sum of B = \(62 \times 50 = 3,100\) meters
5. SIMPLIFY to find the combined sum and mean
- Sum of C = Sum of A + Sum of B = \(2,400 + 3,100 = 5,500\) meters
- Mean of C = \(5,500 \div 125 = 44\) meters
Answer: 44
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students think they should just average the two given means: \((32 + 62)/2 = 47\)
They don't realize that when combining data sets of different sizes, you can't simply average the means. The larger data set (B with 50 buildings) should have more influence on the combined mean than the smaller one (A with 75 buildings), but averaging treats them equally.
This may lead them to calculate 47 as their answer, though this isn't among typical answer choices, causing confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify they need to find sums first, but make arithmetic errors in the multiplication or final division steps.
Common calculation mistakes include:
- \(32 \times 75 = 2,300\) instead of 2,400
- \(62 \times 50 = 3,000\) instead of 3,100
- Final division errors with \(5,500 \div 125\)
This leads to incorrect final answers and selecting wrong choices.
The Bottom Line:
This problem tests whether students understand that combining data sets requires working with totals, not just averaging means. The key insight is recognizing that data sets of different sizes contribute differently to the combined result.