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Difficulty: MediumWeighted Average and Combined Sets

A software development company has two engineering divisions: Cloud Infrastructure and Mobile Applications. The 4545 engineers in the Cloud Infrastructure division completed code reviews with an average duration of 2525 minutes per review, while the engineers in the Mobile Applications division completed code reviews with an average duration of 4040 minutes per review. If the overall average code review duration for all engineers across both divisions combined was 3131 minutes, how many engineers are in the Mobile Applications division?

  1. A
    1818
  2. B
    2727
  3. 3030Answer
  4. D
    4545
  5. E
    7575

Answer

30 engineers
The overall average of a combined set is weighted by the number of elements in each group. Using the distance-to-mean principle, the ratio of the number of engineers in Cloud Infrastructure (NcN_c) to Mobile Applications (NmN_m) is inversely proportional to their distances from the combined mean: Nc:Nm=(40−31):(31−25)=9:6=3:2N_c : N_m = (40 - 31) : (31 - 25) = 9 : 6 = 3 : 2. Given Nc=45N_c = 45, setting up the proportion 45Nm=32\frac{45}{N_m} = \frac{3}{2} yields Nm=30N_m = 30.

Step-by-Step Solution

1
Set up the weighted average equation for the combined set.
45×25+Nm×4045+Nm=31\frac{45 \times 25 + N_m \times 40}{45 + N_m} = 31
The total time across both divisions divided by total engineers equals the combined average time.
2
Multiply both sides by (45+Nm)(45 + N_m) and expand.
1125+40Nm=31(45)+31Nm=1395+31Nm1125 + 40 N_m = 31(45) + 31 N_m = 1395 + 31 N_m
To clear the denominator and group like terms.
3
Solve for NmN_m.
40Nm−31Nm=1395−1125  ⟹  9Nm=270  ⟹  Nm=3040 N_m - 31 N_m = 1395 - 1125 \implies 9 N_m = 270 \implies N_m = 30
Isolating NmN_m gives the exact count of engineers in the Mobile Applications division.

Key Concept

Weighted Average and Combined Sets
Estimated Time:2m 0s
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Topics

Weighted Average and Combined Sets

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