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Difficulty: MediumWeighted Average and Combined Sets

A financial consultancy firm consists of two divisions: Risk Management and Wealth Advisory. The 4040 analysts in the Risk Management division earn an average annual bonus of $15,000\$15,000, while the analysts in the Wealth Advisory division earn an average annual bonus of $25,000\$25,000. If the average annual bonus across all analysts in both divisions combined is $21,000\$21,000, how many analysts work in the Wealth Advisory division?

  1. A
    24
  2. B
    40
  3. C
    50
  4. 60Answer
  5. E
    100

Answer

The Wealth Advisory division has 60 analysts.
The combined mean of two sets is closer to the mean of the larger set. Using the distance-ratio rule for weighted averages, the ratio of the number of Risk Management analysts to Wealth Advisory analysts is inversely proportional to their distances from the combined mean: Ratio=25,000−21,00021,000−15,000=4,0006,000=23\text{Ratio} = \frac{25,000 - 21,000}{21,000 - 15,000} = \frac{4,000}{6,000} = \frac{2}{3}. Given that there are 40 analysts in Risk Management (representing 2 parts), each part equals 20 analysts. Therefore, the 3 parts for Wealth Advisory equal 3×20=603 \times 20 = 60 analysts.

Step-by-Step Solution

1
Set up the weighted average equation with NN representing the number of analysts in the Wealth Advisory division.
Total bonus for Risk Management = 40×15,000=600,00040 \times 15,000 = 600,000. Total bonus for Wealth Advisory = N×25,000N \times 25,000. Total analysts = 40+N40 + N.
The combined mean is equal to the sum of all individual values divided by the total number of items.
2
Formulate the combined average equation and clear the fraction.
\frac{600,000 + 25,000N}{40 + N} = 21,000 \implies 600,000 + 25,000N = 21,000(40 + N)
Equate the weighted average expression to the given overall average of $21,000\$21,000.
3
Expand and solve the linear equation for NN.
600,000 + 25,000N = 840,000 + 21,000N \implies 4,000N = 240,000 \implies N = 60
Isolate the variable NN to find the exact number of analysts in Wealth Advisory.

Key Concept

Weighted Average and Combined Sets
Estimated Time:1m 30s
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Weighted Average and Combined Sets

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