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Difficulty: MediumWeighted Averages in Applied Contexts

A management consulting firm charges clients based on three staff tiers: Junior Consultants, Senior Consultants, and Directors. Junior Consultants are billed at a rate of $120\$120 per hour, Senior Consultants at $210\$210 per hour, and Directors at $390\$390 per hour. On a strategic assignment, the ratio of hours worked by Junior Consultants to Senior Consultants to Directors was 5:3:25 : 3 : 2, respectively. What was the overall average hourly billing rate per consultant hour for this assignment?

  1. $201\$201Answer
  2. B
    $240\$240
  3. C
    $255\$255
  4. D
    $282\$282
  5. E
    $670\$670

Answer

The overall average hourly billing rate per consultant hour for the assignment is $201\$201.
The weighted average is found by multiplying each tier's rate by its corresponding ratio weight, summing these products, and dividing by the sum of the ratio weights: (5×120)+(3×210)+(2×390)5+3+2=600+630+78010=201010=$201\frac{(5 \times 120) + (3 \times 210) + (2 \times 390)}{5 + 3 + 2} = \frac{600 + 630 + 780}{10} = \frac{2010}{10} = \$201.

Step-by-Step Solution

1
Determine the total weight of the ratio elements.
Sum of ratio parts = 5+3+2=105 + 3 + 2 = 10 total parts of hours.
To calculate a weighted average, the sum of all weight parts must be calculated to serve as the denominator.
2
Calculate the total revenue generated for 10 representative hours based on the ratio.
Total revenue = (5×$120)+(3×$210)+(2×$390)=$600+$630+$780=$2,010(5 \times \$120) + (3 \times \$210) + (2 \times \$390) = \$600 + \$630 + \$780 = \$2,010.
Multiplying each tier's hourly billing rate by its respective weight yields the total weighted revenue.
3
Divide the total revenue by the total number of ratio hours.
Weighted average rate = $2,01010=$201\frac{\$2,010}{10} = \$201 per hour.
Dividing total cost/revenue by total units yields the weighted average per unit.

Key Concept

Weighted Average formula: Weighted Average=∑(wi⋅xi)∑wi\text{Weighted Average} = \frac{\sum (w_i \cdot x_i)}{\sum w_i}, where wiw_i represents the weight of each component and xix_i represents the value of each component.
Estimated Time:1m 30s
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Weighted Averages in Applied Contexts

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