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Zorluk: ZorDescriptive Statistics and Data Representations

A technology company has three departments: Sales, Marketing, and Engineering. The table below shows the number of employees in the Sales and Marketing departments, along with the mean annual salary for each of the three departments.

DepartmentNumber of EmployeesMean Annual Salary
Sales88$45,000\$45,000
Marketing1212$55,000\$55,000
EngineeringNN$85,000\$85,000

If the mean annual salary of all employees across all three departments is $68,000\$68,000, what is the total number of employees in the company?

  1. A
    20
  2. B
    24
  3. C
    30
  4. 40Cevap
  5. E
    45

Cevap

40
To find the total number of employees, we first calculate the total salary for each department. The Sales department total is 8×$45,000=$360,0008 \times \$45,000 = \$360,000. The Marketing department total is 12×$55,000=$660,00012 \times \$55,000 = \$660,000. If there are NN employees in Engineering, their total salary is N×$85,000=$85,000NN \times \$85,000 = \$85,000N. The total salary for the entire company is $360,000+$660,000+$85,000N=$1,020,000+$85,000N\$360,000 + \$660,000 + \$85,000N = \$1,020,000 + \$85,000N. The total number of employees is 8+12+N=20+N8 + 12 + N = 20 + N. Since the overall mean salary is $68,000\$68,000, we set up the weighted average equation: 1,020,000+85,000N20+N=68,000\frac{1,020,000 + 85,000N}{20 + N} = 68,000. Solving for NN gives 1,020,000+85,000N=68,000(20+N)1,020,000 + 85,000N = 68,000(20 + N), which simplifies to 1,020,000+85,000N=1,360,000+68,000N1,020,000 + 85,000N = 1,360,000 + 68,000N. Subtracting 68,000N68,000N and 1,020,0001,020,000 from both sides yields 17,000N=340,00017,000N = 340,000, so N=20N = 20. The total number of employees in the company is the sum of the employees in all three departments: 8+12+20=408 + 12 + 20 = 40.

Adım Adım Çözüm

1
Calculate the total combined annual salary for the Sales and Marketing departments.
Sales total: 8×$45,000=$360,0008 \times \$45,000 = \$360,000. Marketing total: 12×$55,000=$660,00012 \times \$55,000 = \$660,000. Combined total: \360,000+$660,000=$1,020,000360,000 + \$660,000 = \$1,020,000.
To find the overall mean, we need the sum of all salaries in the company.
2
Express the total company salary and total number of employees in terms of NN.
Total salary: $1,020,000+85,000N\$1,020,000 + 85,000N. Total employees: 8+12+N=20+N8 + 12 + N = 20 + N.
These terms will be used to set up the weighted average formula.
3
Set up the weighted average equation and solve for NN.
Equation: 1,020,000+85,000N20+N=68,000\frac{1,020,000 + 85,000N}{20 + N} = 68,000. Multiplying both sides by 20+N20 + N gives 1,020,000+85,000N=1,360,000+68,000N1,020,000 + 85,000N = 1,360,000 + 68,000N. Subtracting 68,000N68,000N and 1,020,0001,020,000 from both sides yields 17,000N=340,00017,000N = 340,000, which simplifies to N=20N = 20.
Solving this equation gives the number of employees in the Engineering department.
4
Calculate the total number of employees in the company.
Total employees = 8+12+20=408 + 12 + 20 = 40.
The question asks for the total number of employees, which is the sum of employees in all three departments.

Anahtar Kavram

Weighted Average (Mean) with Variables
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Descriptive Statistics and Data Representations

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