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Zorluk: Çok zorWeighted Averages in Applied Contexts

A financial firm consists of three departments: Analytics, Trading, and Research. The ratio of the number of analysts in the Analytics department to traders in the Trading department is 2:32:3. The average annual bonus is $90,000\$90,000 for an analyst in the Analytics department, $60,000\$60,000 for a trader in the Trading department, and $45,000\$45,000 for a researcher in the Research department. If the Research department has 4040 researchers and the average annual bonus across all employees in all three departments combined is $64,000\$64,000, how many analysts are in the Analytics department?

  1. A
    19
  2. 38Cevap
  3. C
    40
  4. D
    57
  5. E
    95

Cevap

The Analytics department has 38 analysts.
Using the given ratio 2:32:3, let the number of analysts be 2k2k and traders be 3k3k. The weighted average bonus for these two groups is 2k(90,000)+3k(60,000)5k=$72,000\frac{2k(90,000) + 3k(60,000)}{5k} = \$72,000. Combining this sub-group with the 40 researchers earning $45,000\$45,000 gives an overall average of 72,000(5k)+45,000(40)5k+40=64,000\frac{72,000(5k) + 45,000(40)}{5k + 40} = 64,000. Solving 360k+18005k+40=64\frac{360k + 1800}{5k + 40} = 64 yields 360k+1800=320k+2560360k + 1800 = 320k + 2560, so 40k=76040k = 760 and k=19k = 19. The number of analysts is 2(19)=382(19) = 38.

Adım Adım Çözüm

1
Express the number of employees in Analytics and Trading in terms of a variable kk.
Let NAnalytics=2kN_{\text{Analytics}} = 2k and NTrading=3kN_{\text{Trading}} = 3k. The total number of employees in these two departments is 5k5k.
The ratio of analysts to traders is given as 2:32:3.
2
Calculate the combined average bonus for the Analytics and Trading departments.
Combined average = 2k(90,000)+3k(60,000)5k=180,000k+180,000k5k=360,000k5k=$72,000\frac{2k(90,000) + 3k(60,000)}{5k} = \frac{180,000k + 180,000k}{5k} = \frac{360,000k}{5k} = \$72,000.
Weighting each department's average bonus by its relative ratio simplifies the calculation.
3
Set up the weighted average equation combining all three departments.
\frac{72,000(5k) + 45,000(40)}{5k + 40} = 64,000
The total bonus pool divided by the total number of employees across all three departments equals the overall average bonus of $64,000\$64,000.
4
Solve for kk and find the number of analysts.
Dividing all terms by 1,0001,000 yields 360k+18005k+40=64  ⟹  360k+1800=320k+2560  ⟹  40k=760  ⟹  k=19\frac{360k + 1800}{5k + 40} = 64 \implies 360k + 1800 = 320k + 2560 \implies 40k = 760 \implies k = 19. Therefore, NAnalytics=2k=2(19)=38N_{\text{Analytics}} = 2k = 2(19) = 38.
Linear algebraic simplification isolates kk to compute the requested quantity.

Anahtar Kavram

Weighted Averages with Compound Ratios

Alternatif Yöntem

Use the scale-balance method for weighted averages: The combined average of Analytics and Trading is $72,000\$72,000. The Research average is $45,000\$45,000, and the target combined average is $64,000\$64,000. The distances from the target are (72,000−64,000)=8,000(72,000 - 64,000) = 8,000 and (64,000−45,000)=19,000(64,000 - 45,000) = 19,000. The ratio of group sizes (NAnalytics+Trading:NResearch)(N_{\text{Analytics}+\text{Trading}} : N_{\text{Research}}) is inversely proportional to these distances: 5k:40=19,000:8,000=19:85k : 40 = 19,000 : 8,000 = 19 : 8. Cross-multiplying gives 40k=760  ⟹  k=1940k = 760 \implies k = 19, so analysts =2k=38= 2k = 38.
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