In a company with a total of 3540 employees, there are four departments: Management, HR, Production, and Accounts. The number of females in the HR department is 87.5\% of the number of males in the same department. The Production department has 800 males, which is 25\% more than the number of males in the Management department. The number of females in the Management department is equal to the number of males in the Accounts department. In the Accounts department, females are 150 fewer than males and 150 more than females in the Production department. The ratio of the total number of employees in the Production department to those in the Accounts department is 19:15.

What is the difference between the total number of males and females in the company?

Question

In a company with a total of 3540 employees, there are four departments: Management, HR, Production, and Accounts. The number of females in the HR department is 87.5\% of the number of males in the same department. The Production department has 800 males, which is 25\% more than the number of males in the Management department. The number of females in the Management department is equal to the number of males in the Accounts department. In the Accounts department, females are 150 fewer than males and 150 more than females in the Production department. The ratio of the total number of employees in the Production department to those in the Accounts department is 19:15.

What is the difference between the total number of males and females in the company?

Accepted Answer

**Solution:**

1. Let the number of males in Management department be $ M_m $.

2. Given that males in Production department = 800, which is 25\% more than males in Management:
   $$
   $800 = M_m + 0.25 M_m = 1.25 M_m \implies M_m = \frac{800}{1.25} = 640$
   $$

3. Females in HR department are 87.5\% of males in HR. Let males in HR be $ H_m $, then females in HR = $ 0.875 H_m $.

4. Let males in Accounts be $ A_m $ and females in Accounts be $ A_f $.

5. Number of females in Management equals males in Accounts:
   $$
   F_{Management} = A_m
   $$

6. Females in Accounts are 150 less than males in Accounts:
   $$
   A_f = A_m - 150
   $$

7. Females in Accounts are 150 more than females in Production:
   $$
   A_f = F_{Production} + 150
   $$

8. Total employees in Production and Accounts are in ratio 19:15.

Let total employees in Production = $ 19x $ and in Accounts = $ 15x $.

9. Total employees in company = 3540:
   $$
   19x + 15x + (Management + HR) = 3540
   $$

10. Total males and females in Production:
    - Males in Production = 800
    - Females in Production = $ F_{Production} $

So, total in Production:
    $$
  $  800 + F_{Production} = 19x \implies F_{Production} = 19x - 800$
    $$

11. Total in Accounts:
    $$
    A_m + A_f = 15x
    $$

Using $ A_f = A_m - 150 $,
    $$
  $  A_m + (A_m - 150) = 15x \implies 2A_m - 150 = 15x \implies 2A_m = 15x + 150 \implies A_m = \frac{15x + 150}{2}$
    $$

12. From step 7:
    $$
 $   A_f = F_{Production} + 150 \implies A_m - 150 = F_{Production} + 150 \implies A_m - F_{Production} = 300$
    $$

13. Substitute $ A_m $ and $ F_{Production} $ from above:
    $$
   $ \frac{15x + 150}{2} - (19x - 800) = 300$
    $$

Multiply both sides by 2:
    $$
    15x + 150 - 2(19x - 800) = 600
    $$

$$
    15x + 150 - 38x + 1600 = 600
    $$

$$
  $  -23x + 1750 = 600 \implies -23x = 600 - 1750 = -1150 \implies x = \frac{1150}{23} = 50$
    $$

14. Now find total employees in Production and Accounts:
    $$
  $  Production = 19x = 19 \times 50 = 950 $
    $$

$$
    $Accounts = 15x = 15 \times 50 = 750$
    $$

15. Females in Production:
    $$
    F_{Production} = 950 - 800 = 150
    $$

16. Males in Accounts:
    $$
   $ A_m = \frac{15x + 150}{2} = \frac{750 + 150}{2} = \frac{900}{2} = 450$
    $$

17. Females in Accounts:
    $$
    A_f = A_m - 150 = 450 - 150 = 300
    $$

18. Females in Management = Males in Accounts = 450

19. Males in Management = 640 (from step 2)

20. Total employees in Management:
    $$
    640 + 450 = 1090
    $$

21. Total employees in HR:
    $$
    3540 - (950 + 750 + 1090) = 3540 - 2790 = 750
    $$

22. Let males in HR = $ H_m $, females in HR = $ 0.875 H_m $

Total HR employees:
    $$
  $  H_m + 0.875 H_m = 1.875 H_m = 750 \implies H_m = \frac{750}{1.875} = 400$
    $$

Females in HR:
    $$
    0.875 \times 400 = 350
    $$

23. Now, calculate total males and females in the company:

- Total males = Males in Management + Males in HR + Males in Production + Males in Accounts
    $$
    = 640 + 400 + 800 + 450 = 2290
    $$

- Total females = Females in Management + Females in HR + Females in Production + Females in Accounts
    $$
    = 450 + 350 + 150 + 300 = 1250
    $$

24. Difference between total males and females:
    $$
    2290 - 1250 = 1040
    $$

**Answer:** 1040

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