Question

In an organization, there are four departments A, B, C and D with some accountants, managers, stenographers and office boys working together. Department A has 10 accountants, 8 managers, 7 stenographers and 3 office boys. The total monthly salary of all these employees is Rs. 2,37,500. Department B has 5 stenographers, 12 accountants, 6 managers and 7 office boys. The total monthly salary of all these employees is Rs. 2,31,500. Department C has 4 managers, 4 stenographers, 5 office boys and 7 accountants. The total monthly salary of all these employees is Rs. 1,51,000. If all the respective employees are paid equally in all the departments, then find the total monthly salary of 18 accountants, 11 managers, 10 stenographs and 10 office boys of department D?

• A. Rs. 2,85,500
• B. Rs. 3,25,500
• C. Rs. 3,60,500
• D. Rs. 3,85,500

Hint:

You do not need each individual salary; write the target group as a weighted sum of the three department totals.

Answer 1

The Correct Option is D

Step 1: Set up equations from the three departments.
Let \( x_1, x_2, x_3, x_4 \) be the salaries of an accountant, manager, stenographer and office boy.
A: \( 10x_1 + 8x_2 + 7x_3 + 3x_4 = 237500 \)
B: \( 12x_1 + 6x_2 + 5x_3 + 7x_4 = 231500 \)
C: \( 7x_1 + 4x_2 + 4x_3 + 5x_4 = 151000 \)

Step 2: Notice the system has 3 equations for 4 unknowns.
Individual salaries cannot be found, but the target combination \( 18x_1+11x_2+10x_3+10x_4 \) may still be written as \( p \times A + q \times B + r \times C \) for some constants \( p, q, r \).

Step 3: Match coefficients for manager and stenographer.
Manager: \( 8p+6q+4r=11 \). Stenographer: \( 7p+5q+4r=10 \).
Subtracting gives \( p+q=1 \), so \( q = 1-p \).

Step 4: Solve for r using the stenographer equation.
\( 4r = 10-7p-5q = 10-7p-5(1-p) = 5-2p \), so \( r = \dfrac{5-2p}{4} \).

Step 5: Substitute into the accountant equation.
\( 10p+12(1-p)+7\left(\dfrac{5-2p}{4}\right)=18 \)
Multiplying through by 4 and simplifying gives \( -22p+83=72 \), so \( p = \dfrac{1}{2} \).
Then \( q = \dfrac{1}{2} \) and \( r = 1 \).

Step 6: Verify with the office boy coefficient.
\( 3(0.5)+7(0.5)+5(1) = 1.5+3.5+5=10 \), which matches, confirming the values.

Step 7: Compute the target salary total.
\( 0.5(237500) + 0.5(231500) + 1(151000) = 118750+115750+151000 = 385500 \).

Final Answer:
The salary total for department D's group is Rs. 3,85,500. \[ \boxed{Rs. 3,85,500} \]