The average salary of employees in three departments P, Q and R in the company is ₹ 85,000, ₹ 75,000 and ₹ 83,000 respectively. If the average salary of department P and Q is ₹79,000 and the average salary of department Q and R is ₹80,000, then what will be the average salary of the three departments?

Given:

Average salary of department P = ₹85,000

Average salary of department Q = ₹75,000

Average salary of department R = ₹83,000

Average salary of departments P and Q = ₹79,000

Average salary of departments Q and R = ₹80,000

Formula used:

Average salary = (Total salary of the department) / (Number of employees)

Calculation:

Let the number of employees in department P be Pn, in department Q be Qn, and in department R be Rn.

For departments P and Q, the combined average salary is ₹79,000, so:

⇒ (P_n × ₹85,000 + Q_n × ₹75,000) / (P_n + Q_n) = ₹79,000

⇒ P_n × ₹85,000 + Q_n × ₹75,000 = ₹79,000 × (P_n + Q_n)

⇒ Pn × ₹85,000 + Qn × ₹75,000 = ₹79,000 × Pn + ₹79,000 × Qn

⇒ Pn × ₹85,000 - ₹79,000 × Pn =  ₹79,000 × Qn - Qn × ₹75,000

⇒ Pn × ₹6,000 =  ₹4,000 × Qn

⇒ Pn /Qn =  ₹4,000/₹6,000 = 2/3

Similarly, for departments Q and R, the combined average salary is ₹80,000, so:

⇒ (Q_n × ₹75,000 + R_n × ₹83,000) / (Q_n + R_n) = ₹80,000

⇒ Q_n × ₹75,000 + R_n × ₹83,000 = ₹80,000 × (Q_n + R_n)

⇒ Qn × ₹75,000 + Rn × ₹83,000 = ₹80,000 × Qn + ₹80,000 × Rn

⇒ ₹80,000 × Qn - Qn × ₹75,000  = Rn × ₹83,000 - ₹80,000 × Rn

⇒ Qn × ₹5,000  = Rn × ₹3,000 

⇒ Qn/Rn = ₹3,000/₹5,000 = 3/5

On simplifying the ratio we get,

Pn : Qn : Rn = 2 : 3 : 5

Let’s simplify the total average salary of all three departments:

⇒ The total salary of all three departments = (P_n × ₹85,000) + (Q_n × ₹75,000) + (R_n × ₹83,000)

⇒ (2 × ₹85,000) + (3 × ₹75,000) + (5 × ₹83,000)

⇒ ₹1,70,000 + ₹2,25,000 + ₹4,15,000 = 8,10,000

Average salary of the three departments = 8,10,000/(2 + 3 + 5) = 81000

∴ The average salary of the three departments will be ₹81,000.