2.8 Combined Average of Two or More Groups

Already introduced under Weighted Average (Section 2.4), this deserves a dedicated shortcut for the “two groups” case, since it is the most exam-frequent form.

Formula: For two groups of sizes n_1 and n_2 with averages A_1 and A_2: Combined Average = (n_1A_1 + n_2A_2)/(n_1+n_2)

Fast alligation shortcut: The combined average always lies between A_1 and A_2, closer to whichever group is larger. The ratio in which the combined average divides the gap between A_1 and A_2 equals n_2 : n_1 (inverse of the group sizes) — this is the Alligation Rule.

(n_1)/(n_2) = (A_2 - Combined Average)/(Combined Average - A_1)

Solved Example 22: The average score of boys in a class is 70 and of girls is 80. If there are 30 boys and 20 girls, find the combined average score of the class.

Combined average = (30×70 + 20×80)/50 = (2100+1600)/50 = 3700/50 = 74.

Check by alligation: gap = 80−70 = 10, split in ratio n₂:n₁ = 20:30 = 2:3. So combined average = 70 + (2/5)×10 = 70+4 = 74 ✓.

Solved Example 23: Three sections of a class have 20, 25, and 30 students with average marks 60, 65, and 70 respectively. Find the average marks of all 75 students.

Combined average = (20×60 + 25×65 + 30×70)/75 = (1200+1625+2100)/75 = 4925/75 = 65.67 (approx).

Solved Example 23(a): The average salary of all employees in a company is ₹35,000. The average salary of 30 managers is ₹60,000, and the average salary of the remaining 120 non-managerial staff is a certain value. Find the average salary of the non-managerial staff. (Total employees = 150.)

Total salary of all employees = 150×35,000 = 52,50,000. Total salary of managers = 30×60,000 = 18,00,000. Total salary of non-managerial staff = 52,50,000 − 18,00,000 = 34,50,000. Average salary of non-managerial staff = 34,50,000/120 = ₹28,750.

This “find the missing group’s average, given the combined average” variant is common in RRB NTPC and appears frequently as a DI sub-question as well — always isolate the unknown group’s total by subtracting the known group’s total from the grand total.

Solved Example 23(b): The average marks of a class of 100 students is 68. The class has three sections, A, B, and C, with 30, 40, and 30 students respectively. Section A’s average is 60 and Section B’s average is 70. Find Section C’s average.

Total marks of all 100 students = 100×68 = 6800. Section A’s total = 30×60 = 1800. Section B’s total = 40×70 = 2800. Section C’s total = 6800 − 1800 − 2800 = 2200. Section C’s average = 2200/30 = 73.33 (approx).

This extends the “missing group” idea of Example 23(a) to a three-group situation — always subtract off all the known groups’ totals from the grand total to isolate the unknown group’s total, regardless of how many groups there are.

Solved Example 23(c): In a school of 50 people (teachers and students combined), the ratio of the number of teachers to students is 2:3. The average age of the teachers is 40 years, and that of the students is 15 years. Find the combined average age of the entire school.

This is the edge case where weights are given as a ratio rather than as raw counts — a form very common in SSC CGL Tier-II. Ratio 2:3 of a total of 50 gives teachers = (2/5)×50 = 20, students = (3/5)×50 = 30. Combined average = (20×40 + 30×15)/50 = (800+450)/50 = 1250/50 = 25 years.