Read the information and answer the question given below. The information is…

Concept

A linear-equations word problem is solved by assigning a variable to each unknown quantity, translating every given statement into an equation, and then solving the system by substitution. Each “more than / less than / total” phrase becomes one equation; when the number of independent equations equals the number of unknowns, the values are fixed uniquely.

Application

Let lab assistants and professors be: A → La, Pa; B → Lb, Pb; C → Lc, Pc. Translate each statement and solve in order:

1. A total: La + Pa = 82.

2. C total is 24 more than A total: Lc + Pc = 82 + 24 = 106.

3. C lab assistants are 6 less than C professors: Lc = Pc − 6. Substitute into the C total: (Pc − 6) + Pc = 106 → 2Pc = 112 → Pc = 56, so Lc = 56 − 6 = 50.

4. B lab assistants are 4 more than C lab assistants: Lb = Lc + 4 = 50 + 4 = 54.

5. Professors in A are 44 less than lab assistants in B: Pa = Lb − 44 = 54 − 44 = 10.

6. From the A total: La = 82 − Pa = 82 − 10 = 72.

7. Required average = (La + Lc) / 2 = (72 + 50) / 2 = 122 / 2 = 61.

Cross-check

Verify every condition with La=72, Pa=10, Lb=54, Lc=50, Pc=56 and Pb = 102 − Pa − Pc = 102 − 10 − 56 = 36: A total 72+10=82 ✓; C total 50+56=106 = 82+24 ✓; Lb−Lc = 54−50 = 4 ✓; Pc−Lc = 56−50 = 6 ✓; Lb−Pa = 54−10 = 44 ✓; total professors 10+36+56 = 102 ✓. All conditions hold, so the average is 61.