Read the information given below and answer the question that follows. The…

Concept

Word problems with several interlinked totals are solved by translating every English statement into a linear equation in named variables, then eliminating one variable at a time. “P is k more than Q” means P = Q + k, and “P is k less than Q” means P = Q − k. The “total staff” of a college is its lab assistants plus its professors.

Set up the variables

Let the lab assistants be LA, LB, LC and the professors be PA, PB, PC for colleges A, B and C respectively.

Translate each statement

1. LA + PA = 82 (staff in A).

2. LB = LC + 4 (B’s lab assistants are 4 more than C’s).

3. PC = LC + 6 (C’s lab assistants are 6 less than C’s professors).

4. PA = LB − 44 (A’s professors are 44 less than B’s lab assistants).

5. LC + PC = (LA + PA) + 24 = 82 + 24 = 106 (staff in C is 24 more than staff in A).

6. PA + PB + PC = 102 (professors across all colleges).

Solve step by step

1. From (5) and (3): LC + (LC + 6) = 106, so 2LC = 100 and LC = 50; hence PC = 56.

2. From (2): LB = 50 + 4 = 54.

3. From (4): PA = 54 − 44 = 10, and from (1): LA = 82 − 10 = 72.

4. From (6): PB = 102 − PA − PC = 102 − 10 − 56 = 36.

5. Total staff in B = LB + PB = 54 + 36 = 90.

Cross-check

• Staff in A = 72 + 10 = 82 ✓; staff in C = 50 + 56 = 106 = 82 + 24 ✓.

• Professors total = 10 + 36 + 56 = 102 ✓. All conditions hold, so the total staff in B is 90.