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Difficulty: Very hardRatio and Proportion Word Problems

At a corporate law firm, the ratio of Partners to Associates was initially 3:83 : 8, and the ratio of Associates to Paralegals was initially 4:54 : 5. During an internal restructuring, 1515 Associates were promoted to Partners, and 22 Paralegals resigned. As a result of these two changes, the ratio of Partners to Paralegals became 5:85 : 8. Shortly thereafter, the firm conducted a hiring drive, recruiting additional Associates until the ratio of Associates to Paralegals became 7:67 : 6, while the numbers of Partners and Paralegals remained unchanged. What was the total number of staff members (Partners, Associates, and Paralegals) at the firm after the hiring drive?

Answer: 134 staff members

Answer

134
The solution unifies the initial ratios into 3 : 8 : 10, sets up the algebraic equations reflecting both promotions and resignations to find the multiplier k = 5, updates the staff counts, and applies the final ratio to find 56 Associates, yielding a final total of 134 staff members.

Step-by-Step Solution

1
Unify initial two-variable ratios into a continuous three-part ratio.
Partners : Associates : Paralegals = 3 : 8 : 10
Associates is the common term. Scaling 4 : 5 by 2 gives 8 : 10, matching the 8 parts in the Partners-to-Associates ratio.
2
Formulate algebraic expressions for post-restructuring staff counts.
Partners = 3k + 15, Associates = 8k - 15, Paralegals = 10k - 2
Promotions transfer 15 from Associates to Partners, and 2 Paralegals leave.
3
Solve for the ratio multiplier k using the new Partner-to-Paralegal ratio.
k = 5
Setting (3k + 15) / (10k - 2) = 5 / 8 yields 24k + 120 = 50k - 10, so 26k = 130.
4
Calculate exact staff counts after restructuring.
Partners = 30, Associates = 25, Paralegals = 48
Substitute k = 5 into the expressions from Step 2.
5
Determine final Associate count and total staff after hiring drive.
Final Associates = 56; Total Staff = 134
Associates = (7/6) * 48 = 56. Summing all roles gives 30 + 56 + 48 = 134.

Key Concept

Multi-part ratio unification, internal transfer ratio alteration, and sequential proportion scaling
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Topics

Ratio and Proportion Word Problems

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