Question 1

A task requires 120 person-days of work. Worker M can complete it in 30 days, and worker N can complete it in 40 days. M and N work together for 10 days, then M takes a 5-day break while N continues. After M returns, both work together until completion. What is the ratio of wages earned by M to N if they are paid proportionally to work done, and the total wage is ₹12,000?

Accepted Answer

Step 1: M's rate = 120/30 = 4 units/day. N's rate = 120/40 = 3 units/day. Step 2: Phase 1 (10 days, both): Work = 10(4+3) = 70 units. Step 3: Phase 2 (5 days, N only): Work = 5(3) = 15 units. Total so far = 85 units. Remaining = 120 − 85 = 35 units. Step 4: Phase 3 (both together): Days needed = 35 ÷ (4+3) = 35 ÷ 7 = 5 days. Step 5: M's total work = 10(4) + 5(4) = 40 + 20 = 60 units. N's total work = 10(3) + 5(3) + 5(3) = 30 + 15 + 15 = 60 units. Step 6: Both did equal work! Ratio = 60:60 = 1:1. Step 7: M's wage = N's wage = ₹6,000 each.

Question 2

A project is divided into two parts: Part A requires 60 person-days and Part B requires 40 person-days. Worker X completes Part A in 12 days and Part B in 10 days. Worker Y completes Part A in 15 days and Part B in 8 days. If X and Y work together on Part A for 4 days, then X alone finishes Part A, and Y alone finishes Part B, what is the ratio of time taken by X to complete his remaining work to the time taken by Y to complete her remaining work?

Accepted Answer

Step 1: X's rates: Part A = 60/12 = 5 units/day; Part B = 40/10 = 4 units/day. Y's rates: Part A = 60/15 = 4 units/day; Part B = 40/8 = 5 units/day. Step 2: X and Y work together on Part A for 4 days: Work done = 4(5+4) = 36 units. Remaining Part A = 60 − 36 = 24 units. Step 3: X completes remaining Part A alone: Time = 24 ÷ 5 = 4.8 days. Step 4: Y completes Part B alone: Time = 40 ÷ 5 = 8 days. Step 5: Ratio = 4.8 : 8 = 24/5 : 8 = 24 : 40 = 3 : 5.

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