Master the concepts of time and work with our comprehensive guide. Learn about efficiency, work rates, and complex problem-solving techniques.
Work rate is the amount of work done per unit time. It's a fundamental concept in time and work problems.
Basic Formula:
Work = Rate × Time
Rate = Work/Time
Time = Work/Rate
Example 1: Basic Work Rate
If a person can complete a work in 10 days, their work rate is 1/10 of the work per day.
Work Rate = 1/10 work per day
60% of work completed
When multiple people work together, their work rates are added.
Combined Work Rate Formula:
Total Work Rate = Sum of Individual Work Rates
Time Taken = 1/Total Work Rate
Example 2: Combined Work
Person A can complete work in 6 days
Person B can complete work in 8 days
Combined work rate = 1/6 + 1/8 = 7/24 work per day
Time taken together = 24/7 days
Efficiency is the ratio of work done to the time taken, often expressed as a percentage.
Efficiency Formula:
Efficiency = (Work Done/Time Taken) × 100%
Time Taken = Work Done/Efficiency
Example 3: Efficiency Calculation
If a worker completes 3/4 of work in 6 days:
Efficiency = (3/4 ÷ 6) × 100% = 12.5% per day
When comparing efficiencies, we can determine the ratio of work rates.
Efficiency Ratio:
Efficiency Ratio = Time taken by B/Time taken by A
Example 4: Efficiency Comparison
Worker A takes 8 days, Worker B takes 12 days
Efficiency ratio = 12/8 = 3/2
Worker A is 1.5 times more efficient than Worker B
Pipes can fill (inlet) or empty (outlet) a cistern at different rates.
Pipe Work Rate:
Net Work Rate = Sum of Inlet Rates - Sum of Outlet Rates
Time to Fill = 1/Net Work Rate
Example 5: Pipe Problem
Inlet pipe fills in 6 hours
Outlet pipe empties in 8 hours
Net rate = 1/6 - 1/8 = 1/24 per hour
Time to fill = 24 hours
When multiple pipes are involved, we need to consider their combined effect.
Multiple Pipes Formula:
Total Work Rate = Σ(Inlet Rates) - Σ(Outlet Rates)
Example 6: Multiple Pipes
Pipe A fills in 4 hours
Pipe B fills in 6 hours
Pipe C empties in 8 hours
Net rate = 1/4 + 1/6 - 1/8 = 11/24 per hour
Time to fill = 24/11 hours
Wages are distributed based on work done or time taken.
Wage Distribution Formula:
Wage ∝ Work Done
Wage ∝ 1/Time Taken
Example 7: Wage Distribution
Total wage = ₹1200
Worker A takes 6 days
Worker B takes 8 days
Wage ratio = 8:6 = 4:3
Worker A gets ₹685.71
Worker B gets ₹514.29
Wages can be distributed based on efficiency ratios.
Efficiency-Based Wage Formula:
Wage Ratio = Efficiency Ratio
Example 8: Efficiency-Based Wages
Total wage = ₹1000
Worker A is 1.5 times more efficient than Worker B
Wage ratio = 3:2
Worker A gets ₹600
Worker B gets ₹400
When workers take breaks, we need to account for rest periods.
Work and Rest Formula:
Effective Work Rate = Work Rate × (Work Time/Total Time)
Example 9: Work and Rest
Worker works 6 hours and rests 2 hours
Work rate = 1/8 per hour
Effective work rate = (1/8) × (6/8) = 3/32 per hour
Time to complete work = 32/3 hours
When work rates change over time, we need to consider different phases.
Variable Work Rate Formula:
Total Work = Σ(Work Rate × Time) for each phase
Example 10: Variable Work Rate
Worker A: 1/10 for first 4 days, 1/8 for next 4 days
Total work = (1/10 × 4) + (1/8 × 4) = 0.4 + 0.5 = 0.9
Remaining work = 0.1
Time to complete = 0.1 ÷ (1/8) = 0.8 days
Important Theorems:
Example 11: Theorem Application
A takes 6 days, B takes 8 days
Together they take = (6×8)/(6+8) = 48/14 = 24/7 days
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