Total Work
Treat one complete job as 1 unit. Fractions then represent completed or remaining work.
⏱️ Time & Work
Relate work, time and efficiency; combine individual rates; distribute wages; and solve worker, pipe and cistern problems.
🎯 1. Learning Objectives
- Express one-day work as a work rate.
- Find individual and combined completion times.
- Compare efficiency and time ratios.
- Solve workers–days–hours problems.
- Divide wages according to work completed.
- Handle pipes, leaks and alternate-day work.
📖 2. Work, Time and Rate
Work Rate
If a person completes a job in n days, the person completes 1/n of the job per day.
Time
Time = Work ÷ Rate
Work Done
Work = Rate × Time
Key idea: add rates when people or pipes work together. Do not add their completion times.
👤 3. Individual Work
One-Day Work
If time = n days, rate = 1/n
Work in d Days
Work done = d/n
Remaining Work
Remaining = 1 − completed work
Time for a Fraction
Time = required fraction ÷ rate
🤝 4. Combined Work
If A finishes in a days and B in b days, their rates are 1/a and 1/b.
Combined Rate
1/a + 1/b
Combined Time
ab/(a+b)
Find B from A+B
B's rate = combined rate − A's rate
Three Workers
Rate = 1/a + 1/b + 1/c
⚡ 5. Efficiency and Time
Inverse Relationship
For the same work, a more efficient person requires less time.
Efficiency ∝ 1/Time
Ratio Rule
If efficiencies A:B = m:n, then their completion times are reversed.
Time A:Time B = n:m
Example: A is twice as efficient as B, so A needs half of B's time.
👷 6. Workers, Days and Hours
For equally efficient workers and fixed work:
Workers × Days × Hours per day = Constant
| Change | Effect on Time | Relationship |
|---|---|---|
| More workers | Fewer days | Inverse |
| More hours per day | Fewer days | Inverse |
| More work | More time | Direct |
| Higher efficiency | Less time | Inverse |
💵 7. Work and Wages
Wages are divided in the ratio of work completed. When working for the same duration, work is proportional to efficiency.
Same Duration
Wages ratio = efficiency ratio
Different Durations
Wages ratio = rate × time
🚰 8. Pipes and Cisterns
Inlet Pipe
A filling pipe contributes a positive rate.
Rate = +1/filling time
Outlet or Leak
An emptying pipe contributes a negative rate.
Rate = −1/emptying time
Net Rate
Sum of inlet rates − sum of outlet rates
Filling Time
Time = 1 ÷ net rate
🔁 9. Workers Joining, Leaving and Alternating
Join or Leave
Split the timeline into stages. Find work completed in each stage using the active combined rate.
Alternate Days
Calculate one full cycle first, count complete cycles, then handle the remaining fractional day.
🧪 10. Work-Rate Explorer
Choose a problem type, enter the values and click Calculate. Results remain hidden until the button is used.
Method:
🧾 11. Solved Examples
Example 1: Together
Problem: A takes 10 days and B takes 15 days.
Solution: Rate=1/10+1/15=1/6. Together they take 6 days.
Example 2: Find a Partner
Problem: A+B take 6 days; A alone takes 10 days.
Solution: B's rate=1/6−1/10=1/15. B takes 15 days.
Example 3: Workers
Problem: 8 workers finish in 15 days. Find time for 12 workers.
Solution: 8×15=12×d, so d=10 days.
Example 4: Pipe and Leak
Problem: Inlet fills in 8 h; outlet empties in 24 h.
Solution: Net rate=1/8−1/24=1/12. Tank fills in 12 hours.
💻 12. Applications in Programming
Task Scheduling
Estimate when parallel workers or processors will complete a collection of jobs.
Load Balancing
Assign work according to processor speed or worker efficiency.
Project Planning
Calculate team size, completion time and productivity changes.
Network Flow
Model positive input rates, negative output rates and net throughput.
⚠️ 13. Common Mistakes
Adding Times
Add work rates, not the separate completion times.
Ignoring Inverse Ratios
Higher efficiency means lower time for the same work.
Treating an Outlet as Positive
A leak or outlet subtracts from the filling rate.
Skipping Stages
When workers join or leave, calculate every time interval separately.
✍️ 14. Practice Problems
Solve each problem first, then use Show Solution.
1. A completes a job in 12 days. What work does A complete in 3 days?
Work=3×1/12=1/4.
2. A takes 8 days and B takes 24 days. Find their combined time.
Rate=1/8+1/24=1/6, so time=6 days.
3. Efficiencies A:B=4:5. Find their time ratio.
Time ratio is the inverse: 5:4.
4. 10 workers finish a job in 18 days. How many days will 15 workers need?
10×18=15×d, so d=12 days.
5. An inlet fills a tank in 6 hours and a leak empties it in 12 hours. Find the net filling time.
Net rate=1/6−1/12=1/12, so time=12 hours.
📝 15. Quick Revision
- One-day work = 1/completion time.
- Combined rate is the sum of active rates.
- Efficiency ratio is the inverse of time ratio.
- Workers × days × hours is constant for fixed work.
- Wages follow the ratio of completed work.
- Inlets are positive; outlets and leaks are negative.
INTERACTIVE PRACTICE
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⏱️ Time & Work Practice
Complete 20 questions. Score, every option, correct answers and explanations appear only after final submission.