Time and Work made clear.
Think in work completed per day. Once everyone’s rate uses the same whole job, combining workers becomes addition rather than guesswork.
Key concepts in Time and Work
One-person rate
If A completes a job in a days, A completes 1/a of that job per day at a constant rate.
Working together
Add rates, not completion times. Combined time = 1 ÷ combined daily rate.
Efficiency ratio
For the same job, efficiency and time are inversely proportional. Twice the efficiency means half the time.
Pipes and leaks
Treat filling as positive work and emptying as negative work, provided both operate continuously.
Work-rate model: one shared job
| Worker | Units per day |
|---|---|
| A: 12 days for 36 units | 3 |
| B: 18 days for 36 units | 2 |
| Together | 5 |
| Time to finish | 36 ÷ 5 = 7.2 days |
A reliable solving method
- Choose one job or a convenient number of work units as the total.
- Convert every completion time into a rate.
- Separate intervals when the group of workers changes.
- Find remaining work before calculating the final time.
Step-by-step solved examples
A takes 12 days and B takes 18 days. How long together?
- Take total work as LCM(12,18) = 36 units.
- A does 3 units/day; B does 2 units/day. Together they do 5.
- Time = 36/5 = 7.2 days.
A takes 10 days; B takes 15. Both work for 3 days, then B leaves.
- Together they complete 1/10+1/15 = 1/6 per day.
- In 3 days, they finish 1/2. Remaining work is 1/2.
- A needs (1/2)/(1/10) = 5 more days; total time is 8 days.
A pipe fills in 6 hours; a leak empties a full tank in 12 hours.
- Net rate = 1/6−1/12 = 1/12 tank/hour.
- The rate stays positive, so the tank fills.
- Starting empty, it takes 12 hours, assuming constant rates.
Time and Work: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
A takes 8 days and B takes 24. Together?
Show answer and explanation — question 1
Answer B: 6 days
1/8+1/24 = 1/6, so they take 6 days.
A is twice as efficient as B. B takes 18 days. A takes?
Show answer and explanation — question 2
Answer A: 9 days
Time is inversely proportional to efficiency: 18/2 = 9.
A finishes 1/5 of a job daily. Work done in 3 days?
Show answer and explanation — question 3
Answer C: 3/5
3 × 1/5 = 3/5.
Six identical workers finish in 10 days. Five need?
Show answer and explanation — question 4
Answer D: 12 days
Total worker-days = 60; 60/5 = 12.
A fills in 4 hours; B empties in 8. Both opened on an empty tank?
Show answer and explanation — question 5
Answer C: 8 hours
Net filling rate = 1/4−1/8 = 1/8.
A takes 12 days, works alone for 3 days. Fraction left?
Show answer and explanation — question 6
Answer D: 3/4
A completes 3/12 = 1/4, leaving 3/4.
Frequently asked questions
Can I add 12 days and 18 days?
No. Add the work rates 1/12 and 1/18.
Why use the LCM as total work?
It makes daily rates whole numbers; the fraction-of-a-job method gives the same answer.
Choose one job or a convenient number of work units as the total. Find remaining work before calculating the final time.