Time and Work problems deal with the relationship between work, time and efficiency. These questions usually ask how quickly a person, machine or group can complete a particular job.
The most important idea is that a person's work depends on the amount of work completed in a given time.
Efficiency = Work / Time
Time = Work / Efficiency
If the total work is fixed, then efficiency and time are inversely proportional.
Efficiency × Time = Constant
If A can complete a work in x days, then A's work in one day is:
A's 1-day work = 1/10
Therefore, A completes 1/10 of the work per day.
If a person completes a fraction of work every day, total time can be found by dividing the complete work by daily work.
Time = 1 ÷ 1/8
= 8 days
If A can complete a work in x days and B can complete it in y days, their individual one-day works are 1/x and 1/y.
= 1/x + 1/y
= (x + y)/xy
Therefore, time taken together is:
Time = (12 × 18)/(12 + 18)
= 216/30
= 7.2 days
Efficiency tells us how much work a person can complete in a given period of time.
Thus, a person who takes less time has greater efficiency, provided the total work is the same.
Efficiency ratio:
A : B = 20 : 10
= 2 : 1
Therefore, A is twice as efficient as B.
When three persons work together, add their individual one-day works.
One-day work
= 1/12 + 1/15 + 1/20
LCM = 60
= 5/60 + 4/60 + 3/60
= 12/60
= 1/5
Time = 5 days
If A completes a job in x days, then work completed by A in n days is:
Work = 6/15
= 2/5
Remaining work = 1 − 2/5
= 3/5
Sometimes two people start together and one person leaves after working for a certain number of days. First calculate the work already completed and then find the remaining work.
Together one-day work:
= 1/10 + 1/15
= 1/6
Work in 3 days = 3 × 1/6
= 1/2
Remaining work = 1 − 1/2
= 1/2
For workers having the same efficiency, the amount of work is directly proportional to the number of workers and the number of days.
Therefore:
M₁D₁ = M₂D₂
12 × 15 = M × 10
M = 180/10
= 18 men
When the number of working hours per day changes, hours must also be included.
For the same work and efficiency:
M₁D₁H₁ = M₂D₂H₂
10 × 8 × 12 = 12 × 10 × x
960 = 120x
x = 8 days
If A : B efficiency = m : n
then A : B time = n : m
A's efficiency = 150% of B
Efficiency ratio A : B = 3 : 2
Time ratio A : B = 2 : 3
A's time = 12 × 2/3
= 8 days
In many examination questions, it is convenient to assume the total work as the LCM of the individual completion times.
A's efficiency = 24/8 = 3 units/day
B's efficiency = 24/12 = 2 units/day
Together = 3 + 2 = 5 units/day
Time = 24/5
= 4.8 days
When two people work on alternate days, calculate the work completed in one complete cycle and then calculate the remaining work.
A's one-day work = 1/6
B's one-day work = 1/12
Work in 2 days = 1/6 + 1/12
= 1/4
After 8 days:
4 cycles × 1/4 = 1
Therefore, the work is completed in 8 days.
If workers have different efficiencies and work for different periods, the amount of work contributed by each person depends on both efficiency and time.
A's work = 6 × 4 = 24 units
B's work = 8 × 3 = 24 units
Both contribute equal amounts of work.
When workers are paid according to the amount of work they perform, wages are proportional to work done.
Wage ratio = Work ratio
Efficiency ratio = 2 : 1
Wage ratio = 2 : 1
Total parts = 3
A = 900 × 2/3 = ₹600
B = 900 × 1/3 = ₹300
If a worker completes a certain percentage of work per day, the number of days can be calculated using:
Time = 100/8
= 12.5 days
Completed work = 3/7 + 2/7
= 5/7
Remaining = 1 − 5/7
= 2/7
| Quantity | Relationship |
|---|---|
| Work & Time | Work = Efficiency × Time |
| Efficiency & Time | Inverse relationship for fixed work |
| Workers & Work | Direct relationship |
| Workers & Time | Inverse relationship for fixed work |
| Hours & Work | Direct relationship |
| Wages & Work | Direct relationship |
| Concept | Formula |
|---|---|
| Basic Work | W = E × T |
| Efficiency | E = W/T |
| Time | T = W/E |
| A's 1-day Work | 1/x if A completes work in x days |
| A + B | 1/x + 1/y |
| A + B Time | xy/(x+y) |
| A + B + C | 1/x + 1/y + 1/z |
| Men & Days | M₁D₁ = M₂D₂ |
| Men, Days & Hours | M₁D₁H₁ = M₂D₂H₂ |
| Work Done | Efficiency × Time |
| Remaining Work | Total Work − Completed Work |
| Efficiency Ratio | Inverse of Time Ratio |
| Wages | Wages ∝ Work Done |
1-day work = 1/15 + 1/10
= 2/30 + 3/30
= 5/30 = 1/6
Time = 6 days
20 × 18 = M × 12
M = 360/12
= 30 men
Efficiency A : B = 3 : 1
Time A : B = 1 : 3
A's time = 24/3
= 8 days
12 × 8 × 15 = 16 × 6 × x
1440 = 96x
x = 15 days
- A can complete a work in 20 days and B in 30 days. How many days will they take together?
- 15 men complete a work in 24 days. How many men are required to complete the same work in 18 days?
- A is twice as efficient as B. If B takes 30 days, how many days will A take?
- 8 workers working 6 hours a day complete a job in 15 days. How many days will 12 workers working 8 hours a day require?
- A can do a work in 12 days and B in 18 days. Find the fraction of work completed by them together in one day.
- A completes 1/5 of a work in one day. How many days are required for the complete work?
- 10 men complete a work in 16 days. How many men are required to finish it in 8 days?
- A can complete a job in 8 days and B in 24 days. Find their combined time.
- A worker completes 10% of a work per day. Find the time required for the complete work.
- A and B together complete a work in 6 days. If A alone can complete it in 10 days, find the time taken by B alone.
2. 20 men
3. 15 days
4. 7.5 days
5. 5/36 of the work
6. 5 days
7. 20 men
8. 6 days
9. 10 days
10. 15 days
1. Identify the total work.
2. Convert completion time into one-day work.
3. Add efficiencies when people work together.
4. Subtract work when a person leaves.
5. Use LCM when fractions become complicated.
6. For the same work, more efficiency means less time.
7. For equal efficiency, more workers mean fewer days.
8. Include working hours when daily hours change.
9. Check whether the final answer is logically reasonable.
