Time and Work

Work rate, efficiency, combined work

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Time and Work problems deal with the relationship between the time taken to complete a task and the efficiency or rate at which workers operate. These problems typically involve calculating how long individuals or groups take to finish jobs, determining work rates, and solving scenarios involving combined efforts, efficiency variations, and partial work completion.

Key Formulas

Work Rate = 1 / Time (work done per unit time)\text{Work Rate = 1 / Time (work done per unit time)}
Time = Work / Rate\text{Time = Work / Rate}
If A can do a work in ’n’ days, A’s 1 day work = 1/n\text{If A can do a work in 'n' days, A's 1 day work = 1/n}
If A’s 1 day work = 1/n, A can finish the work in ’n’ days\text{If A's 1 day work = 1/n, A can finish the work in 'n' days}
Combined rate of A and B = (1/a + 1/b) where a, b are individual times\text{Combined rate of A and B = (1/a + 1/b) where a, b are individual times}
Time taken by A and B together = (a x b) / (a + b)\text{Time taken by A and B together = (a x b) / (a + b)}
If A is ’x’ times more efficient than B: If B takes ’n’ days, A takes n/x days\text{If A is 'x' times more efficient than B: If B takes 'n' days, A takes n/x days}
Man-Days = Number of men x Number of days (constant for same work)\text{Man-Days = Number of men x Number of days (constant for same work)}
M x D = M x D (for same work with different men and days)\text{M x D = M x D (for same work with different men and days)}
M x D x H = M x D x H (including hours per day)\text{M x D x H = M x D x H (including hours per day)}
Work done = (Days worked) / (Total days to complete)\text{Work done = (Days worked) / (Total days to complete)}
Remaining work = 1 - Work already done\text{Remaining work = 1 - Work already done}

Key Concepts

Basic Work Rate Concept

The fundamental principle is that if a person can complete a work in 'n' days, their daily work rate is 1/n of the total work. For example, if A completes work in 10 days, A does 1/10 of the work each day. This fraction represents the portion of work completed per day and forms the basis for all time and work calculations.

Combined Work (Working Together)

When multiple people work together, their individual work rates add up. If A does 1/a work per day and B does 1/b work per day, together they do (1/a + 1/b) work per day. The time taken together is the reciprocal: (axb)/(a+b). For three people A, B, C: combined rate = 1/a + 1/b + 1/c, and time = 1/(combined rate).

Efficiency and Comparison

Efficiency refers to how quickly someone completes work relative to another. If A is twice as efficient as B, A takes half the time B takes. If A takes 20 days, B takes 40 days. Efficiency ratios help determine time ratios: Time ratio is inverse of efficiency ratio. Higher efficiency means less time required.

Man-Days (Men and Work)

The concept of man-days states that for a fixed amount of work, the product of workers and days remains constant. M x D = M x D . If 10 men take 20 days, then 20 men take 10 days. When hours per day vary: M x D x H = M x D x H . This principle applies to identical work conditions.

Partial Work and Remaining Work

When workers leave or join partway through a project, calculate work done before the change and remaining work after. If A works for 5 days alone (completing 5/a of work), the remaining (1 - 5/a) work is done by others. Track individual contributions and combine them to verify total work equals 1 (whole work).

Alternating Work (A and B Working on Alternate Days)

When two workers alternate, calculate work done in complete cycles. If A works day 1 and B works day 2: in 2 days they complete (1/a + 1/b) work. Find full cycles needed, then determine who finishes the remaining work. For example, if 2-day cycle completes 1/6 work, after 5 cycles (10 days), 5/6 is done; remaining 1/6 is done by A on day 11.

Solved Examples

Problem 1:

A can complete a work in 15 days and B can complete the same work in 20 days. If they work together, how many days will they take to complete the work?

Solution:

Step 1: Find individual work rates.
A's 1 day work = 1/15
B's 1 day work = 1/20

Step 2: Calculate combined work rate.
(A + B)'s 1 day work = 1/15 + 1/20
= 4/60 + 3/60
= 7/60

Step 3: Find time taken together.
Time = 1 / (7/60) = 60/7 days
= 8(4/7) days

Answer: 60/7 days or approximately 8.57 days

Problem 2:

A and B together can complete a work in 12 days. B and C together can complete it in 15 days. C and A together can complete it in 20 days. How many days will A, B, and C take to complete the work together?

Solution:

Step 1: Set up equations from given information.
(A + B)'s 1 day work = 1/12 ... (1)
(B + C)'s 1 day work = 1/15 ... (2)
(C + A)'s 1 day work = 1/20 ... (3)

Step 2: Add all three equations.
2(A + B + C)'s 1 day work = 1/12 + 1/15 + 1/20
= 5/60 + 4/60 + 3/60
= 12/60
= 1/5

Step 3: Find combined rate of all three.
(A + B + C)'s 1 day work = (1/5) / 2 = 1/10

Step 4: Calculate time taken.
Time = 1 / (1/10) = 10 days

Answer: 10 days

Problem 3:

A is twice as efficient as B. Together they complete a work in 18 days. How many days will A alone take to complete the work?

Solution:

Step 1: Establish efficiency relationship.
If B's 1 day work = 1/x, then A's 1 day work = 2/x (since A is twice as efficient)

Step 2: Use combined work information.
(A + B)'s 1 day work = 1/18
2/x + 1/x = 1/18
3/x = 1/18
x = 54

Step 3: Find A's time.
B takes 54 days
A takes 54/2 = 27 days (since twice as efficient)

Verification: 1/27 + 1/54 = 2/54 + 1/54 = 3/54 = 1/18

Answer: 27 days

Problem 4:

20 men can complete a work in 30 days. After working for 10 days, 5 more men join. In how many more days will the work be completed?

Solution:

Step 1: Calculate total work in man-days.
Total work = 20 men x 30 days = 600 man-days

Step 2: Calculate work done in first 10 days.
Work done = 20 men x 10 days = 200 man-days

Step 3: Find remaining work.
Remaining work = 600 - 200 = 400 man-days

Step 4: Calculate new team capacity.
New team = 20 + 5 = 25 men

Step 5: Find days needed to complete remaining work.
Days = 400 man-days / 25 men = 16 days

Answer: 16 more days

Tips & Tricks

  • Always convert 'X can do work in N days' to daily rate = 1/N immediately. This makes calculations easier.
  • For combined work problems, remember the shortcut: If A takes 'a' days and B takes 'b' days, together they take (axb)/(a+b) days.
  • When dealing with efficiency ratios, the time ratio is inverse. If A:B efficiency = 3:2, then time ratio A:B = 2:3.
  • For alternating work problems, calculate work per complete cycle first, then handle the remainder separately.
  • In man-days problems, keep track of the total work constant. M D = M D is your best friend.
  • When workers leave or join mid-project, split the problem: calculate work done before change and work remaining after change.
  • For pipes and cisterns (similar concept), treat filling pipes as positive work and emptying pipes as negative work.
  • Always verify your answer by checking if the computed rates add up correctly to complete 1 unit of work.

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