Time & Work

 Time & Work: In-Depth Explanation

1. Introduction

The topic of Time & Work in quantitative aptitude is critical for understanding how different variables such as time, work, and efficiency interact. This is particularly important for competitive exams, where problem-solving speed and accuracy are crucial.

2. Basic Concepts

1. Work (W): Typically, work is considered as a single unit task or job, but it can also be divided into smaller fractions for complex problems.
2. Time (T): This is the period required to complete a given amount of work. It is usually measured in days, hours, or minutes.
3. Efficiency (E): This represents the rate at which work is done. Higher efficiency means more work completed in a given time. It is often expressed in terms of work units per time unit (e.g., work/day).

3. Key Formulas and Principles

1. Work and Time Relationship:
• If a person (or a machine) can complete a piece of work in T days, then the work done in one day is $\frac{1}{T}$​. 

$Work per day=\frac{1}{T}$

• Conversely, if a person completes $\frac{1}{T}$​ work in one day, the total time taken to complete the work is T days.

$Total time=T$

1. Combined Work:
• If two persons A and B can complete a work in A and B days respectively, their combined one day’s work is:

$\frac{1}{A}+\frac{1}{B}$

• The total time taken for both to complete the work together is:

$Total time=\frac{1}{(\frac{1}{A}+\frac{1}{B})}$

1. Efficiency and Time:
• If person A is x times more efficient than person B, then the time taken by A to do the same work is $\frac{1}{x}$​ times that taken by B. 

$TA=\frac{T_B}{x}$

1. Work Done by Multiple People:
• If A, B, and C working together can complete a work in T days, then their individual work rates can be summed up as:

$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{T}$

4. Sub-Topics and Example Problems

1. Simple Work Problems:
• Concept: Calculating the time taken by one or more workers to complete a job.
• Example: If A can do a piece of work in 5 days and B can do it in 10 days, how long will it take for them to finish it together?
• Solution: Their combined one day’s work is: 
$\frac{1}{5}+\frac{1}{10}+\frac{2}{10}+\frac{1}{10}=\frac{3}{10}$
• Total time to finish the work:
$Total time=\frac{1}{\frac{3}{10}}=\frac{10}{3}\approx 3.33 days$


2. Work Distribution Problems:
• Concept: Dividing the work among different people and calculating their individual times.
• Example: If A, B, and C together can complete a work in 10 days, and A alone can do it in 20 days, B alone in 30 days, find the time taken by C alone.
• Solution: Let C's time to complete the work be C days. Their combined one day’s work is: 
$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{10}$
• Substituting the values:
$\frac{1}{20}+\frac{1}{30}+\frac{1}{C}=\frac{1}{10}$
• Solving for C:
$\frac{1}{C}=\frac{1}{10}-(\frac{1}{20}+\frac{1}{30})$


$\frac{1}{C}=\frac{6-5}{60}=\frac{1}{10}$


$\frac{1}{C}=\frac{1}{10}-\frac{1}{12}$
• Therefore:
C = 60
3. Efficiency-Based Problems:
   • Concept: Comparing the efficiencies of different workers and calculating the time based on their efficiencies.
   • Example: If A is twice as efficient as B and together they complete a work in 12 days, how long will it take for A to do the work alone?
     • Solution: Let B's time to complete the work alone be $T_B$​ days. Therefore, A's time to complete the work alone is $\frac{T_B}{2}$ ​​days.
       • Their combined one day’s work is:
     • $\frac{1}{\frac{T_B}{2}}+\frac{1}{T_B}=\frac{1}{12}$
       • Solving for $T_B$​:
     • $\frac{2}{T_B}+\frac{1}{T_B}=\frac{1}{12}$
     • $\frac{3}{T_B}=\frac{1}{12}$
     • $T_B=36$
       • Therefore, A's time to complete the work alone is:
     • $T_A=\frac{T_B}{\mathrm{2 }}=$18 Days
4. Alternate Work Problems:
   • Concept: Solving problems where workers perform tasks alternately.
   • Example: A and B can complete a work in 10 and 15 days respectively. They work alternately starting with A. How long will it take to complete the work?
     • Solution: A works for the first day, B works for the second day, and so on.
       • Work done in 2 days:
     • Work done by A in one day = $\frac{1}{10}$
     • Work done by B in one day = $\frac{1}{15}$
     • Work done in 2 days = $\frac{1}{10}+\frac{1}{15}=\frac{3}{30}+\frac{2}{30}=\frac{1}{6}$​
       • Time to complete the work:
     • Total cycles (2 days) = $\frac{1}{\frac{1}{6}}=6 \mathrm{cycles}$
       • Remaining work:
     • $1-(\frac{5}{6}\times 6)=1-1=0$
       • Total days taken:
     • 2×6 = 12 days
5. Work with Breaks or Intervals:
   • Concept: Calculating work done when there are interruptions or breaks.
   • Example: A can complete a work in 10 days but takes a break after every 2 days for 1 day. How long will it take to complete the work?
     • Solution: Calculate the effective work done in the cycle.
       • In 3 days, A works for 2 days.
       • Work done in 2 days:
     • $\frac{2}{10}=\frac{1}{5}$​
       • Total work cycles needed:
     • Total cycles = $\frac{1}{\frac{1}{5}}=5 cycles$
       • Total days including breaks:
     • $$5×3 = 15 days

5. Advanced Concepts

1. Work Equivalence:
   • Concept: Problems involving conversion of work units or tasks into equivalent work done by others.
   • Example: If A can complete a work in 12 days, and B is 50% more efficient than A, how long will B take to complete the work?
     • Solution: B’s efficiency is 1.5 times that of A.
       • A’s work rate:
     • $\frac{1}{12} work/day$
       • B’s work rate:
     • $1.5\times \frac{1}{12}=\frac{3}{24}=\frac{1}{8}work/day$
       • Time taken by B:
     • $T_B=\frac{1}{\frac{1}{8}}=8 Days$
2. Fractional Work:
   • Concept: Dealing with problems involving fractions of work completed by different workers.
   • Example: If A can do half the work in 10 days, how long will it take for A to complete the full work?
     • Solution: A completes half the work in 10 days, so:
       • Full work time:
     • $T_A=2\times 10=20 days$
3. Complex Work Scenarios:
• Concept: Solving complex problems involving multiple workers, different efficiencies, and varying time schedules.
• Example: If A, B, and C together can complete a work in 6 days, B and C together can complete it in 10 days, and A and B together in 8 days, find the time taken by each to complete the work individually.
• Solution: Let the times taken by A, B, and C be A, B, and C days respectively.
• Combined work rates:

$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{6}$

$\frac{1}{B}+\frac{1}{C}=\frac{1}{10}$

$\frac{1}{A}+\frac{1}{B}=\frac{1}{8}$

• Solving for $\frac{1}{C}$ :

$\frac{1}{C}=\frac{1}{6}-\frac{1}{8}=\frac{4-3}{24}=\frac{1}{24}$

C = 24 days

• Solving for $\frac{1}{B}$ :

$\frac{1}{B}=\frac{1}{10}-\frac{1}{24}$

$\frac{1}{B}=\frac{24-10}{240}=\frac{14}{240}=\frac{7}{120}$

$B=\frac{120}{7}\approx 17.14 days$

• Solving for $\frac{1}{A}$ :

$\frac{1}{A}=\frac{1}{8}-\frac{7}{120}$

$\frac{1}{A}=\frac{15-7}{120}-\frac{8}{120}=\frac{1}{15}$

A = 15 days

Conclusion

Understanding Time & Work problems involves mastering basic concepts, practicing a variety of problem types, and developing efficient strategies. By referring to the aforementioned books, you can gain a comprehensive understanding and ample practice to excel in this topic.