Time and Work Problems for Competitive Exams

In the realm of competitive exams, the “Time and Work” topic holds significant importance. These problems not only test your mathematical skills but also your ability to think logically and manage time effectively. Whether you’re preparing for banking exams, SSC, or any other government exams, mastering this topic can give you a competitive edge. This article will guide you through the essential concepts, formulas, and strategies to tackle Time and Work problems with confidence.

Understanding the Basics of Time and Work Problems

1. Work and Time Relationship: At its core, Time and Work problems are built on the relationship between the amount of work done, the time taken, and the rate of work. If a person or a machine can complete a task in a specific amount of time, the rate of work is the reciprocal of the time taken. For instance, if a task is completed in 5 days, the work rate is

$\frac{1}{5}$ of the task per day.

2. Total Work Concept: In most problems, the total work is considered as a single unit. The time taken by different individuals or machines to complete the work is then used to determine their respective work rates.

Key Formulas and Concepts

Understanding and memorizing a few key formulas will simplify solving Time and Work problems:

• Work Rate Formula:
  $$\text{Work Rate} = \frac{1}{\text{Time Taken}}$$​If A completes a task in 10 days, A’s work rate is

  $\frac{1}{10}$per day.

• Total Work:
  $$\text{Total Work} = \text{Work Rate} \times \text{Time}$$For example, if A and B together can complete a task in 6 days, and their combined work rate is

  $\frac{1}{4}$​ of the work per day, the total work is 6 units.

• Combined Work Rate:
  When multiple workers are involved, their combined work rate is the sum of their individual work rates: $$\text{Combined Work Rate} = \text{Rate of A} + \text{Rate of B}$$
• Inverse Proportions:
• The time taken by a worker is inversely proportional to their efficiency. Higher efficiency means less time taken to complete the work.

Types of Time and Work Problems

1. Single Worker Problems:
These involve calculating the time required for one individual to complete a task. For example, if A can complete a task in 8 days, how much of the task can A complete in 3 days?

2. Multiple Workers:
Here, two or more workers are involved, and you need to determine how long they will take to finish the work when working together. For example, A can complete a task in 10 days, and B can do it in 15 days. How long will it take for both A and B to complete the work together?

3. Work Efficiency:
These problems focus on comparing the efficiency of different workers. For instance, if A is twice as efficient as B, how long will it take for both to complete the work together?

4. Pipes and Cisterns:
These problems are a variation of Time and Work where pipes fill or empty a tank. The concepts are similar, with the rate of filling or emptying taking the place of work rate.

Time and Work Problem-Solving Strategies

1. Break Down the Problem: Start by identifying the total work and the work rates of the individuals or machines involved. Use the basic formulas to set up equations that represent the problem.

2. Use LCM for Complex Problems: In problems involving multiple workers, especially when the time taken by each is different, use the Least Common Multiple (LCM) to simplify calculations.

3. Cross-Multiplication for Efficiency Problems: When dealing with efficiency problems, cross-multiplication can help compare the rates of work easily.

4. Practice Shortcuts: Many Time and Work problems can be solved quickly using shortcuts and tricks. For example, if A can do a piece of work in ‘x’ days, and B in ‘y’ days, the time taken by both together is:

$$\text{Time Taken}=\frac{xy}{x+y}\text{days}$$

Mastering such formulas can save valuable time during exams.

Example Problems

Let’s work through an example to solidify these concepts:

Problem: A can complete a work in 10 days, and B can do it in 15 days. How long will it take for both A and B to complete the work together?

Solution:

1. Calculate the work rate of A:
   $\frac{1}{10}$ work per day.
2. Calculate the work rate of B:
   $\frac{1}{15}$​  work per day.
3. Combined work rate =
   $\frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}$​  work per day.
4. Time taken by A and B together =
   $\frac{1}{\frac{1}{6}}$$=6$ days.

So, A and B together will take 6 days to complete the work.

Time and Work problems may seem challenging at first, but with a clear understanding of the concepts and regular practice, they become much easier to solve. Remember to practice different types of problems, use shortcuts where applicable, and always double-check your calculations to avoid silly mistakes. By mastering this topic, you’ll not only boost your quantitative aptitude but also improve your overall problem-solving skills, making you better prepared for competitive exams.

Sample Questions on Time and Work

Here are some sample questions on Time and Work with their answers:

Sample Question 1:

Q: A can complete a piece of work in 12 days, and B can do the same work in 16 days. If they work together, how long will it take them to complete the work?

A:

• Work rate of A =
  $\frac{1}{12}$ work/day.
• Work rate of B =
  $\frac{1}{16}$ work/day.
• Combined work rate =
  $\frac{1}{12} + \frac{1}{16} = \frac{4 + 3}{48} = \frac{7}{48}$work/day.
• Time taken =
  $\frac{1}{\frac{7}{48}} = \frac{48}{7}$≈ 6.86 days.

Sample Question 2:

Q: A can finish a task in 10 days, and B can finish it in 15 days. After A and B work together for 4 days, B leaves. How many more days will A take to finish the remaining work?

A:

• Work rate of A =
  $\frac{1}{10}$ work/day.
• Work rate of B =
  $\frac{1}{15}$ work/day.
• Combined work rate =
  $\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ work/day.
• Work done in 4 days =
  $4 \times \frac{1}{6} = \frac{2}{3}$
  
• Remaining work =
  $1 – \frac{2}{3} = \frac{1}{3}$
  
• Time taken by A to finish the remaining work =
  $\frac{1/3}{1/10} = 10/3 = 3 \frac{1}{3}$ days.

Sample Question 3:

Q: C is twice as efficient as D and takes 20 days less than D to complete a work. How many days will it take for C and D to complete the work together?

A:

• Let D take
  $x$days, so C takes $x – 20$ days.
• Work rate of D =
  $\frac{1}{x}$  work/day.
• Work rate of C =
  $\frac{1}{x-20}$ work/day.
• Since C is twice as efficient,
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• Solving,
  $x = 40$ days.
• C’s time = 20 days.
• Combined work rate =
  $\frac{1}{40} + \frac{1}{20} = \frac{3}{40}$ work/day.
• Time taken together =
  $\frac{40}{3}$ days ≈ 13.33 days.

Sample Question 4:

Q: A and B can do a job in 8 days and 12 days, respectively. With the help of C, they finish the job in 4 days. How long would it take for C alone to finish the work?

A:

• Work rate of A =
  $\frac{1}{8}$​  work/day.
• Work rate of B =
  $\frac{1}{12}$​  work/day.
• Let work rate of C =
  $\frac{1}{x}$​  work/day.
• Combined work rate =
  $\frac{1}{8} + \frac{1}{12} + \frac{1}{x} = \frac{1}{4}$.
• 
  $\frac{1}{x} = \frac{1}{4} – \frac{1}{8} – \frac{1}{12}$.
• Solving,
  $x = 24$ days.

Sample Question 5:

Q: A can do a work in 15 days, B can do it in 20 days, and C can do it in 30 days. They start the work together, but B leaves after 4 days. In how many more days will the work be completed?

A:

• Work rate of A =
  $\frac{1}{15}$​, B = $\frac{1}{20}$​, C = $\frac{1}{30}$
• Combined rate for 4 days =
  $4 \times \left( \frac{1}{15} + \frac{1}{20} + \frac{1}{30} \right) = \frac{4 \times 10}{60} = \frac{2}{3}$ of work.
• Remaining work =
  $1 – \frac{2}{3} = \frac{1}{3}$.
• Work rate of A + C =
  $\frac{1}{15} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10}$​
• Time taken =
  $\frac{1/3}{1/10} = \frac{10}{3}$ days ≈ 3.33 days.

Sample Question 6:

Q: 10 men can complete a work in 15 days. If 5 men leave after working for 6 days, how many more days will the remaining men need to complete the work?

A:

• Total work =
  $10 \times 15 = 150$ man-days.
• Work done in 6 days by 10 men =
  $10 \times 6 = 60$ man-days.
• Remaining work = 150 – 60 = 90 man-days.
• Remaining men = 5.
• Time taken =
  $\frac{90}{5} = 18$ days.

Sample Question 7:

Q: A can finish a piece of work in 18 days. B is 50% more efficient than A. How long will it take for B to finish the work?

A:

• Work rate of A =
  $\frac{1}{18}$ work/day.
• Efficiency of B = 1.5 times A.
• Work rate of B =
  $\frac{1.5}{18} = \frac{1}{12}$.
• Time taken by B = 12 days.

Sample Question 8:

Q: A is thrice as efficient as B. If A can complete a task in 12 days less than B, how many days will they take to complete the work together?

A:

• Let B take
  $x$days, so A takes $\frac{x}{3}$ days.
• Given,
  $\frac{x}{3} = x – 12$.
• Solving,
  $x = 18$ days.
• A’s time = 6 days.
• Combined work rate =
  $\frac{1}{6} + \frac{1}{18} = \frac{1}{4.5}$  work/day.
• Time taken together = 4.5 days.

Sample Question 9:

Q: A, B, and C can do a job in 10, 12, and 15 days, respectively. They work together for 2 days, and then C leaves. How many more days will A and B take to complete the work?

A:

• Work rate of A =
  $\frac{1}{10}$​, B = $\frac{1}{12}$​, C = $\frac{1}{15}$​
• Combined rate for 2 days =
  $2 \times \left( \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \right) = \frac{2 \times 37}{180} = \frac{37}{90}$​  of work.
• Remaining work =
  $1 – \frac{37}{90} = \frac{53}{90}$​.
• Work rate of A + B =
  $\frac{1}{10} + \frac{1}{12} = \frac{11}{60}$​  work/day.
• Time taken =
  $\frac{53/90}{11/60} = \frac{53 \times 60}{90 \times 11} = 3.18$days.

Sample Question 10:

Q: A can do a job in 5 days, and B can do it in 8 days. If both A and B work alternately, starting with A, in how many days will the work be completed?

A:

• Work rate of A =
  $\frac{1}{5}$​  work/day.
• Work rate of B =
  $\frac{1}{8}$​  work/day.
• Work done in 2 days (1 cycle) =
  $\frac{1}{5} + \frac{1}{8} = \frac{13}{40}$​  work.
• Work remaining after 3 cycles (6 days) =
  $1 – 3 \times \frac{13}{40} = \frac{1}{40}$  work.
• On the 7th day, A will finish
  $\frac{1}{5} = \frac{8}{40}$  work, so the job will be done in 7 days.

These sample questions cover various aspects of Time and Work

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