Mastering Work Done by Multiple People

This module focuses on a crucial concept in quantitative aptitude for competitive exams like the CAT: understanding how multiple individuals working together affect the time taken to complete a task. We'll explore the principles of work rate and how to combine them effectively.

Understanding Individual Work Rates

The foundation of solving problems involving multiple people is understanding their individual work rates. If a person can complete a job in 'x' days, their work rate is 1/x of the job per day. This concept is fundamental to combining efforts.

If person A can complete a task in 10 days, what is their daily work rate?

1/10 of the task per day.

Combining Work Rates: The Addition Principle

When multiple people work together, their individual work rates add up to form a combined work rate. If person A's rate is R_A and person B's rate is R_B, their combined rate (R_A + R_B) determines how quickly they complete the task together.

Combined work rate is the sum of individual work rates.

If Person A takes 5 days and Person B takes 10 days to complete a job individually, their daily rates are 1/5 and 1/10 respectively. Their combined daily rate is (1/5 + 1/10) = 3/10 of the job per day.

Let's say Person A can complete a job in $D_A$ days and Person B can complete the same job in $D_B$ days. Their individual work rates are $R_A = 1/D_A$ and $R_B = 1/D_B$ jobs per day. When they work together, their combined work rate $R_{combined}$ is the sum of their individual rates: $R_{combined} = R_A + R_B = (1/D_A) + (1/D_B)$ . The total time taken to complete the job together, $D_{combined}$ , is the reciprocal of the combined work rate: $D_{combined} = 1 / R_{combined} = 1 / ((1/D_A) + (1/D_B))$ . This can be simplified to $D_{combined} = (D_A * D_B) / (D_A + D_B)$ .

If Person A completes a job in 6 days and Person B in 3 days, what is their combined daily work rate?

1/6 + 1/3 = 1/2 of the job per day.

Calculating Time Taken Together

Once you have the combined work rate, the time taken to complete the job together is simply the reciprocal of this rate. If the combined rate is 'R' jobs per day, the time taken is '1/R' days.

Imagine a task as a pie. Person A eats 1/5 of the pie per day. Person B eats 1/10 of the pie per day. When they eat together, they consume (1/5 + 1/10) = 3/10 of the pie each day. Therefore, to eat the whole pie (10/10), it will take them 10/3 days.

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Remember: The more people working, the less time it takes, assuming they don't hinder each other.

Handling More Than Two People

The principle extends to any number of people. If you have three people with rates $R_A$ , $R_B$ , and $R_C$ , their combined rate is $R_A + R_B + R_C$ . The time taken together is $1 / (R_A + R_B + R_C)$ .

If Person A takes 10 days, Person B takes 15 days, and Person C takes 30 days, how many days will it take them to complete the job together?

Rates: 1/10, 1/15, 1/30. Combined rate: 1/10 + 1/15 + 1/30 = (3+2+1)/30 = 6/30 = 1/5. Time taken: 5 days.

Efficiency and Variations

Problems might introduce variations like different efficiencies or one person leaving the work midway. Always calculate the work done by each person based on their individual rates and the time they actually worked.

Scenario	Calculation Approach
Two people working together	Add their daily work rates: (1/D1 + 1/D2). Time = 1 / (combined rate).
Multiple people working together	Sum all individual daily work rates: (1/D1 + 1/D2 + 1/D3 + ...). Time = 1 / (total combined rate).
One person leaves midway	Calculate work done by the first person, then calculate remaining work and time for the second person.

Learning Resources

Time and Work - Concepts and Formulas(tutorial)

A comprehensive tutorial covering fundamental concepts and formulas for Time and Work problems, including work done by multiple people.

Time and Work CAT Questions with Solutions(blog)

Provides practice questions and detailed solutions for Time and Work, specifically tailored for CAT aspirants, covering various scenarios.

Understanding Work Rate in Time and Work Problems(documentation)

Explains the core concept of work rate and its application in solving problems involving multiple workers and varying efficiencies.

Time and Work: Work Done by Men, Women, and Children(tutorial)

Focuses on variations in work rates based on different types of workers, a common variation in these problems.

CAT Quantitative Aptitude - Time and Work(video)

A video lecture explaining Time and Work concepts, including how to approach problems with multiple individuals working together.

Time and Work: Advanced Concepts for CAT(video)

Covers more advanced strategies and problem-solving techniques for Time and Work, relevant for competitive exams.

Time and Work Practice Problems(blog)

Offers a set of practice questions with explanations to reinforce understanding of Time and Work concepts.

Time and Work - Concept of Work Rate(documentation)

A detailed breakdown of the work rate concept, essential for solving problems involving multiple workers.

Time and Work: Efficiency and Combined Work(blog)

Explains how to handle problems where efficiency varies among workers and how to calculate combined work.

Time and Work - Work Done by Multiple People(video)

A focused video tutorial specifically addressing how to solve problems involving multiple individuals contributing to a task.