Podcast
Questions and Answers
How is the work rate of a person who can finish a job in 12 days expressed mathematically?
How is the work rate of a person who can finish a job in 12 days expressed mathematically?
What is the combined work rate of A who can do a job in 8 days and B who can do the same job in 12 days?
What is the combined work rate of A who can do a job in 8 days and B who can do the same job in 12 days?
If a job requires 48 hours of work to complete, how many workers are needed if they can each work at a rate of 8 hours per day to complete it in 6 days?
If a job requires 48 hours of work to complete, how many workers are needed if they can each work at a rate of 8 hours per day to complete it in 6 days?
What fraction of the job will be completed by A and B working together for 4 days if A can complete it in 10 days and B can complete it in 15 days?
What fraction of the job will be completed by A and B working together for 4 days if A can complete it in 10 days and B can complete it in 15 days?
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If A and B complete a job together in 15 days, and A alone would take 25 days, how long would B take to complete the job alone?
If A and B complete a job together in 15 days, and A alone would take 25 days, how long would B take to complete the job alone?
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After A completes $ rac{1}{3}$ of a job alone, which takes them 6 days, how many days will it take for A and C, who can complete the job in 18 days together, to finish the remaining work?
After A completes $ rac{1}{3}$ of a job alone, which takes them 6 days, how many days will it take for A and C, who can complete the job in 18 days together, to finish the remaining work?
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What effect does increasing the number of workers have on the total time to complete a job?
What effect does increasing the number of workers have on the total time to complete a job?
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If Worker A can do a job in 40 days and Worker B can do it in 60 days, how long will it take them to finish the job if they work together?
If Worker A can do a job in 40 days and Worker B can do it in 60 days, how long will it take them to finish the job if they work together?
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Study Notes
Time and Work
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Basic Concept:
- Time and work problems involve calculating how long it will take for one or more people or machines to complete a job.
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Work Rate:
- Work rate = Amount of work done / Time taken.
- If one person can complete a job in "x" days, their work rate is ( \frac{1}{x} ) of the job per day.
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Combined Work:
- When multiple people are working together, their combined work rate is the sum of their individual work rates.
- Formula: If A can do a job in "a" days and B can do it in "b" days, then their combined work rate is: [ \text{Combined Rate} = \frac{1}{a} + \frac{1}{b} ]
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Example Calculations:
- If A can complete a job in 10 days and B in 15 days:
- A’s rate: ( \frac{1}{10} )
- B’s rate: ( \frac{1}{15} )
- Combined rate: [ \frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6} ]
- They can complete the job together in 6 days.
- If A can complete a job in 10 days and B in 15 days:
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Work Completion Scenarios:
- If one person finishes a part of a job and another takes over or they work simultaneously, account for both the time taken and the work done.
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Inverse Relationships:
- If the number of workers increases, the time taken to complete the job decreases (and vice versa).
- Formula: ( Time \propto \frac{1}{\text{Number of Workers}} )
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Efficient Workers:
- An efficient worker can do more work in the same amount of time compared to a less efficient worker, affecting the total time taken for the job.
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Example Problem:
- A can complete a job in 20 days, B in 30 days. A starts the job, and after 5 days, B joins A. Calculate the time to finish the job.
- In 5 days, A completes ( \frac{5}{20} = \frac{1}{4} ) of the job.
- Remaining job = ( 1 - \frac{1}{4} = \frac{3}{4} ).
- Combined rate of A and B = ( \frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{5}{60} = \frac{1}{12} ).
- Time to finish ( \frac{3}{4} ) of the job: [ \text{Time} = \text{Work} \div \text{Rate} = \frac{3/4}{1/12} = 9 \text{ days} ]
- Total time = 5 days + 9 days = 14 days.
- A can complete a job in 20 days, B in 30 days. A starts the job, and after 5 days, B joins A. Calculate the time to finish the job.
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Key Takeaway:
- Understanding individual work rates and applying them in scenarios of combined work is crucial to solving time and work problems effectively.
Time and Work Fundamentals
- Time and work problems focus on calculating how long it takes individuals or groups to complete tasks.
- Work Rate is the measure of work done per unit of time. It is calculated by dividing the amount of work by the time taken.
- For an individual completing a job in 'x' days, their work rate is 1/x of the job per day.
Combined Work Rate
- When multiple individuals work together, their combined work rate is the sum of their individual work rates.
- If individual A completes a job in 'a' days and B completes it in 'b' days, their combined work rate is: 1/a + 1/b
Example Calculation
- Scenario: A can complete a job in 10 days and B in 15 days.
- Individual Rates: A's rate = 1/10, B's rate = 1/15
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Combined Rate:
- (1/10) + (1/15) = (3 + 2)/30 = 5/30 = 1/6
- Working together: They can complete the job in 6 days.
Work Completion Scenarios
- If one person partially completes a task and another person joins or takes over, the total time taken for the entire job is calculated by considering the time taken by each person and the amount of work they completed.
Inverse Relationship
- There is an inverse relationship between the number of workers and the time taken to complete a job.
- This means that as the number of workers increases, the time taken to complete the job decreases, and vice versa.
- Formula: Time ∝ 1/Number of Workers
Efficient Workers
- More efficient workers complete more work in the same amount of time compared to less efficient workers. The efficiency of individual workers influences the overall time taken to complete a job.
Example Problem
- Scenario: A can complete a job in 20 days, B in 30 days. A starts the job for 5 days, then B joins.
- A's work in 5 days: A completes (5/20) = 1/4 of the job.
- Remaining work: 1 - (1/4) = 3/4 of the job.
- Combined rate of A and B: (1/20) + (1/30) = (3 + 2)/60 = 5/60 = 1/12.
- Time to finish remaining work: (Work) / (Rate) = (3/4) / (1/12) = 9 days.
- Total time taken: 5 days (A's time) + 9 days (remaining time) = 14 days.
Key takeaway
- Understanding individual work rates and applying them to scenarios involving combined work is crucial for effectively solving time and work problems.
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Description
This quiz focuses on the basic concepts of time and work problems, including work rates and combined work calculations. Test your understanding of how to calculate time taken to complete jobs when working alone or together. Put your knowledge to the test with example problems!
