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A does half as much work as B in one-sixth of the time. If together they take 10 days to complete a work. How much time shall B alone take to do it?
B alone takes 40 days to complete the work.
Let’s analyze the problem step-by-step to find the time B alone takes to complete the work:
A’s Work Rate:
- We know A does half the work B does in one-sixth of the time. This means A’s work rate is:
- Work rate of B * (1/2) / (1/6) = Work rate of B * 3
Combining Work Rates:
- Since A and B work together to complete the work in 10 days, their combined work rate is:
- Total work / Time taken = 1 / 10
Relating Work Rates:
- We can equate their combined work rate to the sum of their individual work rates:
- 1 / 10 = (Work rate of A) + (Work rate of B)
Substituting Work Rates:
- Substitute the expressions from step 1 for A’s work rate:
- 1 / 10 = (Work rate of B * 3) + (Work rate of B)
Solving for B’s Work Rate:
- Combine like terms:
- 1 / 10 = 4 * (Work rate of B)
- Work rate of B = 1 / (10 * 4)
Calculating B’s Time:
- Now that we know B’s work rate, we can find the time B takes to complete the work alone:
- Time taken = Total work / Work rate of B
- Time taken = 1 / (1 / (10 * 4)) = 40 days
Therefore, B alone takes 40 days to complete the work.
Time and Work in Mathematics
“Time and Work” is a concept in applied mathematics that deals with the relationship between the amount of time it takes to complete a task and the number of people or machines working on it. It’s frequently encountered in quantitative aptitude sections of competitive exams and can be applied to various practical scenarios.
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Here are some key aspects of Time and Work:
Basic Formula:
The fundamental relationship between time, work, and rate of work is expressed by the formula:
Work (W) = Time (T) × Rate of Work (R)
This formula can be rearranged to solve for any of the three variables:
- Rate of Work (R) = W / T
- Time (T) = W / R
Key points:
- Work: Measured in units relevant to the task, like painting a wall (square meters) or typing a document (words).
- Time: Usually expressed in days, hours, or minutes.
- Rate of Work: Represents the amount of work completed per unit of time by a single person or machine.
Applications:
- Individual vs. Combined Work: The formula can be used to compare the time taken by individuals or groups to complete a task. For instance, if A can paint a room in 5 days and B can paint the same room in 8 days, working together, they can complete the task in less time.
- Efficiency: It can also be used to calculate the efficiency of workers or machines, which refers to the ratio of actual output to the theoretical maximum output.
Multiple workers: The concept can be extended to situations involving multiple workers with different work rates. The combined rate of work is the sum of individual rates.
Varying work rates: In some cases, the work rate may not be constant throughout the process. The concept can still be applied by considering average rates or dividing the task into smaller segments with assumed constant rates.
By understanding the relationship between time and work, you can solve various practical problems involving task completion times, workforce planning, and efficiency analysis.
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Source: Math Hello Kitty
Categories: Math
