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A can do a piece of work in 10 days and B in 15 days. How long will they take together to finish it?
A and B will take 6 days together to finish the work.
To find out how long it will take A and B to finish the work together, we can use the formula for working together:
Time taken together = Work / (Rate of A + Rate of B)
Where:
- Work = 1 (as they are completing one piece of work)
- Rate of A = 1/10 (since A can complete the work in 10 days, A’s rate per day is 1/10)
- Rate of B = 1/15 (since B can complete the work in 15 days, B’s rate per day is 1/15)
Now, substitute the values into the formula:
Time taken together = 1 / (1/10 + 1/15)
= 1 / ((3 + 2)/30)
= 1 / (5/30)
= 30/5 = 6
So, A and B will take 6 days together to finish the work.
Time and Work in Mathematics
Time and work problems in mathematics deal with calculating the amount of work done by individuals or machines working together or separately over a certain period of time. These problems often involve concepts such as rates of work, efficiency, and the effects of individuals joining or leaving the workforce. Here’s an overview of how to solve time and work problems:
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Understanding the Problem: Carefully read the problem to understand what work needs to be done, who is doing it, and any conditions or constraints given.
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Identify Variables: Identify the variables involved, such as the amount of work to be done, the time taken to complete the work, and the rate at which individuals or machines work.
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Formulate Equations: Use the concept of work = rate × time. If there are multiple workers or machines involved, you may need to sum up their individual rates to find the total rate.
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Solve for Unknowns: Once you have formulated equations representing the given conditions, solve for the unknowns. This often involves basic algebraic manipulation.
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Check the Solution: Ensure that your solution makes sense in the context of the problem. For example, check if the time taken to complete the work is reasonable and if the rates of work are consistent with the given conditions.
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Consider Special Cases: Some problems may involve special cases such as workers joining or leaving the job midway, or varying rates of work over time. Adjust your equations and solutions accordingly.
Here’s a simple example to illustrate:
Example: A can complete a job in 10 days, B can complete the same job in 15 days, and C can complete it in 20 days. How long will it take if all three work together?
Solution: Let’s denote the rates of work of A, B, and C as 1/10, 1/15, and 1/20 respectively (since they complete the job in those many days).
When all three work together, their combined rate is the sum of their individual rates: Combined rate = 1/10 + 1/15 + 1/20 Combined rate = 3/30 + 2/30 + 1/30 = 6/30 = 1/5
So, they can complete 1/5 of the job per day when working together.
To find out how long it takes to complete the whole job, we divide the total work by the combined rate: Time taken = 1/Combined rate = 1/(1/5) = 5 days
So, it will take 5 days for A, B, and C to complete the job together.
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Source: Math Hello Kitty
Categories: Math
