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Discover the math behind trench digging efficiency: 3 men, 3 feet deep, 3 hours/day, 3 days. Now, envision double the workforce, time, and effort. What depth can 6 men reach in 6 hours/day, 6 days?
3 men can dig a trench 3 feet deep working for 3 hours a day in 3 days. How deep a trench can 6 men dig working for 6 hours a day for 6 days?
6 men working for 6 hours a day for 6 days can dig a trench that is 24 feet deep.
To solve this problem, we can use the concept of man-hours, which is a measure of the amount of work performed by one person in one hour.
Given that 3 men can dig a trench 3 feet deep working for 3 hours a day in 3 days, we calculate the total man-hours used to dig this trench:
Total man-hours = 3 men * 3 hours/day * 3 days = 27 man-hours
Now, we need to find out how many man-hours it would take for 6 men to dig a trench of unknown depth in 6 days, working 6 hours a day:
Total man-hours = 6 men * 6 hours/day * 6 days = 216 man-hours
Since the total amount of work (measured in man-hours) remains constant, we can set up a proportion to find the depth of the trench dug by 6 men:
27 man-hours / 3 feet = 216 man-hours / x feet
Cross-multiplying:
27x = 3 * 216
27x = 648
Dividing both sides by 27:
x = 648 / 27
x = 24
So, 6 men working for 6 hours a day for 6 days can dig a trench that is 24 feet deep.
Work and Time in Mathematics
Work and time problems in mathematics typically involve calculating the amount of work done by one or more individuals over a certain period of time. These problems often involve rates of work, such as work per unit of time, and they can be solved using basic principles of algebra and arithmetic. Here’s a general overview of how to approach work and time problems:
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Understanding the Concept: In these problems, “work” generally refers to a task or job to be completed, while “time” refers to the duration it takes to complete that task.
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Identify the Variables: In most work and time problems, you’ll need to identify variables such as: W = amount of work (building the wall)T = time taken (hours)R = rate of work (walls per hour)
Example: If it takes 8 workers 10 hours to build a wall, how long would it take 4 workers to build the same wall?
Solution:
Given:
- W = amount of work (building the wall)
- T = time taken (hours)
- R = rate of work (walls per hour)
Given that 8 workers can complete the wall in 10 hours, the rate of work R is: R = W/T = 1/10 wall per hour
Now, if we have 4 workers, the total rate of work is: R = 8 * (1/10) = 8/10 = 4/5 wall per hour
To find the time it would take for 4 workers to complete the wall, we rearrange the formula: T = W/R = 1 / (4/5) = 5/4 hours = 1.25 hours
So, it would take 4 workers approximately 1.25 hours to build the same wall.
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Source: Math Hello Kitty
Categories: Math
