To solve the problem of how long it would take Chris and Larry to mow the lawn together, we need to understand the concept of the rate of work. This principle is also applicable to calculating the work efficiency of Brad and Kris in a similar scenario. Here’s a detailed step-by-step breakdown:
" "Calculation for Chris and Larry
" "Understanding Individual Rates of Work:
" "Chris: Chris takes 4 hours to mow one lawn.
" "Larry: Larry takes 2 hours to mow one lawn.
" "Thus, their rates of work are as follows:
" "Chris’ Rate: ( frac{1 text{ lawn}}{4 text{ hours}} 0.25 text{ lawns per hour} )
" "Larry’s Rate: ( frac{1 text{ lawn}}{2 text{ hours}} 0.5 text{ lawns per hour} )
" "Next, we add their rates together to find their combined rate:
" "Combined Rate: ( 0.25 0.5 0.75 text{ lawns per hour} )
" "To find the time it takes for them to mow one lawn together:
" "Time: ( frac{1 text{ lawn}}{0.75 text{ lawns per hour}} frac{4}{3} text{ hours} approx 1.33 text{ hours} )
" "Therefore, it would take Chris and Larry approximately 1 hour and 20 minutes to mow the lawn together.
" "Brad and Kris Scenarios
" "Brad’s rate of mowing is half a field in one hour, which is 0.5 field/hour.
" "Kris’s rate of mowing is one third a field in an hour, which is approximately 0.33333 field/hour.
" "If they work together, the equation to find the time they need is:
" "0.5 field/hour * t 0.33333 field/hour * t 1 field
" "Combining the rates:
" "0.833333 * t 1
" "solving for t:
" "t 1.2 hours
" "checking the solution:
" "In 1.2 hours, Brad mows 0.6 field and Kris mows 0.4 field, which totals to 1 field.
" "Utilizing Least Common Multiple (LCM) for Optimal Calculation
" "Let's assume the total work required is the least common multiple (LCM) of 2 and 3, which is 6 units.
" "Brad's work rate: 3 units/hr
" "Kris’s work rate: 2 units/hr
" "Add their rates to get the combined rate:
" "Combined rate: 3 2 5 units/hr
" "Time to complete 6 units of work together:
" "Time: ( frac{6 text{ units}}{5 text{ units per hour}} 1.2 text{ hours} ) or 1 hour 12 minutes
" "Unified Approach
" "Let’s consider the general method of solving the work efficiency problem. If each unit of work is 1, and Brad completes half the work (1/2 part) in one hour, while Kris completes one-third (1/3 part) in one hour, then their combined work per hour is:
" "Brad’s Work per hour: 1/2 part
" "Kris’s Work per hour: 1/3 part
" "Combined per hour work:
" "(1/2 1/3) 5/6 part of the work per hour
" "To complete 1 unit of work, time taken would be:
" "Time: ( frac{1}{5/6} frac{6}{5} 1.2 text{ hours} ) or 1 hour 12 minutes.
" "Conclusion
" "By applying the rate of work concept and understanding the combined rates, we can efficiently determine the time it takes for pairs to complete a task together. This method can be applied to various real-world scenarios where productivity and time management are crucial.