How Long Would It Take Sam Alone to Produce 10 Tables If He Worked With Alex?
Alex and Sam are skilled carpenters who collaborate on various projects. One such project involves producing tables, which they can undertake in record time when working together. However, this collaboration raises the question: if Alex can produce 10 tables in 30 days working solo, and together they can produce 10 tables in 12 days, how long would it take for Sam to produce 10 tables if he worked alone?
Understanding the Efficiency of Alex and Sam Together
The combined effort of Alex and Sam is crucial in determining the individual rate of production. When working together, they can produce 10 tables in 12 days. Therefore, their combined rate of production is:
Combined Rate 10 tables/12 days 5/6 tables per day
Determining Alex's Individual Rate of Production
It is given that Alex can produce 10 tables in 30 days. Therefore, his individual rate of production is:
Alex's Rate 10 tables/30 days 1/3 tables per day
Calculating Sam's Individual Rate of Production
To find Sam's individual rate of production, we need to subtract Alex's rate from the combined rate:
Sam's Rate Combined Rate - Alex's Rate 5/6 tables per day - 1/3 tables per day
Converting 1/3 to sixths:
1/3 2/6
Therefore:
Sam's Rate 5/6 tables per day - 2/6 tables per day 3/6 tables per day 1/2 tables per day
Calculating Time for Sam to Produce 10 Tables Alone
Given that Sam's production rate is 1/2 tables per day, we can calculate the time it would take for him to produce 10 tables:
Time (Sam) 10 tables/1/2 tables per day 10 times; 2 20 days
Therefore, it would take Sam 20 days to produce 10 tables if he worked alone.
Total Work and Combined Rate
The total work involved in producing 10 tables is the same regardless of how it is broken down. When Alex and Sam work together, their combined work per day is 1/12 of the total work, and Alex's work per day is 1/30 of the total work. The remaining work per day, which is Sam's individual contribution, can be calculated as:
Sam's Rate 1/12 - 1/30 3/60 1/20
Therefore, Sam can complete the entire work (producing 10 tables) in:
Time (Sam) 20 days
Conclusion
To summarize, based on the given information and the calculations performed, it would take Sam 20 days to produce 10 tables if he worked alone. This example demonstrates the importance of understanding individual and combined rates of production in carpentry and other collaborative work scenarios.