How Long Will A and B Take to Complete the Work Together?
How Long Will A and B Take to Complete the Work Together?
Have you ever wondered how long it would take for two individuals working together to complete a task? In this article, we will explore the problem of determining how long A and B, who can complete a piece of work in 20 days and 30 days respectively, would take to finish the work together. This is a classic example of a work rate problem. Let's break down the solution step by step.
Understanding the Problem
The problem states that A can complete the work in 20 days and B can do it in 30 days. To find out how long they will take to complete the work together, we need to use the concept of work rates.
Calculating Individual Work Rates
The first step is to determine the work rate of A and B individually. The work rate is the fraction of the work that each person can complete in a single day.
A's Work Rate
Since A can complete the work in 20 days, A's work rate is:
( frac{1}{20} ) of the work per day
B's Work Rate
Similarly, since B can complete the work in 30 days, B's work rate is:
( frac{1}{30} ) of the work per day
Combining Work Rates
When two individuals work together, their work rates add up. Therefore, we add the individual work rates of A and B to find their combined work rate.
( text{Combined work rate} frac{1}{20} frac{1}{30} )
To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.
( frac{1}{20} frac{3}{60} ) and ( frac{1}{30} frac{2}{60} )
Therefore, the combined work rate is:
( text{Combined work rate} frac{3}{60} frac{2}{60} frac{5}{60} frac{1}{12} ) of the work per day
Calculating Time to Complete the Work Together
Now that we have the combined work rate, we can calculate the time it would take for A and B to complete the work together. The time to complete the work is the reciprocal of the combined work rate.
( text{Time} frac{1}{text{Combined work rate}} frac{1}{frac{1}{12}} 12 ) days
Conclusion
A and B together will take 12 days to complete the work. This solution is based on the principle that the combined work rate of two individuals working together is the sum of their individual work rates.
Understanding how to solve work rate problems is valuable in many real-world scenarios, such as project management, timetabling, and resource allocation. Whether you're a student, a professional, or simply curious, mastering these concepts will enhance your problem-solving skills.