Rahul starts a task at 9 AM. Gautam joins him two hours later, and they complete the work together by 5 PM the same day. If both had started at 9 AM, they would have finished the work 30 minutes earlier. How much time would Rahul take to complete the task if he worked alone?
Problem Statement
Rahul started a task at 9 AM. Gautam joined him two hours later, and they completed the work together at 5 PM the same day. If both had started at 9 AM, they would have finished the task 30 minutes earlier. The question is: how much time would Rahul take to complete the task alone?
Step 1: Defining the Variables
Let us define the following variables:
- Total work: 1 unit
- Rahul’s work rate: R (work per hour)
- Gautam’s work rate: G (work per hour)
Step 2: Understanding the Two Scenarios
Scenario 1: Rahul starts at 9 AM, and Gautam joins at 11 AM
Rahul works alone from 9 AM to 11 AM, which is 2 hours. He completes 2R units of work during this time. Then, from 11 AM to 5 PM (6 hours), both work together, completing 6(R + G) units of work. Thus, the total work completed is:
2R + 6(R + G) = 1
Simplifying:
2R + 6R + 6G = 1
8R + 6G = 1 (Equation 1)
Scenario 2: Both start at 9 AM
If both had started at 9 AM, they would have finished the work 30 minutes earlier, i.e., by 4:30 PM. They would have worked together for 7.5 hours. Therefore, the work completed in this case would be:
7.5(R + G) = 1
Simplifying:
R + G = 1 / 7.5 = 2 / 15 (Equation 2)
Step 3: Solving the Equations
We now have two equations:
8R + 6G = 1(Equation 1)R + G = 2 / 15(Equation 2)
From Equation 2, we solve for G:
G = 2/15 - R
Substitute this into Equation 1:
8R + 6(2/15 - R) = 1
Simplifying:
8R + 12/15 - 6R = 1
2R + 12/15 = 1
2R = 1 - 12/15 = 3/15 = 1/5
R = 1/10
Rahul’s work rate is R = 1 / 10, meaning Rahul would take 10 hours to complete the work if he worked alone.
Final Answer
In conclusion, if Rahul were to work alone, he would take 10 hours to complete the task.
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