Alice and Bob are working on a project. Alice can complete a task in 12 hours, while Bob can complete it in 15 hours. If they work together, but Alice starts an hour later than Bob, how long will it take them to complete the task?
["# How Long Will Alice and Bob Take to Complete the Task Together? (A Step-by-Step Breakdown)", "Working on a project with two team members—Alice and Bob—can be more efficient when you understand their individual work rates. In this scenario, Alice completes the task in 12 hours, while Bob takes 15 hours. When they collaborate, but Alice starts one hour after Bob, understanding how their combined effort builds the solution is key to solving real-world scheduling and productivity challenges.", "### Step 1: Calculate Individual Work Rates", "First, convert their task completion times into work rates (tasks per hour):", "- Alice’s rate:\n [\n \frac{1}{12} \ ext{ task/hour}\n ]\n- Bob’s rate:\n [\n \frac{1}{15} \ ext{ task/hour}\n ]", "### Step 2: Understand the Delay in Collaboration", "Since Alice starts 1 hour after Bob, Bob works alone during that hour. Only Bob contributes to the task in those first 60 minutes. After the first hour, both work together for the remaining duration.", "Let’s define:\n- ( t ): total time (in hours) from when Bob starts until the task is complete.", "### Step 3: Break Down Work Done", "- Bob starts early and works alone for 1 hour:\n [\n \ ext{Work done by Bob in 1 hour} = 1 \ imes \frac{1}{15} = \frac{1}{15} \ ext{ of the task}\n ]", "- Alice joins after 1 hour and works for ( t - 1 ) hours with Bob:\n [\n \ ext{Work done by Alice and Bob together} = (t - 1) \ imes \left( \frac{1}{12} + \frac{1}{15} \right)\n ]", "### Step 4: Calculate Combined Work Rate", "[\n\frac{1}{12} + \frac{1}{15} = \frac{5 + 4}{60} = \frac{9}{60} = \frac{3}{20} \ ext{ task/hour}\n]", "### Step 5: Set Up the Total Work Equation", "Total work = work done by Bob in 1 hour + work done by both in ( t - 1 ) hours", "[\n\frac{1}{15} + (t - 1) \cdot \frac{3}{20} = 1\n]", "### Step 6: Solve for ( t )", "[\n(t - 1) \cdot \frac{3}{20} = 1 - \frac{1}{15} = \frac{14}{15}\n]", "[\nt - 1 = \frac{14}{15} \cdot \frac{20}{3} = \frac{280}{45} = \frac{56}{9}\n]", "[\nt = \frac{56}{9} + 1 = \frac{56}{9} + \frac{9}{9} = \frac{65}{9} \approx 7.22 \ ext{ hours}\n]", "### Final Answer", "With Alice starting one hour after Bob, the entire team finishes the task in approximately 7 hours and 13 minutes (or ( \frac{65}{9} ) hours).", "### Why This Matters", "Understanding staggered team workflows helps project managers optimize collaboration, minimize idle time, and improve scheduling efficiency. Even small delays can significantly impact project timelines—especially when one team member joins later.", "---", "Keywords: Alice and Bob, work rate calculation, team collaboration, task completion time, productivity multiplication, staggered start delay, work together calculation, project scheduling", "#### Learn more about efficient team collaboration and time management at [Your Resource Site Link].", "---", "By analyzing individual rates and timing overlaps, even complex projects become manageable—proving that math and teamwork go hand in hand."]
