Armand, Ella, and Amber spent 15 hours collaborating on a study. Amber worked twice as long as Ella, and Ella worked 30% less than Armand. How many hours did Amber work?
["How Many Hours Did Amber Work? Solving a Timeline Collaboration Puzzle", "Armand, Ella, and Amber spent a total of 15 hours collaborating on a challenging study—meticulously planning their research strategy and deep-diving into data analysis. Understanding the time each team member contributed reveals an interesting puzzle of ratios and workload distribution.", "Let’s break down the collaboration hours step by step:", "### Defining the Variables\nLet’s assign variables based on Ella’s hours since her workload connects directly to Amber’s and Armand’s:", "- Let E = Ella’s hours\n- Then Amber’s hours = 2E (since she worked twice as long as Ella)\n- Ella worked 30% less than Armand, meaning Ella’s hours are 70% of Armand’s:\n [\n E = 0.7 \ imes A_{\ ext{Armand}}\n ]\n Or, solving for Armand:\n [\n A_{\ ext{Armand}} = \frac{E}{0.7} = \frac{10}{7}E\n ]", "### Total Hours Equation\nAdding all hours together:\n[\nA_{\ ext{Armand}} + E + A_{\ ext{Ella}} = 15\n]\nSubstitute the expressions:\n[\n\frac{10}{7}E + E + 2E = 15\n]\nCombine like terms:\n[\n\left( \frac{10}{7} + 1 + 2 \right) E = 15\n]\n[\n\left( \frac{10}{7} + \frac{7}{7} + \frac{14}{7} \right) E = 15\n]\n[\n\frac{31}{7}E = 15\n]\nSolve for E:\n[\nE = 15 \ imes \frac{7}{31} = \frac{105}{31} \approx 3.39 \ ext{ hours}\n]", "### Calculating Amber’s Hours\nNow plug back to find Amber’s hours:\n[\nA_{\ ext{Ella}} = 2E = 2 \ imes \frac{105}{31} = \frac{210}{31} \approx 6.77 \ ext{ hours}\n]", "---", "### Final Answer:\nAmber worked approximately 6.77 hours—or exactly 210/31 hours—on the study, making her a significant contributor to this 15-hour collaborative effort.", "This study highlights how careful time analysis can clarify workload distribution, ensuring fairness and efficiency in team projects. Whether you’re a researcher, student, or professional, understanding these ratios helps manage time and responsibilities effectively."]
