Maths Olympiad Prep

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Algebra Difficulty 4.4 AIME Find the answer United States

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00 PM and were able to pack 44, 33, and 33 packages, respectively, every 33 minutes. At some later time, Daria joined the group, and Daria was able to pack 55 packages every 44 minutes. Together, they finished packing 450450 packages at exactly 2:452{:}45 PM. At what time did Daria join the group?

Pick one

Solution

Every 33 minutes, Amy, Bomani, and Charlie together packed 1010 packages. From 1:001{:}00 PM to 2:452{:}45 PM, a span of 60+45=10560 + 45 = 105 minutes, these three packers packed 1053⋅10=350\frac{105}{3} \cdot 10 = 350 packages. This means that Daria must have packed 450−350=100450 - 350 = 100 packages. The time needed for Daria to pack 100100 packages is 1005⋅4=80\frac{100}{5} \cdot 4 = 80 minutes. Therefore Daria joined the group at 1:251{:}25 PM, which is 8080 minutes before 2:452{:}45 PM.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.