AMC 10 A

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 $4$ , $3$ , and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45$ PM. At what time did Daria join the group?

Pick one

Solution

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

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