Question

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?

1:25 PM

1:35 PM

1:45 PM

1:55 PM

2:05 PM

Solution:

From 1:00 to 2:45 is $105$ minutes. Amy, Bomani, and Charlie pack $4 + 3 + 3 = 10$ packages every $3$ minutes, so $\tfrac{10}{3}$ per minute, which is $\tfrac{10}{3} \cdot 105 = 350$ packages. That leaves $450 - 350 = 100$ for Daria, who packs $\tfrac54$ per minute and so needs $100 / \tfrac54 = 80$ minutes. She worked the last $80$ minutes, joining $105 - 80 = 25$ minutes after 1:00. That's 1:25 PM. Therefore, the answer is A.

2024 AMC 10A Problem 8

Solution:

Problem 8 in Other Years