AMC10 2024 A

AMC10 2024 A · Q8

AMC10 2024 A · Q8. It mainly tests Rates (speed), Work problems.

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1:00 \ \mathrm{PM}$ 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 $2:45\ \mathrm{PM}$. At what time did Daria join the group?

Amy、Bomani、Charlie 和 Daria 在一家巧克力工厂工作。周一，Amy、Bomani 和 Charlie 于下午 $1:00$ 开始工作，每 $3$ 分钟分别打包 $4$、$3$ 和 $3$ 个包裹。后来，Daria 加入了他们，Daria 每 $4$ 分钟打包 $5$ 个包裹。他们一起在下午 $2:45$ 正好打包完 $450$ 个包裹。Daria 是何时加入的？

(A) 1:25\text{ PM} 1:25\text{ PM}

(B) 1:35\text{ PM} 1:35\text{ PM}

(D) 1:55\text{ PM} 1:55\text{ PM}

(E) 2:05\text{ PM} 2:05\text{ PM}

Answer

Correct choice: (A)

正确答案：（A）

Solution

Note that Amy, Bomani, and Charlie pack a total of $4+3+3=10$ packages every $3$ minutes. The total amount of time worked is $1$ hour and $45$ minutes, which when converted to minutes, is $105$ minutes. This means that since Amy, Bomani, and Charlie worked for the entire $105$ minutes, they in total packed $\dfrac{105}{3}\cdot10=350$ packages. Since $450$ packages were packed in total, then Daria must have packed $450-350=100$ packages in total, and since she packs at a rate of $5$ packages per $4$ minutes, then Daria worked for $\dfrac{100}{5}\cdot4=80$ minutes, therefore Daria joined $80$ minutes before $2:45$ PM, which was at $\boxed{\textbf{(A) }1:25\text{ PM}}$

注意 Amy、Bomani 和 Charlie 每 $3$ 分钟总共打包 $4+3+3=10$ 个包裹。总工作时间是 $1$ 小时 $45$ 分钟，转换为分钟是 $105$ 分钟。这意味着 Amy、Bomani 和 Charlie 整个 $105$ 分钟都在工作，他们总共打包了 $\dfrac{105}{3}\cdot10=350$ 个包裹。总共打包了 $450$ 个包裹，因此 Daria 必须打包了 $450-350=100$ 个包裹，她每 $4$ 分钟打包 $5$ 个，所以 Daria 工作了 $\dfrac{100}{5}\cdot4=80$ 分钟，因此 Daria 在下午 $2:45$ 前 $80$ 分钟加入，即 $\boxed{\textbf{(A) }1:25\text{ PM}}$

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