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2024 AMC 10A Problem 8

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2024 AMC 10A Problem 8AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00PM1:00 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 2:45PM2:45 PM . At what time did Daria join the group?

  1. A)

    1:25 PM1:25\text{ PM}

  2. B)

    1:35 PM1:35\text{ PM}

  3. C)

    1:45 PM1:45\text{ PM}

  4. D)

    1:55 PM1:55\text{ PM}

  5. E)

    2:05 PM2:05\text{ PM}

Answer

A

Key insight

Amy, Bomani and Charlie pack 10 per 3 minutes for all 105 minutes (350 packages); Daria's remaining 100 at 5 per 4 minutes take 80 minutes, ending at 2:45.

Solution

Amy, Bomani and Charlie together pack 4+3+3=104 + 3 + 3 = 10 packages every 33 minutes. From 1:00 PM to 2:45 PM is 105105 minutes, which is 3535 blocks of 33 minutes, so the three of them pack 35⋅10=35035\cdot 10 = 350 packages.

That leaves 450−350=100450 - 350 = 100 packages for Daria. She packs 55 every 44 minutes, so 100100 packages take 1005⋅4=80\frac{100}{5}\cdot 4 = 80 minutes.

Daria worked the last 8080 minutes, so she joined 8080 minutes before 2:45 PM, at 1:25 PM.

The answer is (A) 1:25 PM\boxed{\textbf{(A)}\ 1{:}25\text{ PM}}.

Why this works

Separate the contributions: the original trio works the entire interval, so their output is fixed; whatever is left over must be Daria's, and dividing by her rate gives her working time. Counting backwards from the known end time avoids setting up an unknown start time at all.

Alternative approach

Let Daria work tt minutes. Then 103⋅105+54t=450\frac{10}{3}\cdot 105 + \frac{5}{4}t = 450, so 54t=100\frac{5}{4}t = 100 and t=80t = 80; she started at 2:45−1:20=1:252{:}45 - 1{:}20 = 1{:}25 PM.

The trap

Applying Daria's rate to the whole 1:00–2:45 window, or miscounting that window as 145 minutes instead of 105.

Common mistakes

  • Applying Daria's rate to the whole 1:00–2:45 window, or miscounting that window as 145 minutes instead of 105.
  • Using per-minute rates carelessly (e.g. 1010 per minute instead of 1010 per 33 minutes).

Techniques

Set up the equation/formula and compute; no special trick needed · Start from the end state / desired conclusion and reverse