Difference between revisions of "2024 AMC 10A Problems/Problem 8"
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+ | == Solution 3 == | ||
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+ | We notice that every <math>3</math> minutes, Amy, Bomani, Charlie pack <math>4 + 3 + 3 = 10</math> packages in total. Thus, over the entire course, they pack <math>105 \cdot \frac{10}{3} = 350</math>. Thus, Daria packaged <math>450 - 350 = 100</math> packages. Thus, Daria worked for <math>100 \cdot \frac{4}{5} = 80</math> minutes, so Daria starting working <math>80</math> minutes before <math>2:45</math> pm, or <math>\boxed{\text{(A) }1:25\text{ PM}}</math>. | ||
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+ | ~andliu766 | ||
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==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|num-b=7|num-a=9}} | {{AMC10 box|year=2024|ab=A|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 16:16, 8 November 2024
Contents
[hide]Problem
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at and were able to pack , , and packages, respectively, every minutes. At some later time, Daria joined the group, and Daria was able to pack packages every minutes. Together, they finished packing packages at exactly . At what time did Daria join the group?
Solution 1
Note that Amy, Bomani, and Charlie pack a total of packages every minutes.
The total amount of time worked is hour and minutes, which when converted to minutes, is minutes. This means that since Amy, Bomani, and Charlie worked for the entire minutes, they in total packed packages.
Since packages were packed in total, then Daria must have packed packages in total, and since he packs at a rate of packages per minutes, then Daria worked for minutes, therefore Daria joined minutes before PM, which was at
~Tacos_are_yummy_1
Solution 2
Let the time, in minutes, elapsed between and the time Daria joined the packaging be . Since Amy packages packages every minutes, she packages packages per minute. Similarly, we can see that both Bomani and Charlie package package per minute, and Daria packages packages every minute.
Before Daria arrives, we can write the total packages packaged as . Since there are hours between and , Daria works with the other three for minutes, meaning for that time there are packages packaged.
Adding the two, we get (The total packaged in the entire time is ). Solving this equation, we get , meaning Daria arrived minutes after , meaning the answer is .
~i_am_suk_at_math_2
Solution 3
We notice that every minutes, Amy, Bomani, Charlie pack packages in total. Thus, over the entire course, they pack . Thus, Daria packaged packages. Thus, Daria worked for minutes, so Daria starting working minutes before pm, or .
~andliu766
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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