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Difference between revisions of "2001 AMC 10 Problems/Problem 8"

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== Problem ==
 
== Problem ==
  
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A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form AB:CD without am or pm, such as 08:00)
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Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?
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<math> \textbf{(A) }42\qquad\textbf{(B) }84\qquad\textbf{(C) }126\qquad\textbf{(D) }178\qquad\textbf{(E) }252</math>
  
 
== Solution ==
 
== Solution ==
  
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We need to find the least common multiple of the three numbers given.  
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By translating the words in the problem into the language of mathematics, the problem is telling us to find the least common multiple of the four numbers given.  
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<math>\textrm{LCM}(3, 4, 6, 7) = \textrm{LCM}(3, 2^2, 2 \cdot 3, 7) = 2^2 \cdot 3 \cdot 7 = 84</math>
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So the answer is <math>\boxed{\textbf{(B) } 84} </math>.
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==Video Solution by Daily Dose of Math==
  
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<math>\textrm{LCM}(15, 20, 25) = \textrm{LCM}(3 \cdot 5, 2^2 \cdot 5, 5^2) = 2^2 \cdot 3 \cdot 5^2 = 300</math>
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https://youtu.be/ts2x9Q0XVM0?si=uxpzYXU3VacX2lKm
  
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300 minutes equals 5 hours. So the bell will ring 5 hours past 12:00, which is 17:00.
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~Thesmartgreekmathdude
  
 
== See Also ==
 
== See Also ==

Latest revision as of 21:30, 13 August 2024

Problem

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

$\textbf{(A) }42\qquad\textbf{(B) }84\qquad\textbf{(C) }126\qquad\textbf{(D) }178\qquad\textbf{(E) }252$

Solution

By translating the words in the problem into the language of mathematics, the problem is telling us to find the least common multiple of the four numbers given.

$\textrm{LCM}(3, 4, 6, 7) = \textrm{LCM}(3, 2^2, 2 \cdot 3, 7) = 2^2 \cdot 3 \cdot 7 = 84$

So the answer is $\boxed{\textbf{(B) } 84}$.

Video Solution by Daily Dose of Math

https://youtu.be/ts2x9Q0XVM0?si=uxpzYXU3VacX2lKm

~Thesmartgreekmathdude

See Also

2001 AMC 10 (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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