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Difference between revisions of "2003 AMC 10B Problems/Problem 22"

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==Problem==
 
==Problem==
  
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A clock chimes once at <math>30</math> minutes past each hour and chimes on the hour according to the hour. For example, at <math>1 \text{PM}</math> there is one chime and at noon and midnight there are twelve chimes. Starting at <math>11:15 \text{AM}</math> on <math>\text{February 26, 2003},</math> on what date will the <math>2003\text{rd}</math> chime occur?
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A clock chimes once at <math>30</math> minutes past each hour and chimes on the hour according to the hour. For example, at <math>1 \text{PM}</math> there is one chime and at noon and midnight there are twelve chimes. Starting at <math>11:15 \text{AM}</math> on <math>\text{February 26, 2003},</math> on what date will the <math>2003^{\text{rd}}</math> chime occur?
  
 
<math>\textbf{(A) } \text{March 8} \qquad\textbf{(B) } \text{March 9} \qquad\textbf{(C) } \text{March 10} \qquad\textbf{(D) } \text{March 20} \qquad\textbf{(E) } \text{March 21}</math>
 
<math>\textbf{(A) } \text{March 8} \qquad\textbf{(B) } \text{March 9} \qquad\textbf{(C) } \text{March 10} \qquad\textbf{(D) } \text{March 20} \qquad\textbf{(E) } \text{March 21}</math>
  
 
==Solution==
 
==Solution==
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First, find how many chimes will have already happened before midnight (the beginning of the day) of <math>\text{February 27, 2003}.</math> <math>13</math> half-hours have passed, and the number of chimes according to the hour is <math>1+2+3+\cdots+12.</math> The total number of chimes is <math>13+78=91</math>
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First, find how many chimes will have already happened before midnight (the beginning of the day) of <math>\text{February 27, 2003}.</math> <math>13</math> half-hours have passed, and the number of chimes according to the hour is <math>1+2+3+\cdots+12.</math> The total number of chimes is <math>13+78=91.</math>
  
 
Every day, there will be <math>24</math> half-hours and <math>2(1+2+3+\cdots+12)</math> chimes according to the arrow, resulting in <math>24+156=180</math> total chimes.
 
Every day, there will be <math>24</math> half-hours and <math>2(1+2+3+\cdots+12)</math> chimes according to the arrow, resulting in <math>24+156=180</math> total chimes.
  
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On <math>\text{February 27},</math> the number of chimes that still need to occur is <math>2003-91=1912.</math> <math>1912/180=10\text{r}112.</math> Round up, and it is <math>11</math> days past <math>\text{February 27},</math> which is <math>\boxed{\mathrm{(B) \ } \text{March 9}}</math>
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On <math>\text{February 26},</math> the number of chimes that still need to occur is <math>2003-91=1912.</math> <math>1912 \div 180=10 \text{R}112.</math> Rounding up, it is <math>11</math> days past <math>\text{February 26},</math> which is <math>\boxed{\textbf{(B) \ } \text{March 9}}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2003|ab=B|num-b=21|num-a=23}}
 
{{AMC10 box|year=2003|ab=B|num-b=21|num-a=23}}
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{{MAA Notice}}

Latest revision as of 00:07, 8 September 2022

Problem

A clock chimes once at $30$ minutes past each hour and chimes on the hour according to the hour. For example, at $1 \text{PM}$ there is one chime and at noon and midnight there are twelve chimes. Starting at $11:15 \text{AM}$ on $\text{February 26, 2003},$ on what date will the $2003^{\text{rd}}$ chime occur?

$\textbf{(A) } \text{March 8} \qquad\textbf{(B) } \text{March 9} \qquad\textbf{(C) } \text{March 10} \qquad\textbf{(D) } \text{March 20} \qquad\textbf{(E) } \text{March 21}$

Solution

First, find how many chimes will have already happened before midnight (the beginning of the day) of $\text{February 27, 2003}.$ $13$ half-hours have passed, and the number of chimes according to the hour is $1+2+3+\cdots+12.$ The total number of chimes is $13+78=91.$

Every day, there will be $24$ half-hours and $2(1+2+3+\cdots+12)$ chimes according to the arrow, resulting in $24+156=180$ total chimes.

On $\text{February 26},$ the number of chimes that still need to occur is $2003-91=1912.$ $1912 \div 180=10 \text{R}112.$ Rounding up, it is $11$ days past $\text{February 26},$ which is $\boxed{\textbf{(B) \ } \text{March 9}}$

See Also

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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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