Difference between revisions of "2016 AMC 10B Problems/Problem 4"
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==Problem== | ==Problem== | ||
| − | Zoey read <math>15</math> books, one at a time. The first book took her <math>1</math> day to read, the second book took her <math>2</math> days to read, the third book took her <math>3</math> days to read, and so on, with each book taking her <math>1</math> more day to read than the previous book. Zoey finished the first book on a | + | Zoey read <math>15</math> books, one at a time. The first book took her <math>1</math> day to read, the second book took her <math>2</math> days to read, the third book took her <math>3</math> days to read, and so on, with each book taking her <math>1</math> more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her <math>15</math>th book? |
<math>\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}</math> | <math>\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}</math> | ||
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| + | ==Solution 1== | ||
| + | The process took <math>1+2+3+\ldots+13+14+15=120</math> days, so the last day was <math>119</math> days after the first day. | ||
| + | Since <math>119</math> is divisible by <math>7</math>, both must have been the same day of the week, so the answer is <math>\boxed{\textbf{(B)}\ \text{Monday}}</math>. | ||
| + | |||
| + | ==Solution 2== | ||
| + | Similar to solution 1, the process took 120 days. <math>120 \equiv 1 \mod 7</math>. Since Zoey finished the first book on Monday and the second book (after three days) on Wednesday, we conclude that the modulus must correspond to the day (e.g., <math>1\mod 7</math> corresponds to Monday, <math>4\mod 7</math> corresponds to Thursday, <math>0\mod 7</math> corresponds to Sunday, etc.). The solution is therefore <math>\boxed{\textbf{(B)}\ \text{Monday}}</math>. | ||
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| + | ==Video Solution (CREATIVE THINKING)== | ||
| + | https://youtu.be/XG5fR4xl1o4 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | |||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/8_xEaEIJZ24 | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2016|ab=B|num-b=3|num-a=5}} | {{AMC10 box|year=2016|ab=B|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 17:22, 30 December 2023
Contents
Problem
Zoey read 
books, one at a time. The first book took her 
day to read, the second book took her 
days to read, the third book took her 
days to read, and so on, with each book taking her more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her th book?
Solution 1
The process took 
days, so the last day was 
days after the first day.
Since is divisible by 
, both must have been the same day of the week, so the answer is 
.
Solution 2
Similar to solution 1, the process took 120 days. 
. Since Zoey finished the first book on Monday and the second book (after three days) on Wednesday, we conclude that the modulus must correspond to the day (e.g., 
corresponds to Monday, 
corresponds to Thursday, 
corresponds to Sunday, etc.). The solution is therefore .
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
| 2016 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
