Difference between revisions of "2017 AMC 8 Problems/Problem 23"
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| + | ==Problem 23== | ||
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| + | Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips? | ||
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| + | <math>\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82</math> | ||
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| + | ==Solution== | ||
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From the question | From the question | ||
<cmath>\dfrac{60}{x},\dfrac{60}{x+5},\dfrac{60}{x+10},\dfrac{60}{x+15}\in \mathbb{N}</cmath> | <cmath>\dfrac{60}{x},\dfrac{60}{x+5},\dfrac{60}{x+10},\dfrac{60}{x+15}\in \mathbb{N}</cmath> | ||
and by simple guess and check, the answer is | and by simple guess and check, the answer is | ||
| − | <cmath>\dfrac{60}{5}+\dfrac{60}{10}+\dfrac{60}{15}+\dfrac{60}{20}=\boxed{25}</cmath> | + | <cmath>\dfrac{60}{5}+\dfrac{60}{10}+\dfrac{60}{15}+\dfrac{60}{20}=\boxed{\textbf{(B)}\ 25}</cmath> |
-kvedula2004 | -kvedula2004 | ||
Revision as of 14:00, 22 November 2017
Problem 23
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
Solution
From the question
![\[\dfrac{60}{x},\dfrac{60}{x+5},\dfrac{60}{x+10},\dfrac{60}{x+15}\in \mathbb{N}\]](http://latex.artofproblemsolving.com/e/3/e/e3e4c1027a9fb86bad9d808aac06281d4e1b7ae8.png)
and by simple guess and check, the answer is
![\[\dfrac{60}{5}+\dfrac{60}{10}+\dfrac{60}{15}+\dfrac{60}{20}=\boxed{\textbf{(B)}\ 25}\]](http://latex.artofproblemsolving.com/c/7/5/c757d44a197490aec0fe090e14fdd9d8ac9d5f4f.png)
-kvedula2004
