Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2017 AMC 8 Problems/Problem 23"

m (→‎Solution)
(→‎Video Solutions)
 
(15 intermediate revisions by 12 users not shown)
Line 1: Line 1:
−
==Problem 23==
+
==Problem==
 +
 
 +
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 <math>5</math> 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?
  
−
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?
 
 
<math>\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82</math>
 
<math>\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82</math>
  
 
==Solution==
 
==Solution==
−
We know that some multiples of 5 go into 60 evenly. Let's check some because the problem specifies 5 minute increase. 5 does, 10 does, 15 does, and 20 does. This is 4. This seems to satisfy the problem! In the first day, she travels 1 mile in 5 minutes, or 12 miles in 60 minutes. In the second day, she travels 1 mile in 10 minutes, or 6 miles in 60 minutes. In the third day, she travels 1 mile in 15 minutes, or 4 miles in 60 minutes. In the fourth day, she travels 1 mile in 20 minutes, or 3 miles in 60 minutes. Adding these up, we get 12+6+4+3, which is 25. So our answer is 25, or <math>C</math>.
+
 
 +
It is well known that <math>\text{Distance}=\text{Speed} \cdot \text{Time}</math>. In the question, we want distance. From the question, we have that the time is <math>60</math> minutes or <math>1</math> hour.  By the equation derived from <math>\text{Distance}=\text{Speed} \cdot \text{Time}</math>, we have <math>\text{Speed}=\frac{\text{Distance}}{\text{Time}}</math>, so the speed is <math>1</math> mile per <math>x</math> minutes. Because we want the distance, we multiply the time and speed together yielding <math>60\text{ mins}\cdot \frac{1\text{ mile}}{x\text{ mins}}</math>. The minutes cancel out, so now we have <math>\dfrac{60}{x}</math> as our distance for the first day. The distance for the following days are:
 +
<cmath>\dfrac{60}{x},\dfrac{60}{x+5},\dfrac{60}{x+10},\dfrac{60}{x+15}.</cmath>
 +
We know that <math>x,x+5,x+10,x+15</math> are all factors of <math>60</math>, therefore, <math>x=5</math> because the factors have to be in an arithmetic sequence with the common difference being <math>5</math> and <math>x=5</math> is the only solution.
 +
<cmath>\dfrac{60}{5}+\dfrac{60}{10}+\dfrac{60}{15}+\dfrac{60}{20}=12+6+4+3=\boxed{\textbf{(C)}\ 25}.</cmath>
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/XixU0JZ5FLk?t=395
 +
 
 +
~ pi_is_3.14
  
 
==See Also==
 
==See Also==

Latest revision as of 09:52, 24 July 2024

Problem

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?

$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82$

Solution

It is well known that $\text{Distance}=\text{Speed} \cdot \text{Time}$. In the question, we want distance. From the question, we have that the time is $60$ minutes or $1$ hour. By the equation derived from $\text{Distance}=\text{Speed} \cdot \text{Time}$, we have $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$, so the speed is $1$ mile per $x$ minutes. Because we want the distance, we multiply the time and speed together yielding $60\text{ mins}\cdot \frac{1\text{ mile}}{x\text{ mins}}$. The minutes cancel out, so now we have $\dfrac{60}{x}$ as our distance for the first day. The distance for the following days are: \[\dfrac{60}{x},\dfrac{60}{x+5},\dfrac{60}{x+10},\dfrac{60}{x+15}.\] We know that $x,x+5,x+10,x+15$ are all factors of $60$, therefore, $x=5$ because the factors have to be in an arithmetic sequence with the common difference being $5$ and $x=5$ is the only solution. \[\dfrac{60}{5}+\dfrac{60}{10}+\dfrac{60}{15}+\dfrac{60}{20}=12+6+4+3=\boxed{\textbf{(C)}\ 25}.\]

Video Solution by OmegaLearn

https://youtu.be/XixU0JZ5FLk?t=395

~ pi_is_3.14

See Also

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems/Problem_23&oldid=223523"