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 "2019 AMC 10B Problems/Problem 18"

(Cleaned up the solution, using notation consistent with the problem statement, and deleting some redundant parts)
(added solution 2)
Line 5: Line 5:
 
<math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}</math>
 
<math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}</math>
  
−
==Solution==
+
==Solution 1==
 
Let the two points that Henry walks in between be <math>P</math> and <math>Q</math>, with <math>P</math> being closer to home. As given in the problem statement, the distances of the points <math>A</math> and <math>B</math> from his home are <math>A</math> and <math>B</math> respectively. By symmetry, the distance of point <math>Q</math> from the gym is the same as the distance from home to point <math>P</math>. Thus, <math>A = 2 - B</math>. In addition, when he walks from point <math>Q</math> to home, he walks <math>\frac{3}{4}</math> of the distance, ending at point <math>P</math>. Therefore, we know that <math>B - A = \frac{3}{4}B</math>. By substuting, we get <math>B - A = \frac{3}{4}(2 - A)</math>. Adding these equations now gives <math>2(B - A) = \frac{3}{4}(2 + B - A)</math>. Multiplying by <math>4</math>, we get <math>8(B - A) = 6 + 3(B - A)</math>, so <math>B - A = \frac{6}{5} = \boxed{\textbf{(C) } 1\frac{1}{5}}</math>.
 
Let the two points that Henry walks in between be <math>P</math> and <math>Q</math>, with <math>P</math> being closer to home. As given in the problem statement, the distances of the points <math>A</math> and <math>B</math> from his home are <math>A</math> and <math>B</math> respectively. By symmetry, the distance of point <math>Q</math> from the gym is the same as the distance from home to point <math>P</math>. Thus, <math>A = 2 - B</math>. In addition, when he walks from point <math>Q</math> to home, he walks <math>\frac{3}{4}</math> of the distance, ending at point <math>P</math>. Therefore, we know that <math>B - A = \frac{3}{4}B</math>. By substuting, we get <math>B - A = \frac{3}{4}(2 - A)</math>. Adding these equations now gives <math>2(B - A) = \frac{3}{4}(2 + B - A)</math>. Multiplying by <math>4</math>, we get <math>8(B - A) = 6 + 3(B - A)</math>, so <math>B - A = \frac{6}{5} = \boxed{\textbf{(C) } 1\frac{1}{5}}</math>.
 +
 +
==Solution 2 (Not rigorous)==
 +
We assume that Henry is walking back and forth exactly between points <math>A</math> and <math>B</math>, with <math>A</math> closer to Henry's home than <math>B</math>. Denote Henry's home as a point <math>H</math> and the gym as a point <math>G</math>. Then <math>HA:AB = 1:3</math> and <math>AB:BG = 3:1</math>, so <math>HA:AB:BG = 1:3:1</math>. Therefore, <math>|A-B| = AB = \frac{3}{1+3+1} \cdot 2 = \frac{6}{5} = \boxed{\textbf{(C) } 1\frac{1}{5}}</math>.
  
 
==Video Solution==
 
==Video Solution==

Revision as of 14:26, 18 February 2019

Problem

Henry decides one morning to do a workout, and he walks $\tfrac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\tfrac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\tfrac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\tfrac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$?

$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}$

Solution 1

Let the two points that Henry walks in between be $P$ and $Q$, with $P$ being closer to home. As given in the problem statement, the distances of the points $A$ and $B$ from his home are $A$ and $B$ respectively. By symmetry, the distance of point $Q$ from the gym is the same as the distance from home to point $P$. Thus, $A = 2 - B$. In addition, when he walks from point $Q$ to home, he walks $\frac{3}{4}$ of the distance, ending at point $P$. Therefore, we know that $B - A = \frac{3}{4}B$. By substuting, we get $B - A = \frac{3}{4}(2 - A)$. Adding these equations now gives $2(B - A) = \frac{3}{4}(2 + B - A)$. Multiplying by $4$, we get $8(B - A) = 6 + 3(B - A)$, so $B - A = \frac{6}{5} = \boxed{\textbf{(C) } 1\frac{1}{5}}$.

Solution 2 (Not rigorous)

We assume that Henry is walking back and forth exactly between points $A$ and $B$, with $A$ closer to Henry's home than $B$. Denote Henry's home as a point $H$ and the gym as a point $G$. Then $HA:AB = 1:3$ and $AB:BG = 3:1$, so $HA:AB:BG = 1:3:1$. Therefore, $|A-B| = AB = \frac{3}{1+3+1} \cdot 2 = \frac{6}{5} = \boxed{\textbf{(C) } 1\frac{1}{5}}$.

Video Solution

For those who want a video solution: https://youtu.be/45kdBy3htOg

See Also

2019 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B_Problems/Problem_18&oldid=103351"