{"status": "success", "data": {"description_md": "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 $|A-B|$?\n\n$\\textbf{(A) } \\frac{2}{3} \\qquad \\textbf{(B) } 1 \\qquad \\textbf{(C) } 1\\frac{1}{5} \\qquad \\textbf{(D) } 1\\frac{1}{4} \\qquad \\textbf{(E) } 1\\frac{1}{2}$", "description_html": "<p>Henry decides one morning to do a workout, and he walks  <span class=\"katex--inline\">\\tfrac{3}{4}</span>  of the way from his home to his gym. The gym is  <span class=\"katex--inline\">2</span>  kilometers away from Henry&#8217;s home. At that point, he changes his mind and walks  <span class=\"katex--inline\">\\tfrac{3}{4}</span>  of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks  <span class=\"katex--inline\">\\tfrac{3}{4}</span>  of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked  <span class=\"katex--inline\">\\tfrac{3}{4}</span>  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  <span class=\"katex--inline\">A</span>  kilometers from home and a point  <span class=\"katex--inline\">B</span>  kilometers from home. What is  <span class=\"katex--inline\">|A-B|</span> ?</p>\n<p> <span class=\"katex--inline\">\\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}</span> </p>\n<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2019 AMC 10B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc10B_p19", "prev": "/problem/19_amc10B_p17"}}