{"status": "success", "data": {"description_md": "In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? \n\n$\\textbf{(A)}$ Her estimate is larger than $x-y$ \n$\\textbf{(B)}$ Her estimate is smaller than $x-y$\n$\\textbf{(C)}$ Her estimate equals $x-y$\n$\\textbf{(D)}$ Her estimate equals $y - x$\n$\\textbf{(E)}$ Her estimate is $0$", "description_html": "<p>In order to estimate the value of  <span class=\"katex--inline\">x-y</span>  where  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span>  are real numbers with  <span class=\"katex--inline\">x &gt; y &gt; 0</span> , Xiaoli rounded  <span class=\"katex--inline\">x</span>  up by a small amount, rounded  <span class=\"katex--inline\">y</span>  down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}</span>  Her estimate is larger than  <span class=\"katex--inline\">x-y</span> <br/>\n <span class=\"katex--inline\">\\textbf{(B)}</span>  Her estimate is smaller than  <span class=\"katex--inline\">x-y</span> <br/>\n <span class=\"katex--inline\">\\textbf{(C)}</span>  Her estimate equals  <span class=\"katex--inline\">x-y</span> <br/>\n <span class=\"katex--inline\">\\textbf{(D)}</span>  Her estimate equals  <span class=\"katex--inline\">y - x</span> <br/>\n <span class=\"katex--inline\">\\textbf{(E)}</span>  Her estimate is  <span class=\"katex--inline\">0</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": 1, "problem_name": "2012 AMC 10B Problem 6", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc10B_p07", "prev": "/problem/12_amc10B_p05"}}