Answer to Question #139563 in Algebra for huehuehe

Question #139563

what is closer to 2/5, 6/21 or 13/29?

Expert's answer

Distance between "\\frac{2}{5}" and "\\frac{6}{21}" is "d\\left(\\frac{2}{5},\\frac{6}{21}\\right)=\\bigl|\\frac{2}{5}-\\frac{6}{21}\\bigr|=\\bigl|\\frac{2}{5}-\\frac{2}{7}\\bigr|=\\bigl|\\frac{2\\cdot 7-2\\cdot 5}{5\\cdot 7}\\bigr|=\\bigl|\\frac{4}{5\\cdot 7}\\bigr|=\\frac{4}{5\\cdot 7}"

Distance between "\\frac{2}{5}" and "\\frac{13}{29}" is "d\\left(\\frac{2}{5},\\frac{13}{29}\\right)=\\bigl|\\frac{2}{5}-\\frac{13}{29}\\bigr|=\\bigl|\\frac{2\\cdot 29-13\\cdot 5}{5\\cdot 29}\\bigr|=\\bigl|\\frac{-7}{5\\cdot 29}\\bigr|=\\frac{7}{5\\cdot 29}"

We have "d\\left(\\frac{2}{5},\\frac{6}{21}\\right)=\\frac{4}{5\\cdot 7}=\\frac{4\\cdot 29}{5\\cdot 7\\cdot 29}>\\frac{7\\cdot 7}{5\\cdot 7\\cdot 29}=\\frac{7}{5\\cdot 29}=d\\left(\\frac{2}{5},\\frac{13}{29}\\right)" .

So "\\frac{13}{29}" is closer to "\\frac{2}{5}" than "\\frac{6}{21}".

Answer: "\\frac{13}{29}"