Question

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

Answer 1

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: $\frac{13}{29}$