Question

Complete the addition-subtraction box:

$\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \square \\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \square \\ |& \square & \square & \square \end{array}$

Answer 1

Consider

$$\begin{gathered} \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \left(\frac{1}{2}+\frac{1}{3}\right)\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \left(\frac{1}{3}+\frac{1}{4}\right)\\ |& \left(\frac{1}{2}-\frac{1}{3}\right)& \left(\frac{1}{3}-\frac{1}{4}\right)& \square \end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \left(\frac{\frac{1}{2}\times 6+\frac{1}{3}\times 6}{6}\right)\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \left(\frac{\frac{1}{3}\times 12+\frac{1}{4}\times 12}{12}\right)\\ |& \left(\frac{\frac{1}{2}\times 6-\frac{1}{3}\times 6}{6}\right)& \left(\frac{\frac{1}{3}\times 12-\frac{1}{4}\times 12}{12}\right)& \square \end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \left(\frac{1\times 3+1\times 2}{6}\right)\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \left(\frac{1\times 4+1\times 3}{12}\right)\\ |& \left(\frac{1\times 3-1\times 2}{6}\right)& \left(\frac{1\times 4-1\times 3}{12}\right)& \square \end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \left(\frac{3+2}{6}\right)\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \left(\frac{4+3}{12}\right)\\ |& \left(\frac{3-2}{6}\right)& \left(\frac{4-3}{12}\right)& \square \end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \square \end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{\frac{1}{6}\times 12+\frac{1}{12}\times 12}{12}\right)\end{array}or\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{\frac{5}{6}\times 12-\frac{7}{12}\times 12}{12}\right)\end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{1\times 2+1\times 1}{12}\right)\end{array}or\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{5\times 2-7\times 1}{12}\right)\end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{2+1}{12}\right)\end{array}or\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{10-7}{12}\right)\end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{3}{12}\right)\end{array}or\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \left(\frac{3}{12}\right)\end{array} \\ \begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \frac{1}{4}\end{array}or\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \frac{1}{4}\end{array} \end{gathered}$$

Hence, $\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \square \\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \square \\ |& \square & \square & \square \end{array}$ is $\begin{array}{cccc}& —& \overline{)+}& —\\ |& \frac{1}{2}& \frac{1}{3}& \frac{5}{6}\\ \overline{)-}& \frac{1}{3}& \frac{1}{4}& \frac{7}{12}\\ |& \frac{1}{6}& \frac{1}{12}& \frac{1}{4}\end{array}$