Question

Complete the addition-subtraction box:

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

Answer 1

Consider

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

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